Memoirs on Differential Equations and Mathematical Physics Volume 21, 2000, 1–122

R. Duduchava and B. Silbermann

BOUNDARY VALUE PROBLEMS IN DOMAINS WITH PEAKS

Abstract. We obtain criteria of solvability of the Dirichlet and the Neumann boundary value problems (BVPs) for the Laplacian in 2D domains with angular points and peaks on the boundary. We start with the correct formulation of BVPs and modify it for domains with outward peaks (classical conditions are incorrect). Boundary integral equations (BIEs), obtained by the indirect potential method, turn out to be equivalent to the corresponding BVPs only when inward peaks are absent. BIEs on boundary curve with angular points are investigated in different weighted function spaces. If boundary curve has a cusp, corresponding to an inward or an outward peak, equations are non-Fredholm in usual spaces and we should impose restrictions on the right-hand sides. The conditions are defined with the Cesaro-type integrals. We consider also equivalent reduction to boundary pseudo-differential equations (BPsDEs) of orders ±1 by the direct potential method. Crucial role in our investigations of BVPs and of corresponding BIEs, PsDOs belongs to the equivalent reduction of BVPs to the Riemann–Hilbert problem for analytic functions on the unit disk. The latter problem can be investigated thoroughly, even when peaks are present and equations have non-closed image by invoking results on convolution equations with vanishing symbols.

2000 Mathematics Subject Classification. 47A68, 35J25, 35J55.

Key words and phrases: Boundary integral equation, convolution equation, non-elliptic symbols, logarithmic potentials, boundary value problems.

Introduction

Let Ω^{+} ⊂ ℂ be a bounded domain in the complex plane with a piecewisesmooth boundary Γ = ∂Ω^{+} and Ω^{−} = ℂ \ Ω^{+} be the complementary outer domain. Let t_{j} ∈ Γ, j = 1, …, n, be all knots on the boundary Γ = ∂Ω^{+} with the angles πγ_{j}, 0 ≤ γ_{j} ≤ 2, j = 1, …, n. Boundary curve might contain cusps γ_{j} = 0, 2 corresponding to an outward (for γ_{j} = 0) and an inward (for γ_{j} = 2) peaks of the domain Ω^{+}. By ν⃗(t) = (ν_{1}(t), ν_{2}(t)) we denote the outer unit normal vector to Γ (with respect to Ω^{+}).

As a model we consider the Dirichlet u^{±}(t) = g(t) (and the Neumann ∂_{ν⃗(t)}u^{±}(t) = f(t), t ∈ Γ) BVPs for harmonic functions

Δu(x) = 0 , x ∈ Ω^{±}  (0.1)

and look for the solution, as common, in the Sobolev space

u ∈ W_{2}^{1}(\overline{Ω^{+}}) or u ∈ W_{2,loc}^{1}(\overline{Ω^{−}}) , u(x) = O(1) , as |x| → ∞ . (0.2)

Applying the potential method, based on the Green formula and its consequence-representation of solution by layer potentials, invoking the Plemelji formulae (see § 1) we get boundary integral equations (BIEs) of logarithmic potential

±\frac{1}{2}φ(t) + \frac{1}{2π} ∫_{Γ} ∂_{ν⃗(τ)} log |t − τ|φ(τ)|dτ| = g(t) ,  (0.3)

±\frac{1}{2}ψ(t) + \frac{1}{2π} ∫_{Γ} ∂_{ν⃗(t)} log |t − τ|ψ(τ)|dτ| = f(t) , t ∈ Γ, (0.4)

which are conjugate to each-other (the indirect method; see [Ma1]). It is rather a classical result, that (0.3) and (0.4) are Fredholm equations provided Γ is smooth and the reduction of BVPs to the corresponding BIEs (0.3) and (0.4) is equivalent.

When Γ has angular points, equations (0.3) and (0.4) have fixed singularities in the kernels (i.e., they are Mellin convolution equations) and are Fredholm except some discrete values of parameters of spaces they are treated in (see Theorems 1.23, 1.24 and cf. [Du1, Du3, Ma1]). It is important that in both mentioned cases equivalence of BVPs with corresponding BIEs still hold.

Piecewise-smooth domains without peaks are particular cases of Lipschitz domains and BVPs for second order equations in such domains were thoroughly investigated recently (mostly in the Hilbert spaces L_{2} and W_{2}^{1}) even for domains in ℝ^{n}, n > 2. For details of these profound investigations as well as for exhaustive survey of vast literature in this field we recommend recent publications [Ke1, MMP1, MMT1, MT1].

Situation changes completely if domain Ω^{±} has peaks. There arise three principal problems.

• If a single outward peak occurs constraints (0.2) become incorrect. Namely, if we look for solution of BVP in the Sobolev space W_{p}^{1}(\overline{Ω^{+}}) for arbitrary fixed value of p ∈ (1, ∞), there exists a compact domain Ω_{2γ+3} ⊂ ℂ^{+} with outward peak at 0 ∈ ∂Ω_{2γ+3} in the first quadrant ℝ^{+}+iℝ^{+} ⊂ ℂ of the complex plane such that the analytic function z^{γ}, z ∈ Ω, with arbitrary 0 < γ < ∞ belongs to the space W_{p}^{1}(\overline{Ω_{2γ+3}}) (details see below in Example 1.2). Therefore a classical solution to BVP u ∈ W_{2}^{1}(\overline{Ω^{+}}) might have non-integrable singularity on the boundary and it is necessary to change constraints on harmonic functions in the domain. Moreover, due to complicated relations between traces of functions on different faces of outward peaks (see, e.g., [Ia1]) it is almost impossible to investigate corresponding BIEs.

• If a single inward peak occurs, equivalence of BVPs (0.1), (0.2) with the corresponding BIEs (0.3), (0.4) fail completely. Such reduction is connected with a representation of harmonic function of the Smirnov class by the Cauchy integral with real valued density. This turned out to be possible if and only if the Riemann–Hilbert BVPs for analytic functions is surjective in the same Smirnov space but for the complementary domain (see Lemmata 1.1 and 1.13). If the domain has an inward peak, the complementary domain has an outward peak and the Riemann–Hilbert BVP is not normally solvable (see Lemma 1.11).

• If a single peak (outward or inward) occurs solvability property of BIEs (0.3) and (0.4) change dramatically: symbols of these convolution-type equations vanish and equations can not be Fredholm in any L_{p}(Γ) or any other space with weight or without (see [MS1]–[MS8] and § 1.6 below). For the space of continuous functions this was noticed already by J.Radon [Ra1].

We start with investigations of correct formulation of the BVPs. Namely, we look for solutions in the weighted Smirnov–Lebesgue space e_{p}(\overline{Ω^{±}}, ρ) (see § 1.2) of harmonic functions written as the real part of analytic functions represented by the Cauchy integrals with densities in the Lebesgue spaces with weight L_{p}(Γ, ρ) (plus constanta for the unbounded domain Ω^{−}). The choice of constraints is justified in the following sense: looking for solutions in more narrow Smirnov–Sobolev space u ∈ w2^{\frac{1}{2}} (\overline{Ω^{±}}) is the same as the common (classical) constraint u ∈ W_{2}^{1}(\overline{Ω^{±}}) provided the domain Ω^{±} has no outward peaks (see Lemma 1.2). Moreover, to raise flexibility of the method we suggest to look for solutions in some other Smirnov spaces: weighted Smirnov–Sobolev w_{p}^{s}(\overline{Ω^{±}}, ρ), 0 ≤ s ≤ 1, Smirnov–Hölder h_{m+μ}^{0}(\overline{Ω^{±}}, ρ) etc. (see § 1.2).

If the boundary curve has cusps (i.e., the domain has peaks) equations (0.3) and (0.4) have non-closed images. Same is true for the Dirichlet and the Neumann BVPs for (0.1) when inward peaks are present. Maz’ya and V.Solov’ev in [MS1]–[MS4] suggested to study BIEs (0.3), (0.4) directly. Namely, they have found conditions on the right-hand sides which ensure existence of solutions and have established properties (smoothness, asymptotic) of such solutions. The method is based on the corresponding results for boundary value problems in domains with peaks, obtained with the help of conformal mappings (see [Wa1, Wa2] for properties of such conformal mappings). In more recent investigations [MS5]–[MS8] for curves with cusps of order μ ∈ ℝ^{+} they have found pairs of Banach spaces where BIEs (0.3), (0.4) are surjective.

Different approach (transformation of the underlying domain which maintains the structure of BVPs) was exploited in [RST1, RST2]. The authors obtained solvability results for BVPs in domains with special cusps when the right-hand sides and solutions are in special weighted spaces.

Essential role in our investigations play an equivalent reduction of the Dirichlet and the Neumann BVPs for (0.1) to the Riemann–Hilbert BVPs for analytic functions on the unit circumference, using the conformal mapping. Namely, we apply the approach exposed in [Mu1, Ch. III] and contributed by I.Vekua in [Ve1]. Obtained BVPs are reduced further to equivalent Cauchy singular integral equations on the unit circumference.

The same method was applyed by G.Khuskivadze and V.Paatashvi li. Namely, they look for solutions of BVPs in the Smirnov–Lebesgue space e_{p}(\overline{Ω^{±}}), 1 < p < ∞. Although the motivation for the choice of constraints, ensuring equivalent reduction to the Riemann–Hilbert problem, was clear justification for the change of conditions in [KKP1] is missing.

For the investigation of the Cauchy singular integral equations on the unit circumference, which arise as an equivalent equation, we apply localization to 2 × 2 systems of convolution equations on the real semi-axes. Local representatives at cusps have vanishing symbols and, by applying results on convolution equations with vanishing symbols of integer order (see [Pr1, § 5.2] and § 3.1 below), we describe the image space by Cesaro-type integrals and find the criteria for the data which ensures unique solvability of the Dirichlet and the Neumann BVPs for (0.1).

Further we prove equivalence of BVPs and of corresponding BIEs (0.3) and (0.4) if inward peaks are absent (see Theorems 1.12 and 1.14). If the boundary curve has no cusps, obtained BIEs are particular cases of general equations studied in § 4 by invoking results from [DLS1]. They are Fredholm with rare exceptions for the parameters of the space. Although such investigations were carried out earlier (see survey in [Ma1]) some results of the present paper are new: we prove boundedness of harmonic (the double and the single) layer potentials and obtain criteria for Fredholm property of equations (0.3) and (0.4) in the spaces of continuous and piecewise-continuous functions C(Γ, κ) and P C(Γ, κ) (in some cases also

in P C^{1}(Γ, κ); see § 1.7) with exponential weight κ(t) = ∏_{j=1}^{n} |t − c_{j}|^{α_{j}}, 0 ≤ α_{j} < 1.

If inward peaks are present equivalence with BVPs fail (see Lemma 1.13) and equations (0.3), (0.4) are investigated by localization. The localization enables replacement of inward peaks by outward ones (see § 5.4). Solvability criteria of equations (0.3) and (0.4) are summarized in Theorems 1.23 and 1.24, which are proved in § 5.4.

Let T_{ow}, T_{iw} be the discrete sets of all outward, all inward peaks and T_{pk} = T_{ow} ∪ T_{iw} be the set of all peaks of Ω^{+}. We define the spaces

L_{p}(Γ, ρ, T_{ow}), L_{p}(Γ, ρ, T_{iw}) ⊂ L_{p}(Γ, ρ, T_{pk}) ⊂ L_{p}(Γ, ρ) ,

with the help of the Cesaro integrals (see (1.76)), where ρ(t) = ∏_{j=1}^{n} |t−c_{j}|^{α_{j}},

−\frac{1}{p} < α_{j} < 1 − \frac{1}{p}, 1 < p < ∞. It is proved that equations (0.3) and (0.4) are Fredholm between spaces L_{p}(Γ, ρ) ⟶ L_{p}(Γ, ρ, T_{pk}) provided the conditions \frac{1}{p} + α_{j} ≠ min \{\frac{1}{γ_{j}} , \frac{1}{2−γ_{j}} \} holds for all t_{j} ∉ T_{pk}. Moreover, if

the inequalities \frac{1}{p} +α_{j} < min \{\frac{1}{γ_{j}} , \frac{1}{2−γ_{j}} \} hold, the mappings are isomorphic.

As for solvability of the Dirichlet BVP for Ω^{+} (for Ω^{−}) it suffices to restrict the data g ∈ L_{p}(Γ, ρ, T_{ow}) (respectively, g ∈ L_{p}(Γ, ρ, T_{iw})) and the solution is unique provided \frac{1}{p} + α_{j} < min \{\frac{1}{γ_{j}} , \frac{1}{2−γ_{j}} \} for all t_{j} ∉ T_{pk} (note, that inward peaks of Ω^{±} have no impact on the corresponding Dirichlet BVP). Similar holds for the Neumann BVPs.

In Lemma 1.22 we formulate sufficient conditions for the inclusion φ ∈ L_{p}(Γ, ρ, T_{iw}), which involves the conformal mapping ζ(z) : Ω^{+} ⟶ D_{1} of the domain Ω^{+} onto the unit disk D_{1} = {ζ ∈ ℂ : |ζ| = 1}. It is possible to write more transparent and explicit condition, but for these we need asymptotic behaviour of the conformal mapping ζ(z) in the vicinity of an outward peak. This we leave for a forthcoming paper.

In our investigations we apply the Cisotti formula, which represents the derivative of the conformal mapping ω : D_{1} → Ω^{+} (see [LS1, Ch. III, § 1, n^{o}. 44, Example 5]):

ω′(z) = ω′(0) exp [\frac{1}{π} ∫_{|τ|=1} \frac{β(τ)dτ}{τ − z} − \frac{1}{π} ∫_{|τ|=1} β(τ)\frac{dτ}{τ} ] , z ∈ D_{1} . (0.5)

Here β(τ) := arg ν⃗(ω(τ)) − argτ and arg ν⃗(ω(τ)) stands for the argument of the outer unit normal vector to the curve Γ = ∂Ω^{+} at the point τ = e^{iϑ} ∈ Γ_{1} := ∂D_{1}. The formula was rediscovered in [PK1] for a piecewise-smooth boundary (see also [KKP1]). We return to the classical approach in [LS1]) which is, above all, very simple and prove the Cisotti formula (0.5) in § 5.1 for a domain with rectifiable Jordan boundary.

Although the conformal mapping is participating implicitly, representation (0.5) simplifies proofs of some classical theorems on conformal mappings^{1)} (see [KKP1, Ch. III] and § 5.1 for the proofs of Lindelöf’s, Kellogg’s, Warschawsky’s theorems). Moreover, using the Cisotti formula we generalize the Kellogg theorem for the Zygmund space (see Theorem 5.9).

In [Po1, Theorem 3.15] the Cisotti formula is rediscovered for a so-called regulated domain, i.e., for a domain for which the inclination α(t), t ∈ Γ of the tangent vector to the boundary has limits α(t ± 0) everywere on the boundary t ∈ Γ.

G.Khuskivadze and V.Paatashvili had applied formula (0.5) to find discontinuities of the coefficient, but the obtained Riemann–Hilbert problems they have found “non-solvable in L_{p}(Γ) spaces in general” when outward cusps are present (see [KKP1, Ch. IY]) and have written sufficient condition of solvability as well as explicit formula for solutions provided the solvability conditions hold.

Applying the representation of solution by layer potentials and the direct method we obtain boundary pseudo-differential equation

\frac{1}{2π} ∫_{Γ} log |t − τ|φ(τ)|dτ| = g_{∗}(t) , t ∈ Γ ,  (0.6)

g_{∗}(t) := −\frac{1}{2}g(t) − \frac{1}{2π} ∫_{Γ} ∂_{ν⃗(τ)} log |t − τ|g(τ)|dτ| ,

of order −1 for the Dirichlet problem for the Laplacian (0.1) and the boundary pseudo-differential equation

\frac{1}{2π} ∫_{Γ} ∂_{ν⃗(t)}∂_{ν⃗(τ)} log |t − τ|φ(τ)|dτ| = f_{∗}(t) , t ∈ Γ ,  (0.7)

f_{∗}(t) := \frac{1}{2}f(t) + \frac{1}{2π} ∫_{Γ} ∂_{ν⃗(t)} log |t − τ|f(τ)|dτ| ,

of order +1 for the Neumann problem. We can formulate criteria of solvability of equations (0.6) and (0.7) based on full equivalence with corresponding BVPs (see Theorems 1.19, 1.20).

All principal theorems on solvability of boundary value problems and boundary integral equations are formulated in § 1.7. Some of them are proved later, mostly in § 5.

Acknowledgments: the authors thank I.Graham (University of Bath) and S.Chandler–Wilde (Brunel University, London) for many encouraging discussions on the subject during the first authors visit to these universities.

1)See [Ga1] for a survey on application of linear and non-linear integral equations in conformal mappings.

We thank also V.Vlasov (Moscow, Computing Center of the Acad. of Sciences) who draw our attention to the Cisotti formula in [LS1, Ch. III, § 1, n^{o}. 44, Example 5].

1 Boundary value problems

In the present section we formulate the Dirichlet and the Neumann boundary value problems for the Laplacian in domains with angular points and peaks; discuss their equivalent reduction to boundary integral equations (the direct potential method), to boundary pseudo-differential equations (the indirect potential method) and to singular integral equations on the unit circumference (Muskhelishvili–Vekua method); we expose properties of harmonic potentials appearing in the method and formulate all principal results.

1.1 Spaces

We start by rigorous definitions of domains and spaces which are necessary for our considerations.

Let Γ be a closed, oriented, simple (i.e., without self-intersection), piecewise-Ljapunov curve on the complex plane ℂ, circumventing a domain Ω^{+} and having knots at t_{1}, …, t_{n} ∈ Γ, i.e.,

Γ = ⋃_{j=1}^{n} Γ_{j} , Γ_{j} = ) ; t_{j}t_{j+1} , t_{n+1} := t_{1} , j = 1, …, n;  (1.1)

[figure: Γ; t_{1}; t_{j}; 0; t_{n}; t_{k}; Ω^{+}; γ_{k} = 2; γ_{n}; γ_{j} = 0; γ_{n}; t_{2}; Ω^{−}; γ_{2}; γ_{1}; t; ν⃗(t); t′(s); Re z; ϑ_{t}]

Fig. 1 here Γ_{j} are ν−smooth, ν > 1, oriented curves connecting knots t_{j} and t_{j+1}. Let πγ_{j} be the angle at t_{j} between Γ_{j−1} and Γ_{j}, measured from Ω^{+}, 0 ≤ γ_{j} ≤ 2, j = 1, …, n. When γ_{j} = 0 or γ_{j} = 2 the domain Ω^{+} has an outward or an inward peak, respectively or, what is the same, the boundary curve Γ has a cusp (see Fig. 1).

We use the following standard notation for spaces.

Write C^{m}(Γ) for the space of functions φ(t), t ∈ Γ with continuous derivatives up to the order m

∂_{t}^{k}φ ∈ C(Γ) , k = 0, 1, …, m , ∂_{t} := \frac{d}{dt} , m ∈ ℕ_{0} := \{0, 1, …\}.

Let us note that invariant (with respect to a parametrisation of the underlying curve Γ) definition of the space C^{m}(Γ) can be provided iff Γ is m-smooth. Therefore for piecewise-smooth curves (with angular points or cusps) we can define only C(Γ) := C^{0}(Γ).

Write H_{μ}(Γ) for the space of Hölder continuous functions ψ(t), t ∈ Γ with the following finite norm

∥ψ|H_{μ}(Γ)∥ := ∥ψ|C(Γ)∥ + sup_{t_{1}≠t_{2}} \frac{|ψ(t_{2}) − ψ(t_{1})|}{|t_{2} − t_{1}|^{μ}} , 1 < μ ≤ 1 .

Write P C(Γ) for the space of functions φ(t) which are continuous on each closed arc between knots t_{1}, …, t_{n} and might have jumps at these knots.

Write P C^{m}(Γ) for the space of functions φ ∈ C^{m−1}(Γ) which have piecewise-continuous last derivative ∂_{t}^{m}φ ∈ P C(Γ) with possible jumps at knots t_{1}, …, t_{n}.

Both, the spaces C^{m}(Γ) and P C^{m}(Γ) are endowed with the uniform norm

∥φ|P C^{m}(Γ)∥ := ∑_{k=1}^{m} sup {|∂^{k}φ(t)| : t ∈ Γ} ,

which makes them into Banach spaces.

Let

ρ(t) = ∏_{j=1}^{n} |t − t_{j}|^{α_{j}}  (1.2)

be a weight function and C^{m}(Γ, ρ) ⊂ P C^{m}(Γ, ρ) denote the weighted spaces of functions:

C^{m}(Γ, ρ) := {φ ∈ C^{m−1}(Γ) : ρ∂^{m}φ ∈ C(Γ)} ,
P C^{m}(Γ, ρ) := {φ ∈ C^{m−1}(Γ) : ρ∂^{m}φ ∈ P C(Γ)} .

These spaces both can be endowed with the weighted norm ∥φ|P C^{m}(Γ, ρ)∥= ∥φ|C^{m−1}(Γ))∥ + ∥ρ∂^{m}φ|P C(Γ))∥.

We write C(Γ), P C(Γ, ρ) etc. when m = 0.

Write H_{m+μ}^{0}(Γ, ρ), 0 < μ < 1, m = 0, 1, …, for the weighted function space

H_{m+μ}^{0}(Γ, ρ) := {φ ∈ C^{m−1}(Γ) : φ^{(m)} := ρ∂^{m}φ ∈ H_{μ}(Γ),
φ^{(m)}(t_{1}) = … = φ^{(m)}(t_{n}) = 0} ,
∥φ|H_{m+μ}^{0}(Γ, ρ)∥ := ∥φ|C^{m−1}(Γ)∥ + ∥ρ∂^{m}φ|H_{μ}(Γ)∥ ,

provided Γ \ {t_{1}, …, t_{n}} is C^{m+μ}-smooth, while Γ itself is P C^{m−1}-smooth. Note, that for piecewise-smooth curve Γ definition is correct only for m = 0, 1.

Write L_{p}(Γ, ρ) for the weighted Lebesgue space endowed with the norm

∥φ|L_{p}(Γ, ρ)∥ := (∫_{Γ} |ρ(t)φ(t)|^{p} |dt|)^{\frac{1}{p}} .

Write W_{p}^{m}(Γ, ρ) for the Sobolev space

W_{p}^{m}(Γ, ρ) := {φ : φ, ∂^{k}φ ∈ L_{p}(Γ, ρ), k = 0, …, m} ,

∥φ|W_{p}^{m}(Γ, ρ)∥ := ∑_{k=0}^{m} ∥∂^{k}φ|L_{p}(Γ, ρ)∥ .

Write W_{p}^{s}(Γ, ρ), s ∈ ℝ, for the weighted Sobolev–Slobodetski space which for s ≥ 0 can be defined by the complex interpolation (see [Tr1]) between the spaces W_{p}^{m}(Γ, ρ) and W_{p}^{0}(Γ, ρ) := L_{p}(Γ, ρ) (s ≤ m ∈ ℕ, while for negative s < 0 can be defined as the dual space to W_{p′}^{−s}(Γ, ρ^{−1}), p′ := p/(p − 1).

Since multiplication by a piecewise-continuous function is a bounded operator in W_{p,loc}^{s}(ℝ) only for s < 1/p, the space W_{p}^{s}(Γ, ρ) on piecewisesmooth curve Γ can be defined correctly only for |s| < 1 + 1/p.

Write E_{p}(\overline{Ω^{+}}, ρ) for the Smirnov–Lebesgue space of analytic functions: if ω : D_{1} → Ω^{+} denotes the conformal mapping of the unit disk D_{1} := \{ζ ∈ ℂ : |ζ| < 1\} onto the domain Ω^{+}, the norm of ψ ∈ E_{p}(\overline{Ω^{+}}, ρ) is defined as follows

∥ψ|E_{p}(\overline{Ω^{+}}, ρ)∥ := sup_{0<r<1} ( ∫_{Γ^{(r)}} |ρ(τ)ψ(τ)|^{p} dτ)^{\frac{1}{p}} ,

where Γ^{(r)} := {z = ω(ζ) : |ζ| = r} are the images of the concentric circumferences of the radius r.

Similarly is defined the Smirnov–Lebesgue space E_{p}(\overline{Ω^{−}}, ρ) for the outer domain Ω^{−}.

An equivalent definition of the Smirnov–Lebesgue spaces E_{p}(\overline{Ω^{±}}, ρ) is the following: u ∈ E_{p}(\overline{Ω^{±}}, ρ) iff u(z) is represented by the Cauchy integral as follows

Φ(z) = c_{0} + C_{Γ}φ(z) , c_{0} = const , φ ∈ L_{p}(Γ, ρ) ,
C_{Γ}φ(z) := \frac{1}{2πi} ∫_{Γ} \frac{φ(τ)dτ}{τ − z} , z ∈ Ω^{±}  (1.3)

(cf. [Pv1] and [Go1, Ch.X, § 5]). In particular, for the compact domain Ω^{+} ⊂ ℂ representation (1.3) can be written also as follows Φ(z) = C_{Γ}φ_{0}(z), φ_{0}(t) = c_{0} + φ(t), t ∈ Γ, while for Ω^{−} we have c_{0} = Φ(∞).

Taking the advantage of the last definition we will introduce the following new spaces, suited for our purposes.

Write W_{p}^{s}(\overline{Ω^{±}}, ρ) for the weighted Smirnov–Sobolev space of functions Φ(z) represented as in (1.3) with a density φ ∈ W_{p}^{s}(Γ, ρ). Note, that the restriction for piecewise-smooth contour Γ is |s| < 1 + \frac{1}{p}.

Write W_{p}^{1}(\overline{Ω^{±}}, ρ) for the space of functions Φ(z) which belong to E_{p}(\overline{Ω^{±}}, ρ) together with their derivatives Φ, ∂Φ ∈ E_{p}(\overline{Ω^{±}}, ρ). This is easy to check with a partial integration.

Due to Theorem 1.8 proved below, we get W_{2}^{s}(\overline{Ω^{±}}) ⊂ W_{2,loc}^{s+ \frac{1}{2}}(\overline{Ω^{±}}). If outward peaks are absent 0 < γ_{j} ≤ 1, the following inverse is also true:

traces of functions from W_{2,loc}^{s+ \frac{1}{2}}(\overline{Ω^{±}}) belong to W_{2}^{s}(Γ). In case of outward peaks the last assertion fails as shown in Example 1.3 (see also [Ia1]). Note that formulated theorem on traces remain valid even in the presence of inward peaks (with interior angle 2π).

Write H_{m+μ}^{0}(\overline{Ω^{±}}, ρ), C^{m}(\overline{Ω^{±}}, ρ) and PC^{m}(\overline{Ω^{±}}, ρ) for the weighted Smirnov–Hölder etc. spaces of functions Φ(z) represented as in (1.3) with a density φ in appropriate spaces H_{m+μ}^{0}(Γ, ρ), in C^{m}(Γ, ρ) (with the restriction m ≤ 1 for a piecewise-smooth contour Γ) or in P C^{m}(Γ, ρ), respectively (with the restriction m ≤ 2 for a piecewise-smooth contour Γ).

Write e_{p}(\overline{Ω^{±}}, ρ) = w_{p}^{0}(\overline{Ω^{±}}, ρ), w_{p}^{s}(\overline{Ω^{±}}, ρ), h_{m+μ}^{0}(\overline{Ω^{±}}, ρ), c^{m}(\overline{Ω^{±}}, ρ) and pc^{m}(\overline{Ω^{±}}, ρ) is used for the spaces of harmonic functions represented as real u(z) = Re Φ(z) (or the imaginary u(z) = Im Φ(z) parts of functions Φ(z) from E_{p}(\overline{Ω^{±}}, ρ) = W_{p}^{0}(\overline{Ω^{±}}, ρ), W_{p}^{s}(\overline{Ω^{±}}, ρ), H_{m+μ}^{0}(\overline{Ω^{±}}, ρ), C^{m}(\overline{Ω^{±}}, ρ) and, respectively, from PC^{m}(\overline{Ω^{±}}, ρ). We use e_{p}(\overline{Ω^{±}} etc for the space e_{p}(\overline{Ω^{±}}, 1) etc.

It is important to have representations of functions (1.3) with a pure real or a pure imaginary density φ(t). Next lemma provides the condition for such representation. Similar considerations can be found in [Mu1, §§ 62–66].

Lemma 1.1 Let 𝕏(Γ) be one of the following spaces: W_{p}^{s}(Γ, ρ), H_{m+μ}^{0}(Γ, ρ), C^{m}(Γ, ρ) or P C^{m}(Γ, ρ) and X (\overline{Ω^{±}})–the corresponding Smirnov space W_{p}^{s}(Γ, ρ), H_{m+μ}^{0}(Γ, ρ) etc.

The function Φ ∈ X (\overline{Ω^{±}}) can be represented by the Cauchy integral as in (1.3) with a pure real φ = Re φ ∈ 𝕏(Γ) or a pure imaginary φ = i Im φ ∈ 𝕏(Γ) density if and only if the Riemann–Hilbert problem for the complementary domain Ω^{∓} = ℂ \ Ω^{±}

Re Ψ^{∓}(t) = g(t), t ∈ Γ , g ∈ 𝕏(Γ) , Ψ(z) → 0 as |z| → ∞

is surjective, i.e., has solution for all right-hand sides in X (\overline{Ω^{∓}}).

For the domain Ω^{+} the same conditions provide the representation Φ(z)= C_{Γ}φ_{0}(z), z ∈ Ω^{+} with a real valued density φ_{0} = Re φ_{0}.

We postpone the proof of the formulated Lemma until Subsection 2.3. Let us conclude this subsection by the following agreements which we

will hold on in the sequel.

I. 𝕏^{s}(Γ, ρ) (or more simple 𝕏(Γ)) is used to denote the spaces W_{p}^{s}(Γ, ρ), H_{s}^{0}(Γ, ρ) C^{s}(Γ, ρ) or P C^{s}(Γ, ρ), where the weight function ρ(t) is defined in (1.2) and X^{s}(\overline{Ω^{±}}, ρ), x^{s}(\overline{Ω^{±}}, ρ)–for the corresponding Smirnov spaces of analytic and harmonic functions.

For the parameters there hold the following constraints:

|s| ≤ 1 , −\frac{1}{p} < α_{j} < 1 − \frac{1}{p} , 1 < p < ∞  for  W_{p}^{s}(Γ, ρ) ,

\{ \begin{matrix}m + s , m=0, 1 , 0<s<1 , \\ s < α_{j} < s + 1 , Γ \ \{t_{1}, …, t_{n}\}∈C^{m+s} \end{matrix}  for H_{s}^{0}(Γ, ρ) ,(1.4)
s=m ∈ ℕ_{0} , 0 < α_{j} < 1 for P C^{m}(Γ, ρ)
and for C^{m}(Γ, ρ)

for j = 0, …, n. Conditions (1.4) are necessary and sufficient for boundedness of the Cauchy singular integral operator

S_{Γ}φ(t) = \frac{1}{πi} ∫_{Γ} \frac{φ(τ)dτ}{τ − t} , t ∈ Γ  (1.5)

(the integral in (1.5) is understood in the Cauchy mean value sense) in the spaces W_{p}^{m}(Γ, ρ) ([GK1, Go2, Kh1, Kh2] and H_{m+μ}^{0}(Γ, ρ) (see [Du1, Du6, Du7, GK1]) and of operators with fixed singularities in the kernel (see § 3.2) in all four spaces W_{p}^{m}(Γ, ρ), H_{m+μ}^{0}(Γ, ρ), C^{m}(Γ, ρ) and in P C^{m}(Γ, ρ) (see [Du1] and § 3.2 below; S_{Γ} is not bounded in C^{m}(Γ, ρ) and in P C^{m}(Γ, ρ)).

II. For a space with weight W_{p}^{m}(Γ, ρ), H_{m+μ}^{0}(Γ, ρ) or P C^{m}(Γ, ρ), if not otherwise stated, the weight function is defined in (1.1) and the exponents satisfy the appropriate conditions (1.4).

1.2 Boundary value problems

For a real valued harmonic function

Δu(x) = 0 , x ∈ Ω^{±},  (1.6)

we consider the Dirichlet

u^{±}(t) = g(t) , g ∈ 𝕏^{s}(Γ, ρ) , 0 ≤ s ≤ 1 , t ∈ Γ ,  (1.7)

and the Neumann

(∂_{ν⃗(t)}u)^{±}(t) = f(t) , f ∈ 𝕏^{s−1}(Γ, ρ) , 0 ≤ s ≤ 1 , t ∈ Γ,  (1.8)

boundary value problems, with some real valued data^{2)} Im g(t)≡ Im f(t)≡ 0, where ∂_{ν⃗(t)} := ν_{1}(t)∂_{t_{1}} + ν_{2}(t)∂_{t_{2}}, t = (t_{1}, t_{2}) ∈ Γ denotes the normal derivative. We hold on the agreement about spaces and weights made in conclusion of § 1.1.

We look for solutions of problem (1.6), (1.7) (of (1.6), (1.8)) in the Smirnov class

u ∈ x^{s}(\overline{Ω^{±}}, ρ) ,  0 ≤ s ≤ 1 .  (1.9)

Let us note that by definition of the Smirnov class a function u ∈ x^{s}(\overline{Ω^{−}}, ρ) automatically possesses a finite limit at the infinity: u(x) = O(1) for x ∈ Ω^{−} as |x| → ∞ (see (1.3)).

Next Lemma and example are a certain justification of the choice of constraints (1.9) instead of (0.2) which is common for domains with a Lipschitz boundary (see [Ke1, Ma1, MT1]).

Lemma 1.2 If (0.2) holds, Ω^{±} has no outward peak and u(z) is a harmonic function (i.e., u(z) solves (1.6)). Then

u ∈ w2^{\frac{1}{2}} (\overline{Ω^{±}}) .  (1.10)

Vice versa, u ∈ w2^{\frac{1}{2}} (\overline{Ω^{±}}) ⊂ e_{2}(\overline{Ω^{±}}) implies (0.2) and u(z) is a harmonic function, also for domains Ω^{±} with outward peaks.

We postpone the proof of the formulated Lemma until Subsection 2.3. Next example shows that under constraints (0.2) solution u(x) of BVPs

(1.6), (1.7) and (1.6), (1.8) might have non-integrable trace u^{+}(t) on the boundary Γ = ∂Ω^{+} as soon as Ω^{±} has a single outward peak.

Example 1.3 Let 0 < σ < ∞, γ > 0 and

Ω_{σ}^{+} := {x_{1} + ix_{2} : 0 ≤ x_{1} ≤ 1 , 0 ≤ x_{2} ≤ x_{1}^{σ+1}} .  (1.11)

2)If we admit complex-valued data Im g ≢ 0 in (1.6) and Im f ≢ 0 in (1.8) but then we have to look for a complex-valued solution u = u_{r} + iu_{i}, u_{r}, u_{i} ∈ x^{s}(\overline{Ω^{±}}, ρ) in (1.9).

Then, choosing the branch of the analytic function φ_{γ}(z) := z^{−γ} appropriately, for the harmonic function φ_{γ}(z) := Re z^{−γ} we get Δφ_{γ} = 0 in Ω_{γ} and φ_{γ} ∈ W_{p}^{1}(\overline{Ω_{σ}^{+}}) provided σ − (γ + 1)p > −2.

In particular, φ_{γ} ∈ W_{2}^{1}(\overline{Ω_{2γ+3}^{+}}).

In fact,

∥φ_{γ} |W_{p}^{1}(Ω_{σ}^{+})∥^{p} = ∫_{0}^{1} dx_{1} ∫_{0}^{x_{1}^{σ+1}}[(x_{1}^{2} + x_{2}^{2})^{− \frac{γ}{2} p} + γ(x_{1}^{2} + x_{2}^{2})^{− \frac{γ+1}{2} p}] dx_{2}

≤ C_{1} ∫_{0}^{1} dx_{1} ∫_{0}^{x_{1}^{σ+1}}(x_{1} + x_{2})^{−(γ+1)p}dx_{2} = C_{2} ∫_{0}^{1} x_{1}^{−(γ+1)p+1}dx_{1}

× ∫_{0}^{x_{1}^{σ}}(1 + t)^{−(γ+1)p}dt ≤ C_{3} ∫_{0}^{1} x_{1}^{σ−(γ+1)p+1}dx_{1} = C_{4} < ∞ .

1.3 Representation of solutions and layer potentials

Applying the Gauss formula on divergence (on “partial integration”)

∫_{Ω^{±}} ∂_{j}u(y)\overline{v(y)}dy = ∫_{Ω^{±}} u(y)\overline{∂_{j}v(y)}dy ± ∮_{Γ} ν_{j}(τ)u(τ)\overline{v(τ)}dτ ,  (1.12)

we readily obtain two well-known Green formulae

∫_{Ω^{±}} Δu(y)\overline{v(y)}dy = ∑_{j=1}^{2} ∫_{Ω^{±}} ∂_{j}u(y)\overline{∂_{j}v(y)}dy ± ∮_{Γ} ∂_{ν⃗(τ)}u(τ)\overline{v(τ)}dτ , (1.13)

∫_{Ω^{±}} [Δu(y)\overline{v(y)} − u(y)\overline{Δv(y)}] dy = ∮_{Γ} [∂_{ν⃗(τ)}u(τ)\overline{v(τ)}

−u(τ)\overline{∂_{ν⃗(τ)}v(τ)}] dτ , u, v ∈ C_{com}^{∞}(\overline{Ω^{±}}) . (1.14)

Invoking the fundamental solution of equation (1.6)

F_{Δ}(z) := \frac{1}{2π} log |z| , ΔF_{Δ}(z) = δ(z) , z ∈ ℝ^{2} ,

where δ is Kroneker’s delta function, we can easily derive from (1.14) the following representation formula for a harmonic function u(x) which meets condition (1.9)

\begin{matrix}χ_{+}(x)u(x) = W_{Γ}u^{+}(x) − V_{Γ}(∂ν⃗u)^{+}(x) , \\ χ_{−}(x)u(x) = u_{∞} − W_{Γ}u^{−}(x) + V_{Γ}(∂ν⃗u)^{−}(x) , \\ x ∈ ℝ^{2} \ Γ = Ω^{−} ∪ Ω^{+} \end{matrix}  (1.15)

(see [Ma1, Ch.1, § 1.2]), where u_{∞} =const, χ_{±} is the characteristic function of the domain Ω^{±} and

W_{Γ}φ(x) = \frac{1}{2π} ∫_{Γ} φ(τ)∂_{ν⃗(τ)} log |τ − x|ds , ds = |dτ| ,

V_{Γ}φ(x) = \frac{1}{2π} ∫_{Γ} φ(τ) log |τ − x|ds , x ∈ Ω^{±},

(1.16)

are the double and the single layer potentials (known as the harmonic or the logarithmic potentials as well).

Let us note, that constants are included into the class of harmonic functions in unbounded domains Ω^{−} (see (1.3) and the second formulae in (1.15)) only in 2-dimensional case (see, e.g., [Ma1, p.216], [Vl1, p.333]).

For the direct values of harmonic potentials (1.16) on Γ we use the notation W_{Γ,0} and V_{Γ,−1} where the additional subscript indices indicate the order of these operators, treated as pseudo-differential operators on the manifold Γ (see Theorem 1.5 below). According this rule we have also S_{Γ} := C_{Γ,0} (see (1.3) and (1.5)).

Lemma 1.4 The following holds:

W_{Γ,0}φ(t) = \frac{1}{4} (S_{Γ} + VS_{Γ}V) φ(t) = \frac{1}{4πi} ∫_{Γ} φ(τ)d log \frac{τ − t}{\overline{τ} − \overline{t}}

= \frac{1}{4πi} ∫_{Γ} φ(τ) [\frac{dτ}{τ − t} − \frac{d\overline{τ}}{\overline{τ} − \overline{t}}] ,  (1.17)

W_{Γ,0}^{∗}φ(t) = \frac{1}{4} (hS_{Γ}\overline{h} + VhS_{Γ}\overline{h}V) φ(t)
= \frac{1}{4πi} ∫_{Γ} φ(τ) [\frac{h(t)}{h(τ)} \frac{dτ}{τ − t} − \frac{\overline{h(t)}}{\overline{h(τ)}} \frac{d\overline{τ}}{\overline{τ} − \overline{t}}] ,  (1.18)

∂_{t}V_{Γ,−1}φ(t) = \frac{i}{4} (S_{Γ} − VS_{Γ}V) φ(t)
= \frac{1}{4π} ∫_{Γ} φ(τ) [\frac{dτ}{τ − t} + \frac{d\overline{τ}}{\overline{τ} − \overline{t}}] , t ∈ Γ ,  (1.19)

where

Vφ(t) := \overline{φ(t)} , h(t) := ie^{iϑ_{t}}  (1.20)

and ϑ_{t} denotes the inclination to the abscissa axes of the outer unit normal vector ν⃗(t) (t ∈ Γ \ {t_{1}, …, t_{n}}; see Fig. 1).

Proof (see [Mu1, § § 12,14]). Let us consider the natural parametrisation of the curve Γ by the arc length parameter

τ(s) : [0, ℓ] ⟶ Γ , τ(0) = τ(ℓ) .

Easy to ascertain that if the derivative τ′(s) exists, coincides with the unit tangent vector to Γ. We have

ν⃗(τ) = (cos ϑ_{τ} , sin ϑ_{τ}) ,
dτ = [cos (\frac{π}{2} + ϑ_{τ} ) + i sin (\frac{π}{2} + ϑ_{τ} )] |dτ| = h(τ)ds  (1.21)

(see Fig. 1 and (1.20)). Therefore

\frac{1}{2π} ∂_{ν⃗(τ)}[log |τ − t|]ds = \frac{1}{2π|τ − t|} [\frac{d|τ − t|}{d Re τ} cos ϑ_{τ} + \frac{d|τ − t|}{d Im τ} sin ϑ_{τ} ] ds

= \frac{Re (τ − t) cos ϑ_{τ} + Im (τ − t) sin ϑ_{τ}}{2π|τ − t|^{2}} ds = \frac{−(τ − t)d\overline{τ} + (\overline{τ} − \overline{t})dτ}{4πi}

= \frac{1}{4πi} [\frac{dτ}{τ − t} − \frac{\overline{dτ}}{\overline{τ} − \overline{t}}] = \frac{1}{4πi}d_{τ} log \frac{τ − t}{\overline{τ} − \overline{t}} ,

which gives (1.17).

Formula (1.18) follows from (1.17) since the adjoint operator S_{Γ}^{∗} to S_{Γ} in (1.5) with respect to the sesquilinear form

⟨φ, ψ⟩ := ∫_{Γ} φ(τ)\overline{ψ(τ)}|dτ|

reads

S_{Γ}^{∗} = VhS_{Γ}h^{−1}V = h^{−1}VS_{Γ}VhI .  (1.22)

In fact, since dτ = h(τ)|dτ| and \overline{h(τ)} = h^{−1}(τ) (see (1.20), (1.21)), we get

⟨S_{Γ}φ, ψ⟩ := ∫_{Γ} S_{Γ}φ(τ)\overline{ψ(τ)}|dτ| = ∫_{Γ} S_{Γ}φ(τ)\overline{h(τ)ψ(τ)}dτ

= − ∫_{Γ} φ(t)S_{Γ}\overline{h(τ)ψ(τ)}dt = ∫_{Γ} φ(t)\overline{V(S_{Γ}Vhψ)(t)}h(t)|dt|

= ∫_{Γ} φ(t)\overline{Vh(t)(S_{Γ}h^{−1}Vψ)(t)}|dt|.

To prove formula (1.19) we proceed as follows:

∂_{t}V_{Γ,−1}φ(t) = \frac{1}{2π} ∫_{Γ} φ(τ)∂_{τ} log |τ − t|ds

= \frac{1}{2π} ∫_{Γ} φ(τ)\frac{− Re (τ − t) sin ϑ_{τ} + Im (τ − t) cos ϑ_{τ}}{|τ − t|^{2}} ds

= \frac{1}{2π} ∫_{Γ} φ(τ) \frac{Re (τ − t) Re dτ + Im (τ − t) Im dτ}{|τ − t|^{2}}

= \frac{1}{4π} ∫_{Γ} φ(τ) [\frac{dτ}{τ − t} + \frac{\overline{dτ}}{\overline{τ} − \overline{t}}] .

Next two theorems deal with boundedness properties of layer potentials. They are based on Lemma 1.4 and justify constraints (1.4) on the weight function ρ(t).

Theorem 1.5 W_{Γ,0} is bounded in the spaces W_{p}^{s}(Γ, ρ) for 0 ≤ s ≤ 1 and in H_{m+μ}^{0}(Γ, ρ) for m = 0, 1.

The operator W_{Γ,0}^{∗} is bounded in the spaces W_{p}^{s}(Γ, ρ) for −1 ≤ s ≤ 0 and in H_{μ}^{0}(Γ, ρ).

V_{Γ,−1} is bounded from L_{p}(Γ, ρ) to W_{p}^{1}(Γ, ρ) and from H_{μ}^{0}(Γ, ρ) to H_{1+μ}^{0}(Γ, ρ).

Theorem 1.6 The operator W_{Γ,0} is bounded in C(Γ, ρ) and in P C^{m}(Γ, ρ) for m = 0, 1.

The operator W_{Γ,0}^{∗} is bounded in P C(Γ, ρ).

We postpone the proofs of the formulated theorems until Subsection 2.3. Here we will prove the following corollary.

Corollary 1.7 Let ρ(t) be defined by (1.2) and (1.4). If Γ is smooth (contains no cusps and no angular points γ_{1} = ⋯ = γ_{n} = 1) operators W_{Γ,0} and W_{Γ,0}^{∗} have weak singular kernels and are compact in the spaces L_{p}(Γ, ρ) and P C(Γ, ρ).

Operator W_{Γ,0} is compact also in spaces W_{p}^{s}(Γ, ρ) for −1 ≤ s ≤ 1, in C(Γ, ρ) and in P C^{1}(Γ, ρ).

Proof. It suffices to prove compactness of W_{Γ,0}, since W_{Γ,0}^{∗} is the adjoint operator and would have weak singular kernel if W_{Γ,0} has.

If Γ = ℝ or Γ ⊂ ℝ then K_{1} = W_{Γ,0} = 0 as it is clear from representations (1.17) and (1.18).

If Γ = Γ_{1} := {ζ ∈ ℂ : |ζ| = 1} is the unit circumference then ϑ_{t} ≡ ϑ, h(t) = e^{iϑ} and τ = e^{iϑ}, t = e^{iλ} (0 ≤ ϑ, λ ≤ 2π) inserted into (1.17) gives

K_{1} = W_{Γ_{1},0}φ(λ) = \frac{1}{4π} ∫_{0}^{2π}φ(ϑ)dϑ;  (1.23)

therefore W_{Γ_{1},0} is one dimensional and compact.

If Γ is arbitrary smooth curve and ω : Γ → Γ′ is a corresponding diffeomorphism where either Γ′ ⊂ ℝ or Γ′ = Γ_{1}, then

W_{Γ,0} = K_{ω} − K_{ω}^{∗} + ω_{∗}^{−1}K_{1}ω_{∗} ,
K_{ω} := ω_{∗}^{−1}S_{Γ}ω_{∗} − S_{Γ} , ω_{∗}φ(t) = φ(ω(t)) , t ∈ Γ ,

with ω^{−1} : Γ → Γ′ standing for the inverse diffeomorphism and K_{ω}^{∗}–for the adjoint to K_{ω}. the integral operator K_{ω} has a weak singular kernel (see [DLS1, § 3.5] or [Kh1, GK1]). As for K_{1}, either K_{1} = 0 or it is a one dimensional operator (see (1.23)).

To accomplish boundedness properties of potential operators and their direct values on the curve we formulate the next result. For a general assertion (layer potentials for partial differential operators with variable coefficient and arbitrary order in ℝ^{n}, provided they have a fundamental solution) we quote [Du10, Theorem 3.2] (for Lipschitz domains see also [MMP1, MMT1, MT1]).

Theorem 1.8 Let s ∈ ℝ and the boundary Γ = ∂Ω^{±} be m-smooth, where m ∈ ℕ_{0}, m ≥ |s|.

The potential operators^{3)}

C_{Γ} : W_{2}^{s}(Γ) ⟶ W_{2,com}^{s+ \frac{1}{2}}(\overline{Ω^{±}}) ,
W_{Γ} : W_{2}^{s}(Γ) ⟶ W_{2,com}^{s+ \frac{1}{2}}(\overline{Ω^{±}}) ,
V_{Γ} : W_{2}^{s}(Γ) ⟶ W_{2,com}^{s+ \frac{3}{2}}(\overline{Ω^{±}})

(1.24)

(see (1.3) and (1.16)) are bounded^{4)}.

In particular, if Γ is piecewise-smooth we should restrict −1 ≤ s ≤ 1.

Proof. For a smooth Γ = ∂Ω^{±} see [Du10, Theorem 3.2].

Let Γ have knots t_{1}, …, t_{n} (see (Fig. 1) and consider C_{Γ}φ(z). The operator C_{Γ} is of the local type, i.e., if

v_{1} ∈ C_{0}^{∞}(\overline{Ω^{+}}) , v_{2} ∈ L_{∞}(Γ) , supp v_{1} ∩ supp v_{2} = ∅ ,

then v_{1}C_{Γ}v_{2}φ ∈ C_{0}^{∞}(\overline{ℂ}). Therefore it suffices to establish continuity (1.24) for vC_{Γ}uI, where v ∈ C_{0}^{∞}(\overline{Ω^{+}}), u ∈ C(Γ) are cut-off functions, equal 1 in some small neighbourhood of a knot t_{j} and vanishing outside another one; in particular, v(t_{k}) = u(t_{k}) = 0 for j ≠ k.

We can suppose that

φ = φ_{1} + φ_{2} , φ_{k} := u_{k}φ ∈ W_{2}^{s}(Γ) , u_{k} := χ_{k}u , k = 1, 2 ,

where χ_{1}(t) and χ_{2}(t) are characteristic functions of the left and right neighbourhoods of t_{j} ∈ Γ and [χ_{1}(t) + χ_{1}(t)]u(t) = u(t). Since χ_{k}(t), k = 1, 2 have discontinuities at t_{j}, for the claimed inclusions φ_{k} ∈ W_{2}^{s}(Γ) we need φ(t_{j}) = 0 if s ≥ \frac{1}{2}. The latter can be provided since (C_{Γ}1)(z) ≡ 1 for z ∈ Ω^{+} and

C_{Γ}φ(z) = C_{Γ}φ_{0}(z) + φ(t_{j}) .

Thus,

v(z)C_{Γ}uφ(z)=u(z)C_{Γ}φ_{1}(z)+u(z)C_{Γ}φ_{2}(z)=u(z)C_{Γ_{1}}φ_{1}(z)+u(z)C_{Γ_{2}}φ_{2}(z), where Γ_{1} and Γ_{2} are smooth and closed contours which have in common either only the point {t_{j}} when γ_{j} < 1, or two points (one of them {t_{j}})

3)For a compact domain we define W_{2,com}^{μ}(\overline{Ω^{±}}) = W_{2}^{μ}(\overline{Ω^{±}}).

4)We have formulated only a particular result–the case p = 2. The general result for 1 < p < ∞ in [Du10] states boundedness between the Bessel potential and the Besov spaces.

when 1 < γ_{j} < 2, or some arc Γ′ = ) ; t_{j} z_{o} when γ_{j} = 2. We can assume φ_{k} ∈ W_{2}^{s}(Γ_{k}) extending functions to Γ_{k} \ (Γ ∩ Γ_{k}) by 0 (k = 1, 2). As noted above, due to smoothness of Γ_{k} we get vC_{Γ_{k}}φ_{k} ∈ W_{2}^{s+ \frac{1}{2}} (\overline{Ω_{k}^{+}}), where Ω_{k}^{+} is the inner domain for Γ_{k}, k = 1, 2. On the other hand,

supp v ∩ \overline{Ω^{+}} = \overline{Ω_{1}} ∪ \overline{Ω_{2}} , Ω_{k} := ( supp v ∩ Ω^{+}) ∩ Ω_{k}^{+} , k = 1, 2 .

Then vC_{Γ}φ_{k} = vC_{Γ_{k}}φ_{k} ∈ W_{2}^{s+ \frac{1}{2}} (\overline{Ω_{1}} ∪ Γ_{2}) since on the common boundary \overline{Ω_{1}} ∩ \overline{Ω_{2}} = Γ_{1} ∩ Γ_{2} ⊂ Ω^{+} ∪ \{t_{j}\}, except t_{j}, functions are C^{∞} smooth.

Therefore, vC_{Γ}uφ = vC_{Γ}φ_{1} + vC_{Γ}φ_{2} ∈ W_{2}^{s+ \frac{1}{2}} (\overline{Ω^{+}}).

The inclusion vC_{Γ}uφ ∈ W_{2,com}^{s+ \frac{1}{2}}(\overline{Ω^{−}}) and other results in (1.24) are proved similarly.

To proceed further we need the Plemelji formulae (the jump relations) for layer potentials, which we formulate next.

Let Φ ∈ C(\overline{Ω^{±}}). By Φ^{±}(t), t ∈ Γ = ∂Ω^{±} is denoted, as usual, nontangential boundary values Φ^{±}(t) = lim_{z∈Ω^{±}, z→t} Φ(z).

Lemma 1.9 Let 1 < p < ∞, −1 ≤ s ≤ 1 and φ ∈ W_{p}^{s}(Γ, ρ), where ρ(t) is defined in (1.2), (1.4). Then

(W_{Γ}φ)^{±}(t) = ±\frac{1}{2}φ(t) + W_{Γ,0}φ(t) , (∂ν⃗V_{Γ}φ)^{±}(t) = ∓\frac{1}{2}φ(t) + W_{Γ,0}^{∗}φ(t) ,

(∂ν⃗W_{Γ}φ)^{+}(t) = (∂ν⃗W_{Γ}φ)^{−}(t) , (C_{Γ}φ)^{±}(t) = ±\frac{1}{2}φ(t) + \frac{1}{2}S_{Γ}φ(t) , (1.25)

for almost all t ∈ Γ (for all t ∈ Γ \ \{t_{1}, …, t_{n}\} provided s > \frac{1}{2} or φ ∈ H_{μ}^{0}(Γ, ρ)).

Proof. The proof can be found e.g. in [Mu1, § § 15,16]) (see the survey [Ma1]). See also [MT1, Appendix C] for the case of Lipschitz domains and [Du10, § 6.4] for much more general operators.

If Γ is a compact curve and φ ∈ L_{1}(Γ) then

W_{Γ}φ(x) = O (\frac{1}{|x|}) as |x| → ∞ .  (1.26)

As for the single layer potential,

V_{Γ}φ(x) = O(1) as |x| → ∞ iff ∫_{Γ} φ(τ)|dτ| = 0  (1.27)

and then

V_{Γ}φ(x) = o(1) as |x| → ∞ .  (1.28)

In fact,

V_{Γ}φ(x) = ∫_{Γ} φ(τ) log \frac{r}{|x|}|dτ| + log |x| ∫_{Γ} φ(τ)|dτ|

= o(1) + log |x| ∫_{Γ} φ(τ)|dτ| as |x| → ∞ .

and (1.27), (1.28) follow.

If

∫_{Γ} (∂_{ν⃗(τ)}u)^{−}(τ)|dτ| = 0,  (1.29)

then in (1.15) we have

u_{∞} = u(∞) = W_{Γ}u^{−}(0) − V_{Γ}(∂ν⃗u)^{−}(0) .  (1.30)

In fact, the first equality u_{∞} = u(∞) follows from (1.15), and (1.26)– (1.28) since (1.29) holds.

Passing to the limit x → t ∈ Γ, x ∈ Ω^{−}, in the representation formula (1.15) and applying the appropriate Plemelji formulae (1.25) we find:

u^{−}(t) = u_{∞} + \frac{1}{2}u^{−}(t) − W_{Γ}u^{−}(t) + V_{Γ}(∂ν⃗u)^{−}(t) , t ∈ Γ .

The obtained formula can be rewritten as follows

u_{∞} = \frac{1}{2}u^{−}(t) + W_{Γ}u^{−}(t) − V_{Γ}(∂ν⃗u)^{−}(t) , t ∈ Γ .

Therefore, the trace of the harmonic function

w(x) = W_{Γ}u^{−}(x) − V_{Γ}(∂ν⃗u)^{−}(x) , x ∈ Ω^{+},

on the boundary Γ = ∂Ω^{+}

w^{+}(t) = \frac{1}{2}u^{−}(t) + W_{Γ}u^{−}(t) − V_{Γ}(∂ν⃗u)^{−}(t) = u_{∞} , t ∈ Γ .

(see the appropriate Plemelji formulae (1.25)) is constant. This implies w(x) ≡=const for the entire domain x ∈ \overline{Ω^{+}} and, therefore, u_{∞} = w(0) = W_{Γ}u^{−}(0) − V_{Γ}(∂ν⃗u)^{−}(0).

The integral W_{Γ}1(x) is known as the Gaussian integral and can be written explicitly:

W_{Γ}1(x) = \frac{1}{2π} ∫_{Γ} ∂_{ν⃗(τ)} log |τ − x||dτ| = \{ \begin{matrix}1 & if & x ∈ Ω^{+} , \\ 0 & if & x ∈ Ω^{−} , \\ \frac{1}{2} & if & x ∈ Γ \end{matrix} (1.31)

(see [Ma1, Chapter I, § 1.1]).

Remark 1.10 The homogeneous equation

−\frac{1}{2}𝔯 + W_{Γ,0}^{∗}𝔯 = 0 ,  (1.32)

has a unique linearly independent solution 𝔯_{0} ≢ 0 in L_{2}(Γ) such that

V_{Γ}𝔯_{0}(x) ≡ 1 , ∫_{Γ} 𝔯_{0}(τ)|dτ| ≡ 1 ,

V_{Γ}𝔯_{0}(x) = O (log |x|) for x ∈ Ω^{−} as |x| → ∞ .

(1.33)

The solution 𝔯_{0} ∈ W_{∞}^{1}(Γ) is known as the Robin function (or the density of the Robin potential; see [Ma1, § 2.2]).

The homogeneous equation

−\frac{1}{2}ψ(t) + W_{Γ,0}ψ(t) = 0

has, due to (1.31), the solution ψ(t) ≡ 1, which is a unique linearly independent solution of this equation in L_{2}(Γ) (see [Ma1, § 2.2]).

Lemma 1.11 The Riemann–Hilbert problem

Re Ψ^{±}(t) = g(t) , t ∈ Γ,  (1.34)

has a solution Ψ ∈ E_{p}(\overline{Ω^{±}}, ρ) for all right-hand sides g ∈ L_{p}(Γ, ρ) (i.e., is surjective under asserted conditions) if and only if:

i. \frac{1}{p} + α_{j} ≠ \{ \begin{matrix}\frac{1}{γ_{j}} & for Ω^{+} , \\ \frac{1}{2 − γ_{j}} & for Ω^{−} , \end{matrix}  (1.35)

ii. the domain has no inward peaks: \{ \begin{matrix}0 ≤ γ_{j} < 2 & for Ω^{+} , \\ 0 < γ_{j} ≤ 2 & for Ω^{−} \end{matrix} (1.36)

for all j = 1, …, n.

Moreover, (1.34) is Fredholm if and only if (1.36) holds and then the index of the corresponding operator reads

Ind A = ∑_{(\frac{1}{p} +α_{j})γ_{j}>1} 1 for Ω^{+} ,

Ind A = ∑_{(\frac{1}{p} +α_{j})(1−γ_{j})>1} 1 for Ω^{−} .

Proof. The proof will be given in § 5.2.

1.4 Reduction to boundary integral equations (the indirect method)

Theorem 1.12 Let conditions (1.35) and (1.36) hold for the complementary domain Ω^{∓}. A harmonic function u ∈ e_{p}(\overline{Ω^{±}}, ρ) solves the Dirichlet problem (1.6), (1.7) if and only if

u(x) = χ_{−}(x)g_{0} + W_{Γ}φ_{±}(x) , x ∈ Ω^{±} ,  (1.37)

where

g_{0} := ∫_{Γ} g(τ)𝔯_{0}(τ)|dτ| ,  (1.38)

𝔯_{0}(τ) is the Robin function (see Remark 1.10) and φ_{±} = Re φ_{±} ∈ L_{p}(Γ, ρ) is some real valued solution of the corresponding boundary integral equation (written separately for the domains Ω^{+} and Ω^{−}, respectively)

A_{+}φ_{+}(t) := \frac{1}{2}φ_{+}(t) + W_{Γ,0}φ_{+}(t) = g(t) , t ∈ Γ ,  (1.39)
A_{−}φ_{−}(t) := −\frac{1}{2}φ_{−}(t) + W_{Γ,0}φ_{−}(t) = g(t) − g_{0} , t ∈ Γ . (1.40)

Proof. Easy to ascertain that formulae (1.17) and (1.19) hold for the corresponding potential operators as well

W_{Γ}φ(z) = \frac{1}{2} (C_{Γ} + VC_{Γ}V) φ(z) = Re [C_{Γ} Re φ(z)] + i Re [C_{Γ} Im φ(z)] ,

∂_{z}V_{Γ}φ(z) = \frac{i}{2} (C_{Γ} − VC_{Γ}V) φ(z) = − Im (C_{Γ} Re φ)(z) + i Im (C_{Γ} Im φ)(z)

= Re (C_{Γ}i Re φ)(z) − i Re (C_{Γ}i Im φ)(z), , z ∈ Ω^{±}  (1.41)

(see (1.3)).

Conditions (1.35), (1.36) provide representation of a solution u ∈ e_{p}(\overline{Ω^{±}}, ρ), by the real part of the Cauchy integral with a real valued density

u(x) = χ_{−}(x)g_{0} + Re [C_{Γ}φ_{±}(x)] , φ ∈ L_{p}(\overline{Ω^{±}}, ρ) , x ∈ Ω^{±}

(see Lemmata 1.1, 1.13 and 1.11) and, due to (1.41) the latter can be rewritten in the form (1.37).

Passing to the limit x → t ∈ Γ, x ∈ Ω^{±} in the representation formula (1.37), applying the appropriate Plemelji formulae (1.25) and inserting u^{±}(t) = g(t) we get equations (1.39) for the density φ_{+} and (1.40) for the density φ_{−}, respectively.

The constant u(∞) in (1.37) is chosen in the form (1.38) to justify the orthogonality condition

∫_{Γ} [g(τ) − g_{0}]𝔯_{0}(τ)|dτ| = ∫_{Γ} g(τ)𝔯_{0}(τ)|dτ| − g_{0} = 0  (1.42)

(see (1.33)) which is necessary and sufficient for the existence of the solution of equation (1.40) provided the equation is Fredholm (see § 1.6, Theorem 1.23).

Vice versa, let φ_{+}, φ_{−} + c_{0} ∈ L_{p}(Γ, ρ), c_{0} =const, be solutions of (1.39), (1.40), respectively (we remind, that homogeneous equation (1.40) has constants as solutions; see Remark 1.10); let u(∞) = g_{0} =const be defined by (1.38). u(x) in (1.37) solves equation (1.6); passing to the limit x → t ∈ Γ, x ∈ Ω^{±} and invoking the appropriate Plemelji formulae (1.25) due to equalities (1.39), (1.40) we get

u^{+}(t) = \frac{1}{2}φ_{+}(t) + W_{Γ,0}φ_{+} = g(t) , t ∈ Γ ,

u^{−}(t) = χ_{−}(t)g_{0} − [(W_{Γ}φ_{−} + c_{0})(x)]^{−} = χ_{−}(t)g_{0} − [(W_{Γ}φ_{−})(x)]^{−}

= χ_{−}(t)g_{0} − \frac{1}{2}φ_{−}(t) + W_{Γ,0}φ_{−} = g(t) , t ∈ Γ

since (W_{Γ}c_{0})(x) ≡ 0 for x ∈ Ω^{−} (see Remark 1.10) and the boundary condition (1.7) holds.

Let us note that representation (1.37) (and, later, a similar one (1.43)) can not be used if inward peak is present. Namely, there holds the following.

Lemma 1.13 The function u ∈ e_{p}(\overline{Ω^{±}}, ρ) (u ∈ w_{p}^{s}(\overline{Ω^{±}})) can be represented by the double layer potential (1.37) with a density φ ∈ L_{p}(Γ, ρ) (in W_{p}^{s}(Γ)) if and only if the Riemann–Hilbert problem for the complementary domain Ω^{∓} (1.34) is surjective (see Lemma 1.11).

Proof. The proof follows from Lemma 1.11. In fact, let Φ(z) = u(z) + iv(z), Φ ∈ E_{p}(\overline{Ω^{±}}, ρ) (Φ ∈ W_{p}^{s}(\overline{Ω^{±}}, ρ)) be the analytic function in the same domain Ω^{±}. Since, due to (1.41), W_{Γ} = \frac{1}{2} (C_{Γ} + VC_{Γ}V), representation (1.37) follows if the representation of the analytic function Φ(z) by the Cauchy integral (1.3) with a pure real φ = Re φ density in L_{p}(Γ, ρ) (in W_{p}^{s}(Γ)) holds.

Vice versa, let φ = Re φ and u = W_{Γ}φ = Re C_{Γ}φ; since Φ = u + iv is defined by u(z) uniquelly modulo a pure imaginary aditive constant ic_{0}, we find Φ(z) = ic_{0} + C_{Γ}φ(z) (cf. (1.3)) with the same density φ = Re φ.

Theorem 1.14 Let conditions (1.35) and (1.36) hold for the complementary domain Ω^{∓}. A harmonic function u ∈ e_{p}(\overline{Ω^{±}}, ρ) solves the Neumann problem (1.6), (1.8) if and only if

\begin{matrix}u(x) = c_{0} + V_{Γ}ψ_{±}(x) , x ∈ Ω^{±} , \\ ∫_{Γ} ψ_{−}(τ)|dτ| = 0 , \end{matrix}  (1.43)

where ψ_{±} ∈ w_{p}^{−1}(Γ, ρ) are solutions of equations (written separately for the domains Ω^{+} and Ω^{−}, respectively)

B_{+}ψ_{+}(t) := −\frac{1}{2}ψ_{+}(t) + W_{Γ,0}^{∗}ψ_{+}(t) = f(t) , t ∈ Γ ,  (1.44)
B_{−}ψ_{−}(t) := \frac{1}{2}ψ_{−}(t) + W_{Γ,0}^{∗}ψ_{−}(t) = f(t) , t ∈ Γ ,  (1.45)

and c_{0} is arbitrary constant.

Proof. Since solution belongs to the Smirnov space u ∈ e_{p}(\overline{Ω^{±}}, ρ) and conditions of Lemmata 1.1, 1.11 hold, we have the following representation

u(x) = c_{0} + Im [C_{Γ}ψ_{±}^{0}(x)] , Im ψ_{±}^{0} = 0 , ψ_{±}^{0} ∈ L_{p}(Γ, ρ) , x ∈ Ω^{±}

(see (1.3)). Due to (1.41) the latter can be rewritten in the form (1.43)

u = Im C_{Γ}ψ_{±}^{0} = i∂_{z}V_{Γ}ψ_{±}^{0} = V_{Γ}[∂_{τ} ψ_{±}^{1}] = V_{Γ}ψ_{±} , ψ_{±} := ∂_{t}ψ_{±}^{1} = i∂_{t}ψ_{±}^{0}

and ψ_{±} ∈ w_{p}^{−1}(Γ, ρ) since ψ_{±}^{0} ∈ e_{p}(Γ, ρ).

Applying the normal derivative ∂_{ν⃗(x)} to the representation (1.43), passing to the limit x → t ∈ Γ, x ∈ Ω^{±} with the help of appropriate Plemelji formulae (1.25) and inserting u^{±} = g we get equations (1.44) for the density ψ_{+}(t) and (1.45) for the density ψ_{−}, respectively.

The second condition in (1.43) provides u(x) = c_{0} + o(1) for x ∈ Ω^{−} as |x| → ∞ (see (1.26), (1.28)).

Vice versa, let ψ_{±} ∈ w_{p}^{−1}(Γ, ρ) be solutions of (1.44), (1.45). Then u(x) in (1.43) solves equation (1.6) and has the asymptotic u(x) = c_{0} + o(1) as |x| → ∞. Applying the normal derivative ∂_{ν⃗(x)}, passing to the limit x → t ∈ Γ, x ∈ Ω^{±} and invoking the appropriate Plemelji formulae (1.25) due to equalities (1.44), (1.45) we get

(∂ν⃗u)^{±}(t) = ∓\frac{1}{2}ψ_{±}(t) + W_{Γ,0}^{∗}ψ_{±} = f(t) , t ∈ Γ,

and the boundary condition (1.8) holds as well.

Lemma 1.15 The homogeneous Dirichlet BVP (1.6), (1.7) with g = 0 and u ∈ w2^{\frac{1}{2}} (\overline{Ω^{±}}) has a unique solution.

The homogeneous Neumann BVP (1.6), (1.8) with f = 0 and u ∈ w2^{\frac{1}{2}} (\overline{Ω^{±}}) has only a constant solution u(x) ≡ const .

Proof. The proof is based on the Green formula (1.13) and is standard. In fact, if u ∈ w2^{\frac{1}{2}} (\overline{Ω^{±}}) then on the boundary u^{±} ∈ W2^{\frac{1}{2}} (Γ). Due to Theorem 1.8 this yields u ∈ W_{2}^{1}(\overline{Ω^{±}}) and ∂_{x_{j}}u ∈ L_{2}(\overline{Ω^{±}}), Δu ∈ W_{2}^{−1}(\overline{Ω^{±}}), j = 1, 2.

Now if Δu(x) = 0 in Ω^{±} and u^{±}(t) = 0 on Γ (see (1.6), (1.7)) by assuming v(x) = u(x) in (1.13) we get

∑_{j=1}^{2} |∂_{j}u(x)|^{2} ≡ 0 for x ∈ Ω^{±} .

Therefore u(x) =const on entire domain and since u(t) = 0 on the boundary, u(x) = 0 everywhere.

For the Neumann BVP (1.6), (1.8) the proof is similar.

1.5 Reduction to Cauchy singular integral equations on the circumference

In the present subsection we reduce the Dirichlet (1.6), (1.7) and the Neumann (1.6), (1.7) BVPs to Riemann–Hilbert BVPs for analytic functions on the unit circumference Γ_{1} or, what is equivalent, to Cauchy singular integral equations (SIEs) on Γ_{1}. Theorems on the Fredholm and the solvability properties for the obtained SIEs will be formulated in § 1.6.

The method goes back to N.Muskhelishvili (see [Mu1, Ch. III]) and I.Vekua [Ve1]; they investigated BVPs in Hölder spaces when domain has smooth boundary (see [Mu1, § § 41,43,75]) and for domains with finite number of cuts (see [Mu1, § 109]). In [Kh1] B.Khvedelidze treated similar problems in the Lebesgue spaces and in [KKP1, Ch. IV] the method was applied to the same BVPs on domains with angular points and cusps in the Smirnov–Lebesgue space e_{p}(\overline{Ω^{±}}) without weight. For the weighted space see [Me1].

Let Ω^{±}, t_{1} …, t_{j} ∈ Γ = ∂Ω^{±} be as in § 1.1 and

ω : D_{1} ⟶ Ω^{±} , ω(ζ_{j}) = t_{j} , j = 1, …, n,  (1.46)

be a conformal mapping of the unit disk

D_{1} = D_{1}^{+} := {z ∈ ℂ : |z| < 1}

onto the domain Ω^{±} (ω(0) = 0, ω′(0) = 1 for the domain Ω^{+} and ω(0) = ∞, ω′(0) = 1 for the domain Ω^{−}; see § 5 for further details). By ζ(x) we denote the inverse mapping

ζ : Ω^{±} ⟶ D_{1} , ζ(ω(z)) ≡ z , ω(ζ(x)) ≡ x .  (1.47)

Then

ω′(ζ(z)) = [ζ′(z)]^{−1} ,
ζ(0) = 0, ζ′(0) = 1 for Ω^{+} ,
ζ(∞) = 0, ζ′(∞) = 1 for Ω^{−} .

Let D_{1}^{−} := {z ∈ ℂ : |z| > 1} be the domain outer to the unit disk D_{1} = D_{1}^{+} and

ρ_{0}(z) := ∏_{j=1}^{n}(z − t_{j})^{α_{j}} for z ∈ Ω^{+} ⊂ ℂ  (1.48)

denote the analytic function in the domain Ω^{+}, which is the extension of the weight function; namely, ρ_{0}(x) is analytic in the complex plane ℂ cut along some curves connecting knots t_{1}, …, t_{n} ∈ ∂Ω^{±} with infinity and do not crossing the domain Ω^{+}.

Theorem 1.16 A harmonic function u ∈ e_{p}(\overline{Ω^{±}}, ρ) solves the Dirichlet problem (1.6), (1.7) if and only if

u(x) = Re [\frac{[ζ′(x)]^{\frac{1}{p}}}{2πρ_{0}(x)} \{ ∫_{|τ|=1} \frac{φ(τ)dτ}{τ − ζ(x)} − \frac{i}{2} ∫_{−π}^{π} φ(e^{iϑ})dϑ\}]  (1.49)

for x ∈ Ω^{±},. where ζ(x) is the conformal mapping from (1.47). φ = Re φ ∈ L_{p}(Γ_{1}) in (1.49) is a real-valued solutions of the following singular integral equation on the unit circumference

Aφ(ζ) := P_{Γ_{1}}^{+}φ(ζ) + G(ζ)P_{Γ_{1}}^{−}φ(ζ) + \frac{G(ζ) − 1}{2}Kφ = g_{0}(ζ) , ζ ∈ Γ_{1} ,

Kφ := \frac{1}{2π} ∫_{−π}^{π} φ(e^{iϑ})dϑ , P_{Γ_{1}}^{±} := \frac{1}{2}(I ± S_{Γ_{1}}) ,  (1.50)

where the coefficient G ∈ P C(Γ_{1}) (see § 5.2) and the right-hand side g_{0} ∈ L_{p}(Γ_{1}) are defined as follows:

\begin{matrix}G(ζ) := −\frac{ρ_{0}(ω(ζ))}{\overline{ρ_{0}(ω(ζ))}} [\frac{ω′(ζ)}{\overline{ω′(ζ)}}]^{\frac{1}{p}} , \\ g_{0}(ζ) := −2iρ_{0}(ω(ζ))[ω′(ζ)]^{\frac{1}{p}} g(ω(ζ)) , ζ ∈ Γ_{1} . \end{matrix}  (1.51)

The solution has the following asymptotic at infinity

u(∞) = Re [(2π)^{−1} ∏_{j=1}^{n}(−t)^{α_{j}} ∫_{−π}^{π} φ(e^{iϑ})dϑ] .  (1.52)

Proof. The Dirichlet problem (1.6), (1.7) can be written as follows

Re [Ψ^{±}(t)] = g(t) , t ∈ Γ_{1} ,
u(x) = Re Ψ(x) , Ψ ∈ E_{p}(\overline{Ω^{±}}, ρ_{0}) , x ∈ Ω^{±} .

(1.53)

Then for the analytic function

Φ(z) := \{ \begin{matrix}ρ_{0}(ω(z))[ω′(z)]^{\frac{1}{p}} Ψ(ω(z)) & for & |z| < 1 , \\ \overline{ρ_{0} (ω (\frac{1}{\overline{z}}))} [ \overline{ω′ (\frac{1}{\overline{z}})} ]^{\frac{1}{p}} \overline{Ψ (ω (\frac{1}{\overline{z}}))} & for & |z| > 1 \end{matrix} (1.54)

(see [Mu1, § 39] and [KKP1, Ch. IV, § 1]), where ρ_{0}(x) is defined in (1.48), boundary condition (1.53) acquires the form

Re [Ψ^{±}(ω(ζ))] = \frac{1}{2} [\frac{Φ^{+}(ζ)}{ρ_{0}(ω(ζ))[ω′(ζ)]^{\frac{1}{p}}} + \frac{Φ^{−}(ζ)}{\overline{ρ_{0}(ω(ζ))}[\overline{ω′(ζ)}]^{\frac{1}{p}}} ] = g(ω(ζ)) ,

which can also be written as follows

Φ^{+}(ζ) − G(ζ)Φ^{−}(ζ) = g_{0}(ζ) , ζ ∈ Γ_{1} ,  (1.55)

with G(ζ) and g_{0}(ζ) defined in (1.51). Since Φ ∈ E_{p}(\overline{D_{1}^{−}}) ∩ E_{p}(\overline{D_{1}^{+}}) it is represented by the Cauchy integral

Φ(z) = −\frac{χ_{Ω_{+}}(z)}{2}Kiφ + C_{Γ_{1}}iφ(z) = −\frac{χ_{Ω_{+}}(z)}{2π} ∫_{−π}^{π} φ(e^{iϑ})dϑ

+\frac{1}{2π} ∫_{|τ|=1} \frac{φ(τ)dτ}{τ − z}  (1.56)

for all |z| ≠ 1 with a pure imaginary density iφ, φ ∈ L_{p}(Γ_{1}). If we apply the Plemelji formulae for the Cauchy integral (1.25) we get

Φ^{±}(ζ) = −\frac{i}{2} [χ_{Ω_{+}}(z)Kφ ± φ(ζ) + S_{Γ_{1}}φ(ζ)] = −\frac{iχ_{Ω_{+}}(z)}{2}Kφ ± iP_{Γ_{1}}^{±}φ(ζ)

for ζ ∈ Γ_{1} and inserting this into (1.55) we get equation (1.50) for the density φ ∈ L_{p}(Γ_{1}).

Let us remind that we need only the real-valued solution φ = Re φ of (1.50). To this end let us check that if ψ ∈ L_{p}(Γ_{1}) is a solution, than \overline{ψ} is a solution as well. In fact, applying the relations

\overline{ζ} = \frac{1}{ζ} , \overline{τ} = \frac{1}{τ} , d\overline{τ} = \frac{dτ}{τ^{2}} , \frac{dτ}{τ} = idϑ for τ = e^{iϑ}, |ζ| = 1, −π < ϑ < π

we find that

\overline{G(ζ)} = G^{−1}(ζ) , \overline{g_{0}(ζ)} = G^{−1}(ζ)g_{0}(ζ) , \overline{g} = g ,

\overline{P_{Γ_{1}}^{±}ψ(ζ)} = \frac{1}{2}\overline{ψ(ζ)} ∓ \frac{1}{2πi} ∫_{|τ|=1} \frac{\overline{ψ(τ)}\overline{dτ}}{\overline{τ} − \overline{ζ}} = \frac{1}{2}\overline{ψ(ζ)} ∓ \frac{1}{2πi} ∫_{|τ|=1} \frac{ζ}{τ} \frac{\overline{ψ(τ)}dτ}{τ − ζ}

= P_{Γ_{1}}^{∓} \overline{ψ}(ζ) ± \frac{1}{2πi} ∫_{|τ|=1} \overline{ψ(τ)}\frac{dτ}{τ} = P_{Γ_{1}}^{∓} \overline{ψ}(ζ) ± K\overline{ψ} .  (1.57)

Now, if ψ ∈ L_{p}(Γ_{1}) is a solution of equation (1.50), taking the complex conjugate and invoking (1.57) we get the same equality for \overline{ψ}:

G(ζ)\overline{Aφ(ζ)} := P_{Γ_{1}}^{+} \overline{ψ}(ζ) + G(ζ)P_{Γ_{1}}^{−} \overline{ψ}(ζ) + \frac{G(ζ) − 1}{2}K\overline{ψ} = g_{0}(ζ) , ζ ∈ Γ_{1}.

Therefore, the real-valued function ψ := Re ψ = \frac{1}{2}(ψ + \overline{ψ}) is a solution we look for.

With a solution φ Re φ of (1.50) at hand we find Φ(z) from (1.56), but the latter might have the following symmetry property

Φ_{∗}(z) := \overline{Φ (\frac{1}{\overline{z}})} = Φ(z) , z ∈ Ω^{+} ∪ Ω^{−} ,

originating from the definition (1.54). This property is proved similarly to (1.57):

Φ(z) = \overline{Φ (\frac{1}{\overline{z}})} = \frac{i}{2}Kφ + \frac{1}{2π} ∫_{|τ|=1} \frac{φ(τ)\overline{dτ}}{\overline{τ} − \frac{1}{z}} = \frac{i}{2}Kφ + \frac{1}{2π} ∫_{|τ|=1} \frac{z}{τ} \frac{φ(τ)dτ}{τ − z}

= −\frac{i}{2}Kφ + \frac{1}{2π} ∫_{|τ|=1} \frac{φ(τ)dτ}{τ − z} = −\frac{i}{2}Kφ + iC_{Γ_{1}}φ(z) = Φ(z) . (1.58)

Inserting Φ(z) in (1.54) we find first Ψ(x) and afterwards u = Re Ψ. The result is written in (1.49).

Vice versa, if φ(ζ) is a solutions of (1.58) we easily ascertain that Ψ(z) found in (1.56) and (1.54) solves BVP (1.53) and u(x) (see (1.49)) solves BVP (1.6), (1.7).

Asymptotic (1.52) results from (1.47)–(1.49) and from the following asymptotic of the weight function

\overline{ρ_{0} (ω (\frac{1}{\overline{z}}))} = ∏_{j=1}^{n}(−t)^{−α_{j}} + O(|z|^{−1}) as |z| → ∞ .

Theorem 1.17 A harmonic function u ∈ w_{p}^{1}(\overline{Ω^{±}}, ρ) solves the Neumann problem (1.6), (1.8) if and only if

u(x) = c_{0} + Re \{∫_{x_{0}}^{x} \frac{[ζ′(y)]^{\frac{1}{p}}}{2πρ_{0}(y)} ∫_{|τ|=1} \frac{ψ^{−}(τ)dτ}{τ − ζ(y)} dy\} ,  (1.59)

u(x) = c_{0} + Re \{∫_{x_{0}}^{x} \frac{[ζ′(y)]^{\frac{1}{p}}}{2πρ_{0}(y)} [ ∫_{|τ|=1} \frac{ψ^{+}(τ)dτ}{τ − ζ(y)} − \frac{i}{2} ∫_{−π}^{π} ψ^{+}(e^{iϑ})dϑ] dy\}

for x ∈ Ω^{−} and x ∈ Ω^{+}, respectively; x_{0} ∈ Ω^{±} is some fixed point, c_{0} ∈ ℝ is a real constant and ζ(x) is the conformal mapping from (1.46). ψ^{±} = Re ψ^{±} ∈ L_{p}(Γ_{1}) are real-valued solutions of the following singular integral equations on the unit circumference

B^{−}ψ^{−}(ζ) := \{ \begin{matrix}P_{Γ_{1}}^{+}ψ^{−}(ζ) + F (ζ)P_{Γ_{1}}^{−}ψ^{−}(ζ) = f_{0}(ζ) , ζ ∈ Γ_{1} , \\ Kψ^{−} = \frac{1}{2π} ∫_{−π}^{π} ψ^{−}(e^{iϑ})dϑ = 0 , \end{matrix} (1.60)

B^{+}ψ^{+}(ζ) := P_{Γ_{1}}^{+}ψ^{+}(ζ) + F (ζ)P_{Γ_{1}}^{−}ψ^{+}(ζ) + \frac{F (ζ) − 1}{2}Kψ^{+} = f_{0}(ζ) .

The coefficient F ∈ P C(Γ_{1}) (see § 5.2) and the right-hand side f_{0} ∈ L_{p}(Γ_{1}) are defined as follows:

\begin{matrix}F (ζ) := \frac{ρ_{0}(ω(ζ))}{\overline{ρ_{0}(ω(ζ))}} [\frac{ω′(ζ)}{\overline{ω′(ζ)}}]^{\frac{1}{p} −1} , \\ f_{0}(ζ) := −2iρ_{0}(ω(ζ))[ω′(ζ)]^{\frac{1}{p}} f(ω(ζ)) , ζ ∈ Γ_{1} . \end{matrix}  (1.61)

Proof. The Neumann problem (1.6), (1.8) can be written as follows (see [Mu1, § § 74,75])

Re [e^{iϑ_{t}}(Ψ′)^{±}(t)] = f(t) , t ∈ Γ_{1} ,

u(x) = Re Ψ(x) , Ψ ∈ W_{p}^{1}(\overline{Ω^{±}}, ρ_{0}) x ∈ Ω^{±} , (1.62)

where ϑ(ζ) = ϑ_{t} denotes the inclination of the outer unit normal vector ν⃗(t) to the abscissa axes at t = ω(ζ) ∈ Γ \ {t_{1}, …, t_{n}} (see Fig. 1). In fact, since

Ψ = u + iv ∈ W_{p}^{1}(\overline{Ω^{±}}, ρ_{0}) , Ψ′ := \frac{∂u}{∂x} − i\frac{∂u}{∂y} ∈ E_{p}(\overline{Ω^{±}}, ρ_{0}) , (1.63)

∂_{ν⃗(t)}u(t) = cos ϑ_{t} \frac{∂u}{∂x} + sin ϑ_{t} \frac{∂u}{∂y} , cos ϑ_{t} + i sin ϑ_{t} = e^{iϑ_{t}}

(see (1.21)), we get

Re [e^{iϑ_{t}}(Ψ′)^{±}(t)] = cos ϑ_{t} [\frac{∂u}{∂x}]^{±} + sin ϑ_{t} [\frac{∂u}{∂y}]^{±} = (∂_{ν⃗(t)}u)^{±}(t)

and (1.62) follows.

Similarly to (1.54) (see also [Mu1, § 39] and [KKP1, Ch. IV, § 2]) for the analytic function

Φ(z) := \{ \begin{matrix}ρ_{0}(ω(z))[ω′(z)]^{\frac{1}{p}} Ψ′(ω(z)) & for & |z| < 1, \\ \overline{ρ_{0} (ω (\frac{1}{\overline{z}}))} [\overline{ω′ (\frac{1}{\overline{z}})}]^{\frac{1}{p}} \overline{Ψ′ (ω (\frac{1}{\overline{z}}))} & for & |z| > 1, \end{matrix} (1.64)

which belongs to the space E_{p}(\overline{D_{1}^{+}}), we get the following BVP:

Φ^{+}(ζ) − F (ζ)Φ^{−}(ζ) = f_{0}(ζ) , ζ ∈ Γ_{1} ,  (1.65)

where f_{0}(ζ) is defined in (1.61) and

F (ζ) := −e^{−2ϑ(ζ)i} \frac{ρ_{0}(ω(ζ))}{\overline{ρ_{0}(ω(ζ))}} [\frac{ω′(ζ)}{\overline{ω′(ζ)}}]^{\frac{1}{p}} = e^{−2α(ζ)i} \frac{ρ_{0}(ω(ζ))}{\overline{ρ_{0}(ω(ζ))}} [\frac{ω′(ζ)}{\overline{ω′(ζ)}}]^{\frac{1}{p}}

with α(ζ) = ϑ(ζ) + \frac{π}{2} denoting the inclination of the tangent to Γ vector to the abscissa axes at t = ω(ζ) ∈ Γ \ {t_{1}, …, t_{n}} (see Fig. 1). Let us recall that ω′(z) has an angular (i.e., non-tangential) boundary limits →^{∧} for almost all ζ ∈ Γ_{1} and

ω′(ζ) = e^{α(ζ)i}|ω′(ζ)|  (1.66)

(see, e.g., [Go1, p.p. 405–411] and [Ks1, Ch. I, II]). Therefore

e^{−2α(ζ)i} = (\frac{ω′(ζ)}{\overline{ω′(ζ)}})^{−1}

and by inserting this into the foregoing formula we get F (ζ) as written in (1.61). In (1.64), (1.65) Φ ∈ E_{p}(\overline{D_{1}^{+}}) and, therefore, it can be represented by the Cauchy integral with a pure imaginary density for the problems in the domains Ω^{+} and Ω^{−}, respectively (cf. (1.56))

Φ(z) := −\frac{i}{2}Kψ^{+} + iC_{Γ_{1}}ψ^{+}(x) = −\frac{i}{4π} ∫_{−π}^{π} ψ^{+}(e^{iϑ})dϑ + \frac{1}{2π} ∫_{|τ|=1} \frac{ψ^{+}(τ)dτ}{τ − z} ,

Φ(z) := iC_{Γ_{1}}ψ^{−}(x) = −\frac{1}{2π} ∫_{|τ|=1} \frac{ψ^{−}(τ)dτ}{τ − z} , ψ^{±} = Re ψ^{±} ∈ L_{p}(Γ_{1}) (1.67)

for all |z| ≠ 1 because for the domain Ω^{−} we should require in addition (see the condition in (1.60))

Kψ^{−} = \frac{1}{2π} ∫_{−π}^{π} ψ^{−}(e^{iϑ})dϑ = 0 .

To justify the latter we remind that Ψ ∈ W_{p}^{1}(\overline{Ω^{±}}, ρ_{0}) and, due to representation (1.3) the derivative should vanish at the infinity Ψ′(∞) = 0; therefore (see (1.64), (1.67))

∫_{−π}^{π} ψ^{−}(e^{iϑ})dϑ = 2πΦ(0) = 2πρ_{0}(ω(0))[ω′(0)]^{\frac{1}{p}} Ψ′(ω(0)) = 0

because ω(0) = ∞ (see (1.46)–(1.47)).

Let us note that Φ(z) in (1.67) has the symmetry property Φ_{∗}(z) = Φ(z)(cf. (1.58)).

Since we need only real-valued solutions ψ^{±} = Re ψ^{±} of (1.60), we check, based on the properties similar to (1.57) that along with ψ^{±} equations (1.60) have solutions \overline{ψ^{±}}. Therefore the real-valued solutions ψ^{±} = Re ψ^{±} = \frac{1}{2}(ψ^{±} + \overline{ψ^{±}}) are those we look for.

Vice versa, if ψ^{±} = Re ψ^{±} are real-valued solutions of (1.59), (1.60), we find easily that Φ(z), defined in (1.65) solves BVP (1.67), which implies that u(x) in (1.59) solves BVP (1.6), (1.8).

Remark 1.18 Similar resulta about equivalent reduction of the Dirichlet (1.6), (1.7) and the Neumann (1.6), (1.8) BVPs to BIEs (1.50) and (1.60) can be carried out in the spaces of continuous C(\overline{Ω^{∓}}, ρ) and piecewisecontinuous P C(\overline{Ω^{∓}}, ρ) functions. Transition to the unit disk is clear and smooth, but is senseless because the Cauchy SIO is unbounded in these spaces, even on the unit circumference.

Solvability results we possess e.g. for the Hölder spaces with weight h_{μ}^{0}(\overline{D_{1}}, ρ) on the unit disk (see § 4), but transformation of the Riemann– Hilbert problem for Ω^{+} to the unit disk (similar to (1.53)–(1.59)) is not implemented so far.

1.6 Reduction to boundary pseudo-differential equations (the direct method)

Theorem 1.19 Let 𝕏^{s}(Γ, ρ) stand for one of the following spaces: W_{p}^{s}(Γ, ρ) with 0 ≤ s ≤ 1 or for H_{μ+1}^{0}(Γ, ρ), P C^{1}(Γ, ρ). x^{s}(\overline{Ω^{±}}, ρ) is used for the corresponding Smirnov space of harmonic functions. ρ(t) is defined in (1.2) and inequalities (1.4) hold.

A harmonic function u ∈ x^{s}(\overline{Ω^{±}}, ρ) solves the Dirichlet problem (1.6), (1.7) if and only if

u(x) = χ_{−}(x)[W_{Γ}g(0) − V_{Γ}φ_{−}(0)] ± W_{Γ}g(x) ∓ V_{Γ}φ_{±}(x) ,  (1.68)

where φ ∈ 𝕏^{s−1}(Γ, ρ) is a solution of the following pseudo-differential equation of order −1 (written separately for the domains Ω^{+} and Ω^{−}, respectively)

V_{Γ,−1}φ_{+}(t) := \frac{1}{2π} ∫_{Γ} log |t − τ|φ_{+}(τ)|dτ| = g_{+}(t) , t ∈ Γ, (1.69)

\{ \begin{matrix}V_{Γ,−1}φ_{−}(t) := \frac{1}{2π} ∫_{Γ} log |\frac{t − τ}{τ} | φ_{−}(τ)|dτ| = g_{−}(t) , t ∈ Γ , \\ ∫_{Γ} φ_{−}(τ)|dτ| = 0 \end{matrix} (1.70)

and

g_{+}(t) := −\frac{1}{2}g(t) − \frac{1}{2π} ∫_{Γ} ∂_{ν⃗(τ)} log |t − τ|g(τ)|dτ| , t ∈ Γ ,

g_{−}(t) := \frac{1}{2}g(t) − \frac{1}{2π} ∫_{Γ} ∂_{ν⃗(τ)} log |\frac{t − τ}{τ} | g(τ)|dτ| , t ∈ Γ .

Proof. Solution u(x) of the the Dirichlet problem (1.6), (1.7) has the form (1.68) (see (1.15) and (1.30)). Taking the trace on Γ from Ω^{±}, invoking the Plemelji formulae (1.25), inserting u^{±}(t) = g(t) from (1.7) and choosing the function φ_{±}(t) := (∂_{ν⃗(t)}u)^{±}(t) for an unknown, we get equations (1.69) and the first equation in (1.70), because

W_{Γ}φ(x) − W_{Γ}φ(0) = \frac{1}{2π} ∫_{Γ} ∂_{ν⃗(τ)} log |\frac{x − τ}{τ} | φ(τ)|dτ| ,

V_{Γ}φ(x) − V_{Γ}φ(0) = \frac{1}{2π} ∫_{Γ} log |\frac{x − τ}{τ} | φ(τ)|dτ| , x ∈ Ω^{±} .

The second equation in system (1.70) is necessary for boundedness of solution (1.68) at infinity (see (1.27), (1.28)).

Vice versa, if u(x) is written in the form (1.68), it is obviously harmonic and u ∈ x^{s}(\overline{Ω^{±}}, ρ). In fact, φ_{±} = Re φ_{±} ∈ 𝕏^{s−1}(Γ, ρ) and, due to (1.41),

u(x) = χ_{−}(x)u(∞)±W_{Γ}g(x)∓V_{Γ}φ_{±}(x)=χ_{−}(x)u(∞)+ Re C_{Γ}[±g∓iφ_{±}](x).

Further, u(x) has finite limit u(∞) = W_{Γ}g(0) − V_{Γ}φ_{−}(0) at infinity and it remains to check the boundary condition (1.7). To this end it suffices to take the trace in (1.68), applying the Plemelji formulae (1.25), and remember that equations (1.69) and (1.70) hold. We easily get:

u^{±}(x) = χ_{−}(x)[W_{Γ}g(0) − V_{Γ}φ_{−}(x)] + \frac{1}{2}g(t)

±W_{Γ,0}g(t) ∓ V_{Γ}φ_{±}(t) = g(t).

Theorem 1.20 Let 𝕏^{s}(Γ, ρ) stand for one of the following spaces: W_{p}^{s}(Γ, ρ) with 0 ≤ s ≤ 1 or for H_{μ+1}^{0}(Γ, ρ), P C^{1}(Γ, ρ). x^{s}(\overline{Ω^{±}}, ρ) is used for the corresponding Smirnov space of harmonic functions. ρ(t) is defined in (1.2) and inequalities (1.4) hold.

A harmonic function u ∈ x^{s}(\overline{Ω^{±}}, ρ) solves the Neumann problem (1.6), (1.8) if and only if

u(x) = C_{0} ± W_{Γ}ψ_{±}(x) ∓ V_{Γ}f(x) ,  (1.71)

where C_{0} is arbitrary constant, ψ ∈ 𝕏^{s}(Γ, ρ) is a solution of the following pseudo-differential equation of order +1

D_{Γ,+1}ψ_{±}(t) := \frac{1}{2π} ∫_{Γ} ∂_{ν⃗(t)}∂_{ν⃗(τ)} log |t − τ|ψ_{±}(τ)|dτ| = f_{±}(t), t ∈ Γ, (1.72)

and

f_{±}(t) := ±\frac{1}{2}f(t) + \frac{1}{2π} ∫_{Γ} ∂_{ν⃗(t)} log |t − τ|f(τ)|dτ| , t ∈ Γ .

For the outer domain problem Ω^{−} the data f(t) should meet the additional constraint

∫_{Γ} f(τ)|dτ| = 0 .  (1.73)

Proof. Solution u(x) of the the Neumann problem (1.6), (1.8) has the form (1.71) (see (1.15)) and to be bounded in the outer domain condition (1.73) should hold (see (1.27), (1.28)). Taking the trace on Γ from Ω^{±}, invoking the Plemelji formulae (1.25), inserting (∂_{ν⃗(t)}u)^{±}(t) = f(t) from (1.8) and anounceing ψ_{±}(t) := u^{±}(t) as an unknown function, we get equations (1.72).

Vice versa, if u(x) is written in the form (1.71), it is obviously harmonic and u ∈ x^{s}(\overline{Ω^{±}}, ρ). In fact, ψ_{±} = Re ψ_{±} ∈ 𝕏^{s}(Γ, ρ) and, due to (1.41),

u(x) = C_{0} ± W_{Γ}ψ_{±}(x) ∓ V_{Γ}f(x) = C_{0} + Re C_{Γ}[±ψ_{±}(x) ∓ if](x) .

Further, u(x) has finite limit u(∞) = C_{0} at infinity (see (1.73) and recall (1.27), (1.28)). It remains to check the boundary condition (1.8). To this end it suffices to take the trace in (1.71), applying the Plemelji formulae (1.25), and remember that equations (1.72) hold. We easily get:

(∂_{ν⃗(t)}u)^{±}(x) = ±D_{Γ,+1}ψ_{±}(t) + \frac{1}{2}g(t) ∓ V_{Γ}f(t) = f(t) .

1.7 Statement of the principal results

In the present subsection we formulate principal results on BIEs (1.39), (1.40), (1.44), (1.45), (1.50), (1.60) (see Theorems 1.26 and 1.29), which we prove later in § 5.3–§ 5.4. We also formulate (and prove) their immediate consequences-solvability results for corresponding BVPs (see Theorems 1.28 and 1.30). Theorems are formulated separately for the case of absence of cusps because in such a case equations can be studied directly and not only the weighted Lebesgue space L_{p}(Γ, ρ), but in the weighted spaces of continuous, piecewise-continuous and Hölder functions. Moreover, equations are Fredholm in usual spaces, in contrast to the case of cusps, when we have to introduce special image spaces to make operators Fredholm. The approaches to the cases are substantially different (cf. § 5.3 and § 5.4).

Before formulating theorems on solvability of boundary integral equations and boundary value problems let us recall [DNS1, Lemma 19] which will be quoted later and which is useful in establishing additional smoothness properties of solutions to BVPs (e.g., Hölder continuity with weight).

A pair of Banach spaces {𝕏_{0}, 𝕏_{1}} embedded in some topological space 𝔼 is called an interpolation pair. For such a pair we can introduce the following two spaces 𝕏_{min} = 𝕏_{0} ∩ 𝕏_{1} and 𝕏_{max} = 𝕏_{0} + 𝕏_{1} := {x ∈ 𝔼 : x = x_{0} + x_{1}, x_{j} ∈ 𝕏_{j}, j = 0, 1}; 𝕏_{min} and 𝕏_{max} become Banach spaces if they are endowed with the norms

∥x|𝕏_{min}∥ = max{∥x|𝕏_{0}∥, ∥x|𝕏_{1}∥},

∥x|𝕏_{max}∥ = inf{∥x_{0}|𝕏_{0}∥ + ∥x_{1}|𝕏_{1}∥ : x = x_{0} + x_{1}, x_{j} ∈ 𝕏_{j}, j = 0, 1},

respectively.

Besides, we have the continuous embedding

𝕏_{min} ⊂ 𝕏_{0}, 𝕏_{1} ⊂ 𝕏_{max}.

For any interpolation pairs {𝕏_{0}, 𝕏_{1}} and {𝕐_{0}, 𝕐_{1}} the space L({𝕏_{0}𝕏_{1}}, {𝕐_{0}𝕐_{1}}) consists of all linear operators from 𝕏_{max} into 𝕐_{max} whose restrictions to 𝕏_{j} belong to L(𝕏_{j}, 𝕐_{j}) (j = 0, 1). The notation L(𝕏, 𝕐) is used for the space of all linear bounded operators A : 𝕏 → 𝕐.

Lemma 1.21 (see [DNS1, Lemma 19]). Assume {𝕏_{0}, 𝕏_{1}} and {𝕐_{0}, 𝕐_{1}} are interpolation pairs and the embedding 𝕏_{min} ⊂ 𝕏_{max}, 𝕐_{min} ⊂ 𝕐_{max} to be dense. Let an operator A ∈ L(𝕏_{0}, 𝕐_{0}) ∩ L(𝕏_{1}, 𝕐_{1}) have a common regularizer: R ∈ L(𝕐_{0}, 𝕏_{0}) ∩ L(𝕐_{1}, 𝕏_{1}) and RA − I ∈ L(𝕏_{0}𝕏_{0}) ∩ L(𝕏_{1}, 𝕏_{1}) be compact. Then

A : 𝕏_{min} → 𝕐_{min}, A : 𝕏_{max} → 𝕐_{max}

are Fredholm operators and

Ind_{𝕏_{min}→𝕐_{min}}A = Ind_{𝕏_{max}→𝕐_{max}}A = Ind_{𝕏_{j}→𝕏_{j}}A, j = 0, 1.

If y ∈ 𝕐_{j}, then any solution x ∈ 𝕏_{max} of the equation Ax = y belongs to 𝕏_{j}. In particular,

Ker_{𝕏_{min}}A = Ker_{𝕏_{j}}A = Ker_{𝕏_{max}}A , j = 0, 1.

Let

\begin{matrix}T := \{t_{1}, … t_{n}\} , T_{pk} := T_{ow} ∪ T_{iw} , \\ T_{ow} := \{t_{j} ∈ T : γ_{j} = 0\} , T_{iw} := \{t_{j} ∈ T : γ_{j} = 2\} \end{matrix}  (1.74)

be the collections of all knots, of all peaks, of all outward and all inward peaks on Γ.

Let us define the following Cesaro-type mean value integral on the contour Γ (cf. (1.90))

V_{t_{j}}φ(t) := (c) ∫_{t_{j}t^{⌣}} (\frac{ζ(t)}{ζ(τ)})^{\frac{1}{p}} (\frac{ζ(τ) − ζ(t_{j})}{ζ(t) − ζ(t_{j})})^{\frac{1}{p}} \frac{ρ(τ)}{ρ(t)} \frac{φ(τ)ζ′(τ)dτ}{ζ(τ) − ζ(t_{j})}  (1.75)

= lim_{η→t_{j}} ∫_{η t^{⌣}} \begin{matrix}log & \frac{ζ(τ)−ζ(t_{j})}{ζ(η)−ζ(t_{j})} \\ log & \frac{ζ(t)−ζ(t_{j})}{ζ(η)−ζ(t_{j})} \end{matrix} (\frac{ζ(t)}{ζ(τ)})^{\frac{1}{p}} (\frac{ζ(τ) − ζ(t_{j})}{ζ(t) − ζ(t_{j})})^{\frac{1}{p}} \frac{ρ(τ)}{ρ(t)} \frac{φ(τ)ζ′(τ)dτ}{ζ(τ) − ζ(t_{j})} ,

where ζ is the conformal mapping from (1.47). Obviously,

V_{t_{j}}φ(t) := ∫_{t_{j}t^{⌣}} (\frac{ζ(t)}{ζ(τ)})^{\frac{1}{p}} (\frac{ζ(τ) − ζ(t_{j})}{ζ(t) − ζ(t_{j})})^{\frac{1}{p}} \frac{ρ(τ)}{ρ(t)} \frac{φ(τ)ζ′(τ)dτ}{ζ(τ) − ζ(t_{j})}

if the latter (usual) integral exists.

Let t_{j} ∈ T_{pk} be a peak and t ∈ Γ. The points

t = ω(ζ) and t_{t_{j}}^{∗} := ω(\overline{ζ(t)}ζ_{j}^{2}) , ω(ζ_{j}) = t_{j}

are the images of equidistant points |ζ(t) − ζ_{j}| = |\overline{ζ(t)}ζ_{j}^{2} − ζ_{j}| on the unit circumference under the conformal mapping (1.46). Points t ∈ Γ and t_{t_{j}}^{∗} ∈ Γ

are on different sides from the outward peak t_{j} ∈ ) ; t_{t_{j}}^{∗}t . Let Γ_{t_{j}} ⊂ Γ be, similarly to Γ_{1ζ_{j}} ⊂ Γ_{1}, a sufficiently small fixed neighbourhood of t_{j} ∈ Γ such that Γ_{t_{k}} ∩ Γ_{t_{j}} = ∅ (and, therefore, t_{k} ∉ Γ_{t_{j}}) for k ≠ j. Let Γ_{t_{j}} = Γ_{t_{j}}^{−} ∪ Γ_{t_{j}}^{+} be the decomposition of the neighbourhood of t_{j} into the semiclosed left and right neighbourhoods and χ_{t_{j}} be the characteristic function of Γ_{t_{j}}. We define the space

L_{p}(Γ, ρ, T_{pk}) := {φ ∈ L_{p}(Γ, ρ) : Ṽ_{t_{j}}φ ∈ L_{p}(Γ, ρ) , t_{j} ∈ T_{pk}} , (1.76)

Ṽ_{t_{j}}φ := V_{t_{j}}φ_{t_{j}}^{∗} , φ_{t_{j}}^{∗}(t) := ε_{j}φ(t) − φ(ω(\overline{ζ(t)}ζ_{j}^{2})), , ε_{j} := e^{\frac{π}{p} (γ_{j}−1)i} ,

∥φ|L_{p}(Γ, ρ, T_{pk})∥ :=∥φ|L_{p}(Γ, ρ)∥+ ∑_{t_{j}∈T_{pk}}∥Ṽ_{t_{j}}χ_{t_{j}}φ|L_{p}(Γ_{t_{j}}^{+}, (t − t_{j})^{α_{j}})∥

and ^{5)} ε_{j} = e^{− \frac{π}{p} i} for t_{j} ∈ T_{ow}, ε_{j} = e^{\frac{π}{p} i} for t_{j} ∈ T_{iw}. Similarly is defined the space L_{p}(Γ, ρ, T_{ow}) ⊂ L_{p}(Γ, ρ, T_{pk}).

5)Non-equal rights of left and right neighbourhoods and differences for outward and inward peaks in the definition of the space L_{p}(Γ, ρ, T_{pk}) are explained in Remark 5.12.

Lemma 1.22 Let Γ be a piecewise-Ljapunov curve. If ψ ∈ L_{p}(Γ, ρ) and log[ζ(t) − ζ(t_{j})]ψ_{t_{j}}^{∗} ∈ L_{p}(Γ, ρ) for all t_{j} ∈ T_{pk}, then ψ ∈ L_{p}(Γ, ρ, T_{pk}).

Let a ∈ L_{∞}(Γ) and

a(t) = a(t_{j}) + O (| log[ζ(t) − ζ(t_{j})]|^{−1})  (1.77)

for all t_{j} ∈ T_{pk} as t → t_{j}. Then the operators

\begin{aligned}aI & : & L_{p}(Γ, ρ, T_{pk}) ⟶ L_{p}(Γ, ρ, T_{pk}) , \\ [a − a_{0}(t)]I & : & L_{p}(Γ, ρ) ⟶ L_{p}(Γ, ρ, T_{pk}) \end{aligned}  (1.78)

are bounded, where a_{0}(t) := ∑_{t_{j}∈T_{pk}} a(t_{j})χ_{j}(t) and χ_{j}(t) denotes the charac-

teristic function of Γ_{t_{j}}.

Proof. The proof is an easy consequence of Lemmata 1.25 and 1.27.

Note, that if Γ has no cusps, 0 < γ_{j} < 2 for all j = 11, …, n, than

log[ζ(t) − ζ(t_{j})] ∼ log[t − t_{j}] , t ∈ Γ

(see Corollary 5.10). For a curve with cusps this is not valid any more.

Theorem 1.23 Let T_{pk} = ∅ and 𝕏^{m}(Γ, ρ) be one of the following spaces W_{p}^{m}(Γ, ρ), H_{μ+m}^{0}(Γ, ρ), C^{m}(Γ, ρ) or P C(Γ, ρ), m = 0, 1.

Equations (1.39) and (1.40) are Fredholm in the space 𝕏^{m}(Γ, ρ) if and only if

β_{j} ≠ \{ \begin{matrix}γ_{j}^{0} & if m = 0 , \\ 1 − γ_{j}^{0} & if m = 1 , \end{matrix} γ_{j}^{0} := min \{\frac{1}{γ_{j}} , \frac{1}{2 − γ_{j}} \}

for all j = 1, …, n, where

β_{j} := \{ \begin{matrix}\frac{1}{p} + α_{j} & for & 𝕏^{m}(Γ, ρ) = W_{p}^{m}(Γ, ρ) , \\ α_{j} & for & P C^{m}(Γ, ρ) , C(Γ, ρ), \\ α_{j} − μ_{j} & for & H_{μ+m}^{0}(Γ, ρ) . \end{matrix}  (1.79)

If T_{pk} ≠ ∅ or β_{j} = γ_{j}^{0} when m = 0, β_{j} = 1 − γ_{j}^{0} when m = 1, then the operators A_{±} in (1.39), (1.40) have non-closed images in W_{p}^{m}(Γ, ρ).

Equations (1.39) and (1.40) with φ ∈ L_{p}(Γ, ρ), g ∈ L_{p}(Γ, ρ, T_{pk}) are Fredholm, i.e., the operators

A_{±} : L_{p}(Γ, ρ) → L_{p}(Γ, ρ, T_{pk})  (1.80)

are bounded and are Fredholm if and only if β_{j} ≠ γ_{j}^{0} for all t_{j} ∉ T_{pk}; the following formulae hold for the index, kernel and cokernel in the space 𝕏^{m}(Γ, ρ) when T_{pk} = ∅ or in the pairs (1.80) when T_{pk} ≠ ∅

Ind_{𝕏^{0}(Γ,ρ)}A_{±} = ∑_{\begin{matrix}t_{j}∉T_{pk} \\ β_{j}>γ_{j}^{0} \end{matrix}} 1 ,  Ind_{𝕏^{1}(Γ,ρ)}A_{±} = − ∑_{\begin{matrix}t_{j}∈T \\ β_{j}<1−γ_{j}^{0} \end{matrix}} 1 , (1.81)

dim_{𝕏^{0}(Γ,ρ)} Ker A_{±} = ε_{±} + Ind_{𝕏^{0}(Γ,ρ)}A_{±} , dim Coker_{𝕏^{0}(Γ,ρ)}A_{±} = ε_{±} ,

dim Ker_{𝕏^{1}(Γ,ρ)}A_{±} = ε_{±} , dim Coker_{𝕏^{1}(Γ,ρ)}A_{±} = ε_{±} − Ind_{𝕏^{1}(Γρ)}A_{±}

with ε_{+} = 0 and ε_{−} = 1.

In particular, if

0 < β_{j} < γ_{j}^{0} for all t_{j} ∉ T_{pk} ,  (1.82)

then equations (1.39) and (1.40) have solutions for all right-hand sides g(t) in L_{p}(Γ, ρ, T_{pk}) (in C(Γ, ρ) and in^{6)} H_{μ}^{0}(Γ, ρ) when T_{pk} = ∅), while for

T_{pk} = ∅ , 1 − γ_{j}^{0} < β_{j} < 1 for all t_{j} ∈ T  (1.83)

they have solutions in W_{p}^{1}(Γ, ρ), in P C^{1}(Γ, ρ) and in H_{μ+1}^{0}(Γ, ρ) for the right hand sides in the same spaces. Equation (1.39) has a unique solution in these spaces, while homogeneous equation (1.40), g(t) ≡ 0 has a single linearly independent solution φ_{−}(t) ≡ 1.

Proof. The proof is postponed to § 5.4.

Theorem 1.24 Let T_{pk} = ∅ and 𝕏^{m}(Γ, ρ) be either W_{p}^{m}(Γ, ρ) (m = 0, −1) or H_{μ}^{0}(Γ, ρ).

Equations (1.44) and (1.45) are Fredholm in 𝕏(Γ, ρ) if and only if

β_{j} ≠ \{ \begin{matrix}1 − γ_{j}^{0} & for L_{p}(Γ, ρ), H_{μ}^{0}(Γ, ρ) , \\ γ_{j}^{0} & for W_{p}^{−1}(Γ, ρ) , \end{matrix} γ_{j}^{0} := min \{\frac{1}{γ_{j}} , \frac{1}{2 − γ_{j}} \}

for all j = 1, …, n, where β_{j} is defined in (1.79).

If either T_{pk} ≠ ∅ or β_{j} = γ_{j}^{0} when m = 0, β_{j} = 1 − γ_{j}^{0} when m = 1 then the operators B_{±} in (1.44), (1.45) have non-closed images in W_{p}^{m}(Γ, ρ).

Equations (1.44) and (1.45) with ψ ∈ L_{p}(Γ, ρ), f ∈ L_{p}(Γ, ρ, T_{pk}) are Fredholm, i.e., the operators

B_{±} : L_{p}(Γ, ρ) → L_{p}(Γ, ρ, T_{pk})  (1.84)

are bounded and are Fredholm if and only if β_{j} ≠ 1 − γ_{j}^{0} for all t_{j} ∉ T_{pk}; the following formulae hold for the index, kernel and cokernel in the space 𝕏^{m}(Γ, ρ) when T_{pk} = ∅ or in the pairs (1.84) when T_{pk} ≠ ∅

6)Absence of additional solvability condition for equation (1.40) under constraints (1.82) and (1.83), which are inevitable since dim Coker A_{−} = 1 (see Remark 1.10), is due to the special right-hand side g(t) − g_{0}, which already satisfies the orthogonality condition (1.42).

Ind_{𝕏^{0}(Γ,ρ)}B_{±} = ∑_{\begin{matrix}t_{j}∉T_{pk} \\ β_{j}>1−γ_{j}^{0} \end{matrix}} 1 ,  (1.85)

Ind_{𝕏^{1}(Γ,ρ)}B_{±} = − ∑_{\begin{matrix}t_{j}∈T \\ β_{j}<γ_{j}^{0} \end{matrix}} 1 ,  (1.86)

dim Ker_{𝕏^{0}(Γ,ρ)}B_{±} =ε_{±} + Ind_{𝕏^{0}(Γ,ρ)}B_{±} , dim Coker_{𝕏^{0}(Γ,ρ)}B_{±} =ε_{±} ,

dim Ker_{𝕏^{−1}(Γ,ρ)}B_{±} = ε_{±} , dim Coker_{𝕏^{−1}(Γ,ρ)}B_{±} = ε_{±} − Ind_{𝕏^{−1}(Γ,ρ)}B_{±} ,

where ε_{+} = 0 and ε_{−} = 1.

In particular, if

0 < β_{j} < 1 − γ_{j}^{0} for all t_{j} ∉ T_{pk} ,  (1.87)

then equation (1.45) has solution for all right-hand sides f(t) in L_{p}(Γ, ρ, T_{pk}) (in H_{μ}^{0}(Γ, ρ) when T_{pk} = ∅), if and only if (1.73) holds. If

T_{pk} = ∅ , γ_{j}^{0} < β_{j} < 1 for all t_{j} ∈ T  (1.88)

then, again, equation (1.45) has a solution for all right-hand sides f(t) in W_{p}^{−1}(Γ, ρ), while equation (1.44) has a solution if and only if condition (1.73) holds.

Equation (1.45) has a unique solution in these spaces, while homogeneous equation (1.44), f(t) ≡ 0 has a single linearly independent solution ψ_{−} = 𝔯_{0} (see Remark 1.10).

Proof. For the cases T_{pk} = ∅ and W_{p}^{m}(Γ, ρ) (m = 0, −1) the proof follows from the foregoing Theorem 1.23 because equations (1.39), (1.40) in W_{p}^{m}(Γ, ρ) (m = 0, 1) and equations (1.45), (1.44) in W_{p′}^{−m}(Γ, ρ^{−}1) are pairwise conjugate.

As for equations (1.44) and (1.45) in the Hölder spaces H_{μ}^{0}(Γ, ρ) and the case T_{pk} ≠ ∅ (see (1.84)), the assertion is proved word to word as Theorem 1.23 (see § 5.4).

Let

Ξ := {ζ_{1}, …, ζ_{n}} ⊂ Γ_{1} , Ξ_{pk} := Ξ_{ow} ∪ Ξ_{iw} ,  (1.89)
Ξ_{ow} := {ζ_{j} = {ω^{−1}(t_{j}) : t_{j} ∈ T_{ow}} ,
Ξ_{iw} := {ζ_{j} = {ω^{−1}(t_{j}) : t_{j} ∈ T_{iw}}

be the images on the unit circumference of the discrete sets T , T_{pk}, T_{ow} T_{iw} (see (1.74)) under the inverse conformal mapping ω^{−1}(ζ) in (1.46)–(1.47).

We define the following Cesaro-type mean value integral

V_{ζ_{j}}φ(ζ) := (c) ∫_{ζ_{j}ζ^{⌣}} (\frac{τ − ζ_{j}}{ζ − ζ_{j}} )^{\frac{1}{p}} φ(τ)\frac{dτ}{τ − ζ_{j}}

:= lim_{η→ζ_{j}} [log \frac{ζ − ζ_{j}}{η − ζ_{j}} ]^{−1} ∫_{η ζ^{⌣}} \frac{dλ}{λ − ζ_{j}} ∫_{λζ^{⌣}} (\frac{τ − ζ_{j}}{ζ − ζ_{j}} )^{\frac{1}{p}} φ(τ)\frac{dτ}{τ − ζ_{j}}

= lim_{η→ζ_{j}} ∫_{η ζ^{⌣}} \begin{matrix}log & \frac{τ−ζ_{j}}{η−ζ_{j}} \\ log & \frac{ζ−ζ_{j}}{η−ζ_{j}} \end{matrix} (\frac{τ − ζ_{j}}{ζ − ζ_{j}} )^{\frac{1}{p}} φ(τ)\frac{dτ}{τ − ζ_{j}} .  (1.90)

Obviously,

V_{ζ_{j}}φ(ζ) := ∫_{ζ_{j}ζ^{⌣}} (\frac{τ − ζ_{j}}{ζ − ζ_{j}} )^{\frac{1}{p}} φ(τ)\frac{dτ}{τ − ζ_{j}}  (1.91)

if the latter (usual) integral exist (cf. (3.4), (3.5)).

Let us fix a neighbourhood Γ_{1ζ_{j}} ⊂ Γ_{1} of ζ_{j} ∈ Γ_{1} such that Γ_{1ζ_{k}} ∩Γ_{1ζ_{j}} = ∅ (which implies ζ_{k} ∉ Γ_{1ζ_{j}}) for k ≠ j and decompose Γ_{1ζ_{j}} into the left and the right neighbourhoods Γ_{1ζ_{j}} = Γ_{1ζ_{j}}^{−} ∪ Γ_{1ζ_{j}}^{+}. χ_{ζ_{j}} be the characteristic function of Γ_{1ζ_{j}}. We define the space (see (1.89))

L_{p}(Γ_{1}, Ξ_{ow}) := {φ ∈ L_{p}(Γ_{1}) : Ṽ_{ζ_{j}}φ ∈ L_{p}(Γ_{1ζ_{j}}^{+}) , ζ_{j} ∈ Ξ_{ow}} , (1.92)
Ṽ_{ζ_{j}}φ := V_{ζ_{j}}φ_{ζ_{j}}^{∗} , φ_{ζ_{j}}^{∗}(ζ) := e^{− \frac{π}{p} i}φ(ζ) − φ(\overline{ζ}ζ_{j}^{2}) ,

∥φ|L_{p}(Γ_{1}, Ξ_{ow})∥ := ∥φ|L_{p}(Γ_{1})∥ + ∑_{ζ_{j}∈Ξ_{ow}} ∥Ṽ_{ζ_{j}}χ_{ζ_{j}}φ|L_{p}(Γ_{1ζ_{j}}^{+})∥ .

Below, in Lemma 1.22, there is given a sufficient condition for the inclusion φ ∈ L_{p}(Γ_{1}, Ξ_{ow}) and for the boundedness of a multiplication operator aI).

Let us note, that if ζ ∈ Γ_{1ζ_{j}}^{±}, then the point \overline{ζ}ζ_{j}^{2} belongs to the different half-neighbourhood \overline{ζ}ζ_{j}^{2} ∈ Γ_{1ζ_{j}}^{∓} (i.e., points are on different sides of ζ_{j} ∈ Γ_{1ζ_{j}}), but are equidistant from ζ_{j}: |ζ − ζ_{j}| = |\overline{ζ}ζ_{j}^{2} − ζ_{j}|.

Lemma 1.25 If ψ ∈ L_{p}(Γ_{1}) and log(ζ − ζ_{j})ψ_{ζ_{j}}^{∗} ∈ L_{p}(Γ_{1}) for all ζ_{j} ∈ Ξ_{pk}, then ψ ∈ L_{p}(Γ_{1}, Ξ_{pk}).

Let a ∈ L_{∞}(Γ_{1}) and

a(ζ) = a(ζ_{j}) + O (| log(ζ − ζ_{j})|^{−1}) for all ζ_{j} ∈ Ξ_{pk} as ζ → ζ_{j} .

Then the operators

\begin{aligned}aI & : & L_{p}(Γ_{1}, Ξ_{pk}) ⟶ L_{p}(Γ_{1}, Ξ_{pk}) , \\ [a − a_{0}(ζ)]I & : & L_{p}(Γ_{1}) ⟶ L_{p}(Γ_{1}, Ξ_{pk}) \end{aligned}

are bounded, where a_{0}(ζ) := ∑_{ζ_{j}∈Ξ_{pk}} a(ζ_{j})χ_{j}(ζ) and χ_{j}(ζ) denotes the char-

acteristic function of Γ_{ζ_{j}}.

The proof will be given later at the end of § 3.3.

Theorem 1.26 The operator

A : L_{p}(Γ_{1}) ⟶ L_{p}(Γ_{1}, Ξ_{ow}) ,  (1.93)

where A is defined in (1.50), is bounded and is Fredholm (i.e., equation (1.50) is Fredholm if g_{0} ∈ L_{p}(Γ_{1}, Ξ_{ow}) and we look for a solution φ ∈ L_{p}(Γ_{1})) if and only if

ν_{j} := (\frac{1}{p} + α_{j}) γ_{j} ≠ 1 for all ζ_{j} ∉ Ξ_{ow} .  (1.94)

If conditions (1.94) hold,

Ind A = ∑_{\begin{matrix}ζ_{j}∉Ξ_{ow} \\ ν_{j}>1 \end{matrix}} 1 ,

dim Ker A = Ind A , dim Coker A = 0 ,

(1.95)

Proof. The proof is postponed to § 5.3.

Let (cf. (1.64))

Z_{ω}φ(ζ) := ρ(ω(ζ))[ω′(ζ)]^{\frac{1}{p}} φ(ω(ζ)) .  (1.96)

Lemma 1.27 Z_{ω} defines an isomorphism of spaces

Z_{ω} : L_{p}(Γ, ρ) ⟶ L_{p}(Γ_{1}) ,
: L_{p}(Γ, ρ, T_{ow}) ⟶ L_{p}(Γ_{1}, Ξ_{ow}),
: L_{p}(Γ, ρ, T_{pk}) ⟶ L_{p}(Γ_{1}, Ξ_{pk})

(1.97)

and the inverse operator reads

Z_{ω}^{−1}ψ(t) := ρ^{−1}(t)[(ω^{−1})′(t)]^{\frac{1}{p}} ψ(ω^{−1}(t)) .  (1.98)

The Cesaro-type operators V_{t_{j}} in (1.75), Ṽ_{t_{j}} in (1.76) and V_{ζ_{j}} in (1.90), Ṽ_{ζ_{j}} in (1.92) are related as follows

Z_{ω}χ_{t_{j}}V_{t_{j}}χ_{t_{j}}Z_{ω}^{−1} = V_{ζ_{j}} ,  Z_{ω}χ_{t_{j}} Ṽ_{t_{j}}χ_{t_{j}}Z_{ω}^{−1} = Ṽ_{ζ_{j}} .  (1.99)

Proof. The proof is direct and follows from the definitions.

Theorem 1.28 The Dirichlet problem (1.6), (1.7) with

u ∈ e_{p}(\overline{Ω^{+}}, ρ) and g ∈ L_{p}(Γ, ρ, T_{ow})  (1.100)

is Fredholm if and only if the conditions

ν_{j} := (\frac{1}{p} + α_{j}) γ_{j} ≠ 1 for all ζ_{j} ∉ T_{iw} .  (1.101)

hold. If this is the case, the problem has solution for each right hand-side in (1.100) and the homogeneous problem has exactly

κ := ∑_{ν_{j}>1} 1  (1.102)

solutions (i.e., the index of the corresponding operator is κ). In particular, if conditions

ν_{j} = ν_{j}^{0} := (\frac{1}{p} + α_{j}) γ_{j} < 1 for all ζ_{j} ∉ T_{iw}  (1.103)

hold, the problem has a unique solution.

Moreover, if T_{ow} = ∅ the Dirichlet problem (1.6), (1.7) with

u ∈ w_{p}^{1}(\overline{Ω^{+}}, ρ)  and g ∈ W_{p}^{1}(Γ, ρ) , ν_{j}^{1} := (\frac{1}{p} + α_{j} − 1) γ_{j} ,
u ∈ pc^{m}(\overline{Ω^{+}}, ρ)  and g ∈ P C^{m}(Γ, ρ) ,
ν_{j}^{m} := (α_{j} − m)γ_{j} , m = 0, 1 ,
u ∈ c(\overline{Ω^{+}}, ρ)  and g ∈ C(Γ, ρ) , ν_{j}^{0} := α_{j}γ_{j} ,
u ∈ h_{μ+m}^{0}(\overline{Ω^{+}}, ρ) and g ∈ H_{μ+m}^{0}(Γ, ρ) ,
ν_{j}^{m} := (α_{j} − μ_{j} − m)γ_{j} , m = 0, 1,

(1.104)

is Fredholm if and only if the condition

ν_{j}^{m} ≠ (−1)^{m}  (1.105)

holds for all j = 1, …, n. If this is the case, the problem has the following index

Ind A := ∑_{|ν_{j}^{m}|>1}(−1)^{m}.  (1.106)

and either the kernel (when Ind A ≤ 0) or the cokernel (when Ind A ≥ 0) is trivial. For Ind A = 0 both kernel and cokernel are trivial and the problem has a unique solution for all right-hand sides (see (1.103)).

The same holds for the domain Ω^{−} with the obvious replacements: T_{ow} by T_{iw} and γ_{j} by 1 − γ_{j}.

Proof. The first part of the theorem (1.100)–(1.103) follows from Theorems 1.16 and 1.26.

The second half of the theorem, when T_{ow} = ∅, follows from equivalence of the Dirichlet problem and of the corresponding singular integral equation (1.39)–(1.40) in appropriate space, which can be proved as in Theorem 1.12, and from appropriate assertions on singular integral equations in § 4.

Theorem 1.29 The operator

B^{+} : L_{p}(Γ_{1}) ⟶ L_{p}(Γ_{1}, Ξ_{ow})  (1.107)

(see (1.60)) is bounded and is Fredholm (i.e., equation (1.60) is Fredholm if f_{0} ∈ L_{p}(Γ_{1}, Ξ_{ow}) and we look for a solution φ ∈ L_{p}(Γ_{1})) if and only if the conditions

μ_{j} := (1 − \frac{1}{p} − α_{j}) γ_{j} ≠ 1 for all t_{j} ∉ T_{ow} .  (1.108)

hold. If conditions (1.107) hold,

Ind B^{+} = −1 + ∑_{\begin{matrix}ζ_{j}∉Ξ_{ow} \\ μ_{j}>1 \end{matrix}} 1 ,

dim Ker B^{+} = Ind B^{+} , dim Coker B = 1^{+} .

(1.109)

Proof. The proof follows word in word the proof of Theorem 1.26 (see § 5.3) with obvious modifications (including substitution of \frac{1}{p} by \frac{1}{p} − 1, as seen from (1.51) and (1.61)). The only difference which we have found worth explaining is the appearance of “−1” and “1” in the index formulae (1.106): the second condition in (1.60) obviously increases dim Coker B^{+} by 1 and diminishes Ind B^{+} also by 1.

Theorem 1.30 The Neumann problem (1.6), (1.8) for Ω^{+} has solutions u(x) + c_{0}, where c_{0} =const is arbitrary and, u ∈ w_{p}^{1}(\overline{Ω^{+}}, ρ) for f ∈ L_{p}(Γ, ρ, T_{ow}) if and only if conditions (1.108) hold and the solution is unique modulo a constant if μ_{j} < 1 for t_{j} ∉ T_{ow}. The index of the problem is given by the formula

Ind B^{+} := 1 + ∑_{μ_{j}>1} 1.  (1.110)

If T_{ow} = ∅ the Neumann problem (1.6), (1.8) Ω^{+} with

u ∈ pc^{1}(\overline{Ω^{+}}, ρ)  and g ∈ P C^{m}(Γ, ρ) , μ_{j} := (1 − α_{j})γ_{j} ,
u ∈ h_{μ+1}^{0}(\overline{Ω^{+}}, ρ)  and g ∈ H_{μ+1}^{0}(Γ, ρ) ,
μ_{j} := (1 − α_{j} + μ_{j})γ_{j} , m = 0, 1

(1.111) is Fredholm if and only if the condition μ_{j} ≠ 1 holds for all j = 1, …, n. If this is the case, the problem has the same index (1.110).

The same holds for the domain Ω^{−} with the obvious replacements: T_{ow} by T_{iw} and γ_{j} by 1 − γ_{j}.

Proof. The first part of the theorem (1.110) follows from Theorems 1.17 and 1.29.

The second half of the theorem, when T_{ow} = ∅, follows from equivalence of the Neumann problem to the corresponding singular integral equation (1.43)–(1.44) in appropriate space, which can be proved as in Theorem 1.14, and from appropriate assertions on singular integral equations in § 4.

Remark 1.31 Fredholm and solvability properties of pseudodifferential equations (1.69), (1.70), (1.72) can easily be derived from Theorems 1.28 and 1.30 (see Theorems 1.19 and 1.20). To save the space we leave this to readers.

2 Convolutions with elliptic symbols

2.1 Boundedness properties

C_{0}^{∞}(ℝ) denotes the Frechet space of all infinitely differentiable functions on ℝ := (−∞, ∞) with compact supports supp φ and D′(ℝ) – the dual space of distributions.

The convolution operator W_{a}^{0} with a symbol a ∈ L_{∞}(ℝ) is defined as follows

W_{a}^{0}φ := F^{−1}aFφ , φ ∈ C_{0}^{∞}(ℝ),  (2.1)

where

Fφ(ξ) = ∫_{ℝ} e^{iξx}φ(x)dx  and

F^{−1}ψ(x) = (2π)^{−n} ∫_{ℝ} e^{−ixξ}ψ(ξ)dξ, x, ξ ∈ ℝ^{n},  (2.2)

are the Fourier transforms.

𝕄_{p}(ℝ) denotes, as usual (see [Du1, Hr1], the class of Fourier L_{p}- multipliers, i.e., the class of all those symbols a(λ) ∈ L_{∞}(ℝ) for which the operator W_{a}^{0} admits a bounded extension

W_{a}^{0} : L_{p}(ℝ) → L_{p}(ℝ)  (2.3)

for all 1 < p < ∞ (see [BS1, Du1, RS1]).

In particular, if symbol a(λ) has: one of the following properties: i. bounded total variation a ∈ V_{1}(ℝ) (B.Stechkin theorem),

ii. if

a ∈ C^{1}(ℝ \ {0}) , |a(t)| ≤ M_{0} < ∞ , |ta′(t)| ≤ M_{0} < ∞ (2.4)

(J.Marcinkievicz theorem),

iii. belongs to the Wiener algebra

a ∈ W (ℝ) := {a(λ) = c + Fk(λ) : k ∈ L_{1}(ℝ)} ,

then a ∈ 𝕄_{p}(ℝ). Moreover, in the case (iii.) W_{a}^{0} is written as an integral convolution

W_{a}^{0}φ(x) = cφ(x) + ∫_{−∞}^{∞} k(x − y)φ(y)dy ,

while in general case convolution has distributional kernel (see [Du1, Hr1, St1] for details).

Let ℝ̇ and ℝ̈ denote one point and two point compactifications of the real axes

ℝ̇ := ℝ ∪ {∞} , or ℝ̈ := ℝ ∪ {±∞}

respectively and P C(ℝ̇) denote the space of all piecewise-continuous functions on ℝ̇, i.e., the space of all functions a(λ) on ℝ which have finite limits a(λ ± 0) for all λ ∈ ℝ̇. The space P C(ℝ̇) coincides with the closure of all piecewise-constant functions on ℝ̇ with respect to the uniform norm (in L_{∞}(ℝ̇); see [Du1]). Let P C_{p}(ℝ̇) be the same closure of all piecewiseconstant functions with respect to the multiplier norm ∥a|𝕄_{p}(ℝ)∥ := ∥W_{a}^{0}| L(L_{p}(ℝ))∥. Then

P C_{2}(ℝ̇) = P C(ℝ̇) , V_{1}(ℝ), W (ℝ) ⊂ ⋂_{1<p<∞} P C_{p}(ℝ̇) .

For a matrix symbol a ∈ P C_{p}^{N×N}(ℝ̇) invertibility criteria of the operator W_{a}^{0} in L_{p}(ℝ) space reads

inf_{λ∈ℝ} | det a(λ)| > 0 ,  (2.5)

which yields a^{−1} ∈ P C_{p}^{N×N}(ℝ̇) and the inverse operator is W_{a^{−1}}^{0} (see [Du1, Hr1] for these and other properties of multipliers).

Moreover, we can take 1 ≤ p ≤ ∞ and involve new spaces. Namely W_{a}^{0} has bounded extensions in the following spaces of smooth functions: C_{0}^{m}(ℝ̇), C^{m}(ℝ̇), C_{0}^{m}(ℝ̈) for all m = 0, 1, … (see [Kr1]). These spaces are defined as follows.

Let 𝕏 be either one point or two point compactifications of the real axes and C^{m}(𝕏) denote the Banach space of continuous functions on the compact Hausdorff set 𝕏, which have continuous derivatives up to the order m and is endowed with the appropriate uniform norm

∥φ∥ = ∑_{k=0}^{m} sup_{t∈ℝ} |(\frac{d^{k}}{dt^{k}} φ) (t)|

(C^{m}(𝕏) is even a Banach algebra with pointwise multiplication). Note, that a function φ ∈ C^{m}(ℝ̇) and its derivatives have equal limits at infinity (d^{k}/dt^{k})φ(∞) = (d^{k}/dt^{k})φ(±∞) while function φ ∈ C^{m}(ℝ̈) might have two different limits (d^{k}/dt^{k})ψ(±∞) for all k = 0, 1, …, m.

C_{0}^{m}(ℝ̇) denotes the subspace (the sub-algebra) of C^{m}(ℝ̇) of those functions φ(x) which vanish at infinity with all derivatives up to the order m:

C_{0}^{m}(ℝ̇) := \{φ ∈ C^{m}(ℝ̇) : φ(∞) = ⋯ = (\frac{d^{m}}{dt^{m}} φ) (∞) = 0\} .

Let

W_{a}φ := r_{+}W_{a}^{0}ℓ_{0}φ ,  φ ∈ C_{0}^{∞}(ℝ^{+}) ,  (2.6)

where r_{+} denotes the restriction to ℝ^{+} from ℝ, while ℓ_{0}–the right inverse to r_{+} which extends functions by 0 from ℝ^{+} to ℝ. Let L_{p}(ℝ^{+}, ρ), ρ(x) ≥ 0, denote the weighted Lebesgue space endowed with the standard norm ∥φ|L_{p}(ℝ^{+}, ρ)∥ := ∥ρφ|L_{p}(ℝ^{+})∥.

Lemma 2.1 (see [Du1, Sc1]). Let a ∈ V_{1}(ℝ) and

1 < p < ∞ , −\frac{1}{p} < α, γ < 1 − \frac{1}{p} .  (2.7)

Then

W_{a} : L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α}) ⟶ L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α})  (2.8)

is continuous.

Let β ∈ ℝ and

L_{1}^{⟨β⟩}(ℝ) := \{ \begin{matrix}L_{1}(ℝ^{+}) ∩ L_{1}(ℝ^{−}, (1 − x)^{−β}) & for β < 0 , \\ L_{1}(ℝ^{−}) ∩ L_{1}(ℝ^{+}, (1 + x)^{β}) & for β > 0 , \end{matrix}

where ℝ^{−} := (−∞, 0]. Let further

W_{β}(ℝ) := {a(λ) = c + Fk(λ) : c = const , k ∈L_{1}(ℝ, (1 + |x|)^{|β|})} , (2.9)
W_{1}^{⟨β⟩}(ℝ) := {a(λ) = c + Fk(λ) : c = const , k ∈ L_{1}^{⟨β⟩}(ℝ)} (2.10)

and endow them with the appropriate norms

∥a|W_{β}(ℝ)∥ := |c| + ∥k|L_{1}(ℝ, (1 + |x|)^{|β|})∥ for ± β > 0 ,
∥a|W^{⟨β⟩}(ℝ)∥ := |c| + ∥k|L_{1}^{⟨β⟩}(ℝ)∥ = |c| + ∥k|L_{1}(ℝ^{∓})∥
+∥k|L_{1}(ℝ^{±}, (1 + |x|)^{|β|})∥ for ± β > 0

provided a(λ) = c + Fk(λ). Obviously, W_{β}(ℝ) ⊂ W^{⟨β⟩}(ℝ).

Let C(ℝ̇^{+}) denote the restriction of the space C(ℝ̇) to the semi-axes ℝ^{+} and C(ℝ̇^{+}, (1 + x)^{β}) denote the weighted space of functions φ(x) on the semi-axes ℝ^{+} for which (1 + x)^{β}φ(x) belong to C(ℝ̇^{+}).The space is endowed with the appropriate weighted norm ∥φ|C(ℝ̇^{+}, (1 + x)^{β})∥ := ∥(1 + x)^{β}φ(x)|C(ℝ̇^{+})∥.

Lemma 2.2 Let a ∈ W^{⟨β⟩}(ℝ) and β ∈ ℝ. Then the operator

W_{a} : C(ℝ̇^{+}, (1 + x)^{β}) ⟶ C(ℝ̇^{+}, (1 + x)^{β})  (2.11)

is continuous^{7)} and

lim_{x→∞}(1 + x)^{β}W_{a}φ(x) = a(0) lim_{x→∞}(1 + x)^{β}φ(x) ,  (2.12)
∥W_{a}|C(ℝ̇^{+}, (1 + x)^{β}∥ ≤ ∥a|W^{⟨β⟩}(ℝ)∥ .  (2.13)

Proof. For a(λ) = c we have W_{a} = cI and the assertion is trivial. Thus, we can take a = Fk, k ∈ L_{1}^{⟨β⟩}(ℝ).

The integral

W_{a}^{0} = ∫_{−∞}^{∞} k(x − y)φ(y)dy = ∫_{−∞}^{∞} k(y)φ(x − y)dy

is continuous function for a continuous φ ∈ C(ℝ̇^{+}, (1 + x)^{β}) and we should check only (2.12)–(2.13).

Obviously,

\begin{matrix}∥W_{a}|C(ℝ̇^{+}, (1 + x)^{β})∥ ≤ K_{β} , \\ K_{β} = sup_{x∈ℝ^{+}} ∫_{0}^{∞}(\frac{1 + x}{1 + y})^{β} |k(x − y)|dy . \end{matrix}  (2.14)

If β < 0, applying the inequality

1 + x ≤ (1 + |x − y|)(1 + y) , x, y ∈ ℝ^{+} ,  (2.15)

7)See similar assertions in [GF1, Pr1, PS1].

we proceed as follows

K_{β} ≤ sup_{x>0} [∫_{0}^{x} (\frac{1 + x}{1 + y})^{β} |k(x − y)|dy + ∫_{x}^{∞}(\frac{1 + y}{1 + x})^{−β} |k(x − y)|dy]

≤ sup_{x>0} [∫_{0}^{∞}|k(x − y)|dy + ∫_{x}^{∞}(1 + |x − y|)^{−β}|k(x − y)|dy]

≤ ∫_{0}^{∞}|k(t)|dt + ∫_{−∞}^{0} (1 + |t|)^{−β}|k(t)|dt = ∥a|W^{⟨β⟩}(ℝ)∥ .

Now let β > 0. Similarly to the foregoing case we find (see (2.14) and (2.15))

K_{β} ≤ sup_{x>0} [∫_{0}^{x} (\frac{(1 + x}{1 + y})^{β} |k(x − y)|dy + ∫_{x}^{∞}(\frac{(1 + x}{1 + y})^{β} |k(x − y)|dy]

≤ sup_{x>0} [∫_{0}^{x}(1 + |x − y|)^{β}|k(x − y)|dy + ∫_{x}^{∞}|k(x − y)|dy]

≤ sup_{x>0} [∫_{0}^{x}(1 + t)^{β}|k(t)|dt + ∫_{−∞}^{0} |k(t)|dt] = ∥a|W_{β}(ℝ)∥ .

To prove (2.12) (for arbitrary β ∈ ℝ) we represent

φ_{β}(x) := (1 + x)^{β}φ(x) = φ_{β}(∞) + φ_{β}^{0}(x) , φ_{β}^{0}(∞) = 0

and suppose that both φ_{β}^{0}(x) and k(t) have compact supports

supp φ_{β}^{0} ⊂ [0, c_{1}] , supp k ⊂ [−c_{2}, c_{2}] .

Since such functions are dense in appropriate spaces, this does not restricts generality. Then

lim_{x→∞}(1 + x)^{β}W_{a}φ(x) = lim_{x→∞} ∫_{0}^{∞}(\frac{1 + x}{1 + y})^{β} k(x − y)φ_{β}(y)dy

= lim_{x→∞} ∫_{0}^{c_{1}}(\frac{1 + x}{1 + y})^{β} k(x − y)φ_{β}^{0}(y)dy

+φ_{β}(∞) lim_{x→∞} ∫_{0}^{∞}(\frac{1 + x}{1 + y})^{β} k(x − y)dy = a(0)φ_{β}(∞) ,

since

k(x − y)φ_{β}^{0}(y) = 0 if x ≥ c_{1} + c_{2},

lim_{x→∞} ∫_{0}^{∞}(\frac{1 + x}{1 + y})^{β} k(x − y)dy = lim_{x→∞} ∫_{−c_{2}}^{c_{2}} (\frac{1 + x}{1 + x − t})^{β} k(t)dt

= ∫_{−c_{1}}^{c_{2}} k(t)dt = ∫_{−∞}^{∞} k(t)dt = a(0) .

This accomplishes the proof.

2.2 Fredholm properties

Lemma 2.3 Let β ∈ ℝ. Then W_{β}(ℝ) ⊂ W_{0}(ℝ) = W (ℝ) ⊂ C(ℝ̇) is an inverse closed Banach algebra in C(ℝ̇), which reads: the element a ∈ W_{β}(ℝ) is invertible if and only if it is invertible in C(ℝ̇), i.e., iff inf_{λ∈ℝ} |a(λ)| > 0,

and then a^{−1} ∈ W_{β}(ℝ).

Proof. The proof see in [GRS1, § 18].

Let for a matrix-function a = [a_{jk}]_{N×N} with entries a_{jk} ∈ 𝔄 use the same notation a ∈ 𝔄.

Lemma 2.4 Let β ∈ ℝ and a matrix-function a ∈ W_{β}(ℝ) be elliptic

inf_{λ∈ℝ} | det a(λ)| > 0 .  (2.16)

Then a(λ) has the following factorization

a(λ) = a_{−}(λ) diag \{(\frac{λ − i}{λ + i})^{κ_{1}} , …, (\frac{λ − i}{λ + i})^{κ_{N}} \} a_{+}(λ)  (2.17)

where the matrix-functions a_{−}^{±} ∈ W_{β}(ℝ) and a_{+}^{±} ∈ W_{β}(ℝ) have uniformly bounded analytic extensions a_{−}^{±}(λ − iσ) and a_{+}^{±}(λ + iσ) in the lower and upper σ > 0 complex half-planes, respectively. The integers κ_{1}, …, κ_{N} are known as the partial indices of the factorization (2.17).

Proof. For the algebra W (ℝ) = W_{0}(ℝ) the proof is well-known (see, e.g., [GF1]) and we follow the same scheme: if all rational functions are dense in W_{β}(ℝ) (a rationally dense algebra) and the Hilbert transform

S_{ℝ}φ(x) = \frac{1}{πi} ∫_{−∞}^{∞} \frac{φ(y)dy}{y − x} = −W _{sign λ}^{0} φ(x) , x ∈ ℝ  (2.18)

(see [Du1, Lemma 1.35]) is bounded there (a decomposable algebra), then according to the general theorem proved in [BG1] (see also [CG1, GF1]) all invertible elements of W_{β}(ℝ) would possess factorization (2.17). Invertibility of a ∈ W_{β}(ℝ) under condition (2.16) is provided by Lemma 2.3.

Rational density of W_{β}(ℝ) follows since the Laguerre polynomials are dense in L_{1}(ℝ, (1 + |x|)^{|β|}) (see, e.g., [GF1, § 8]).

W_{β}(ℝ) is a decomposable because FS_{ℝ}F^{−1}ψ(λ) = − sign λ ψ(λ) (see (2.18)) is a bounded operator in L_{1}^{⟨β⟩}(ℝ) and W_{β}(ℝ) = const +FL_{1}(ℝ, (1+ |x|)^{|β|})) (see (2.8)).

Let us consider a = c + Fk ∈ W_{β}(ℝ) and the corresponding equation

W_{a}^{0}φ(x) = c φ(x) + ∫_{−∞}^{∞} k(x − y)φ(y)dy = f(x) , x ∈ ℝ^{+}  (2.19)

(cf. (2.5)).

Theorem 2.5 Equation (2.19) in the space C(ℝ̇^{+}, (1 + x)^{β}), β ∈ ℝ is Fredholm if and only if the symbol a(λ) is elliptic (see (2.16)). If this is the case, then

Ind W_{a} = − ind a .

If, in addition, (2.19) is a scalar equation N = 1, then:

i. equation (2.19) is uniquely solvable for all f ∈ C(ℝ̇^{+}, (1+x)^{β}) provided ind a = 0;

ii. if κ = ind a < 0 equation (2.19) has a solution φ ∈ C(ℝ̇^{+}, (1 + x)^{β}) for all f ∈ C(ℝ̇^{+}, (1 + x)^{β}) and the homogeneous equation f = 0 has exactly −κ linearly independent solutions;

iii. if κ = ind a > 0 equation (2.19) has a solution φ ∈ C(ℝ̇^{+}, (1 + x)^{β}) only for those right-hand sides f ∈ C(ℝ̇^{+}, (1 + x)^{β}) for which

∫_{0}^{∞}f(y)\overline{g_{j}(y)}dy = 0 , j = 1, …, κ ,

where g_{1}, …, g_{κ} are all solutions to the dual homogeneous equation

c g(x) + ∫_{−∞}^{∞} k(y − x)g(y)dy = 0  (2.20)

in the dual space C(ℝ̇^{+}, (1 + x)^{−β}).

If the solution exists it is unique.

Proof. The proof is standard and based on Lemmata 2.3, 2.4 (see [Du1, GF1, GK1, Kr1] for similar proofs, except the last claim).

Concerning the last claim–we replaced the adjoint space C^{∗}(ℝ̇^{+}, (1+x)^{β}) by the dual one C(ℝ̇^{+}, (1 + x)^{−β}); this is possible since the equation (2.20) has the same solutions in these two spaces (see [Du5] for a similar assertion).

The last claim follows also from Lemma 1.21. which states that equation has the same solutions in two spaces 𝔅_{1} ⊂ 𝔅_{2} provided the embedding is dense and the equation has a common regularizer in 𝔅_{1} and in𝔅_{2}.

Now let a ∈ V_{1}(ℝ); then W_{a} can be written as integral convolution (2.19) only conventionally–the kernel k(t) is a distribution. If a(λ) possesses a single jump, operator W_{a} is not bounded in C(ℝ̇^{+}, (1 + x)^{β}) because the Hilbert transform (2.18) is not bounded in these space.

Thus, we should consider equation (2.19) with a ∈ P C_{p}(ℝ)in the Lebesgue space L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α}) with weight under conditions (2.7). With equation (2.19) we associate the symbol

a_{ω}(λ, ξ) := \frac{1 + coth π[iβ(λ) + ξ]}{2}a(λ − 0)

+\frac{1 − coth π[iβ(λ) + ξ]}{2}a(λ + 0) , λ ∈ ℝ̇ ,  ξ ∈ ℝ , (2.21)

where (note, that a ∈ P C_{p}(ℝ) has limits a(λ ± 0), λ ∈ ℝ̇ including infinity a(∞ ± 0) := a(∓∞)). ω := (p, α, γ) reminds the space and

β(λ) := \{ \begin{matrix}\frac{1}{p}, & if & λ ≠ 0, ∞ , \\ \frac{1}{p} + α, & if & λ = 0, \\ \frac{1}{p} + γ, & if & λ = ∞ . \end{matrix}

Theorem 2.6 Let a ∈ P C_{p}(ℝ); the weight ρ(t) be defined by (1.2) and satisfy appropriate (namely the first) condition in (1.4).

Equation (2.32) is Fredholm in the space L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α}) if and only if the symbol a_{ω}(λ, ξ) is elliptic

inf_{λ∈ℝ̇, ξ∈ℝ} | det a_{ω}(λ, ξ)| > 0 .

If this is the case, then

Ind W_{a} = −\frac{1}{2πi} ∑_{j=1}^{∞} \{[arg a(λ)]_{λ∈[λ_{j},λ_{j+1}]} + [arg a_{ω}(λ_{j}, ξ)]_{ξ∈ℝ}\} , (2.22)

where {λ_{j}}_{j=1}^{∞} ⊂ ℝ̇ denotes the set of all points where a ∈ V_{1}(ℝ) has jumps a(λ_{j} − 0) ≠ a(λ_{j} + 0) and^{8)} [arg g(t)]_{t∈I} denotes the increment of any continuous branch of arg g(t) as t ranges through I in the positive direction.

If, in addition, (2.19) is a scalar equation N = 1, then:

i. equation (2.19) is unuquelly solvable for all f ∈ L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α}) provided Ind W_{a} = 0;

ii. if κ = Ind W_{a} > 0 (2.19) has a solution φ ∈ L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α}) for all f ∈ L_{p}(ℝ^{+}, x^{α}(1+x)^{γ−α}) and the homogeneous equation f = 0 has exactly κ linearly independent solutions;

iii. if κ = ind W_{a} < 0 (2.19) has a solution φ ∈ L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α}) only for those right-hand sides f ∈ L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α}) for which

∫_{0}^{∞}f(y)\overline{g_{j}(y)}dy = 0 , j = 1, …, −κ ,

where g_{1}, …, g_{−κ} are all solutions of the dual homogeneous equation

c g(x) + ∫_{−∞}^{∞} k(y − x)g(y)dy = 0  (2.23)

in the dual space L_{p′}(ℝ^{+}, x^{−α}(1 + x)^{−γ+α}) with p′ := \frac{p}{p − 1}.

If solution exists it is unique.

Proof. For the proof we quote [Du1] (the case α = γ = 0) and [Sc1] (the case α ≠ 0, β ≠ 0).

2.3 Some proofs

Proof of Lemma 1.1. Let, for definiteness, consider the domain Ω^{+}. Since Φ ∈ X (\overline{Ω^{+}}) we have

Φ(z) = c_{0} + \frac{1}{2πi} ∫_{Γ} \frac{Φ_{0}^{+}(τ)dτ}{τ − z} , z ∈ Ω^{+} ,

where Φ_{0}^{+} ∈ 𝕏(Γ) is the trace of Φ_{0}(z) := Φ(z) − c_{0} on Γ from Ω^{+}. On the other hand

Φ(z) = c_{0} + \frac{1}{2πi} ∫_{Γ} \frac{φ(τ)dτ}{τ − z} , z ∈ Ω^{+} ,

8)The set {λ_{j}}_{j=1}^{∞} is at most countable and the sum in (2.22) is convergent (see, e.g., [Du1]).

for some φ ∈ 𝕏(Γ) (see (1.3)) and we get

∫_{Γ} \frac{Φ_{0}^{+}(τ) − φ(τ)}{τ − z}dτ ≡ 0 , z ∈ Ω^{+} .

The obtained equality means

Φ_{0}^{+}(t) − φ(t) = Ψ^{−}(t) , t ∈ Γ ,  (2.24)

where Ψ ∈ X (\overline{Ω^{−}}); therefore,

Ψ(z) = −C_{Γ}φ(z) := −\frac{1}{2πi} ∫_{Γ} \frac{φ(τ)dτ}{τ − z} , z ∈ Ω^{−},

and Ψ(z) → 0 as |z| → ∞, since C_{Γ}Φ_{0}^{+}(z) ≡ 0 for z ∈ Ω^{−}. In fact, P_{Γ}^{+}Φ_{0}^{+} = Φ_{0}^{+} yields P_{Γ}^{−}Φ_{0}^{+} = 0 (we remind that P_{Γ}^{−} + P_{Γ}^{+} = I; see (1.50)). On the other hand, due to the Plemelji formula for C_{Γ}φ in (1.25) (C_{Γ}Φ_{0}^{+})^{−} = P_{Γ}^{−}Φ_{0}^{+} = 0 and the analytic function C_{Γ}Φ_{0}^{+}(z), z ∈ Ω^{−} vanishing on the boundary vanishes everywhere in Ω^{−}.

(2.24) can be written as follows

Re (−iΨ^{−})(t) = Im Ψ^{+}(t) = Im Φ_{0}^{+}(t) , t ∈ Γ

if φ(t) = Re φ(t) = Re Φ_{0}^{+}(t) − Re Ψ^{−}(t) is pure real and

Re Ψ^{−}(t) = Re Φ_{0}^{+}(t) , t ∈ Γ,

if φ(t) = i Im φ(t) = i Im Φ_{0}^{+}(t) − i Im Ψ^{−}(t) is pure imaginary. Since Φ_{0}^{+}(t) is known, solvability of the obtained Riemann–Hilbert problems is equivalent to the claimed representations.

Proof of Lemma 1.2. If (0.2) holds and Ω^{±} has no outward peak (T_{ow} = ∅), u^{±} ∈ W2^{\frac{1}{2}} (Γ) due to theorem on traces (see, e.g., [Tr1])). Although ∂ν⃗u ∈ W_{2,com}^{0}(Ω^{±}), we can not claim (∂ν⃗u)^{±} ∈ W_{2}^{− \frac{1}{2}} (Γ) because the trace does not exists. But u is harmonic Δu(z) = 0 in Ω^{±} and from the Green formula (1.13) we get

∮_{Γ} ∂_{ν⃗(τ)}u(τ)\overline{v(τ)}dτ = ± ∑_{j=1}^{2} ∫_{Ω^{±}} ∂_{j}u(y)\overline{∂_{j}v(y)}dy .  (2.25)

Taking arbitrary v ∈ W_{2,com}^{1}(\overline{Ω^{±}}), which implies v^{±} ∈ W2^{\frac{1}{2}} (Γ) due to theo-

rem on traces, by duality of spaces from (2.25) follows (∂ν⃗u)^{±} ∈ W_{2}^{− \frac{1}{2}} (Γ).

Since u(z) is a harmonic function, due to representation formula (1.15)

u(z) = Re u(z) = χ_{−}(z)u(∞) ± W_{Γ}u^{±}(z) ∓ V_{Γ}(∂ν⃗u)^{±}(x)

= χ_{−}(z)u(∞) ± W_{Γ}u^{±}(z) ∓ Re (∂_{z}V_{Γ})v^{±}(z)  (2.26)

= χ_{−}(z)u(∞) + Re (C_{Γ}u_{±})(z) , u_{±}(t) := ±u^{±}(t) + i Re v^{±}(t) ,

v^{±}(t) := ∫_{ct^{⌣}} (∂_{ν⃗(τ)}u)^{±}(τ)dτ , z ∈ Ω^{±} , t ∈ Γ .

From (2.26) we get the inclusion into the Smirnov class u ∈ w2^{\frac{1}{2}} (\overline{Ω^{±}}) with the complex valued density u_{±} ∈ W2^{\frac{1}{2}} (Γ) because u^{±}, v^{±} ∈ W2^{\frac{1}{2}} (Γ).

Vice versa, u ∈ w2^{\frac{1}{2}} (\overline{Ω^{±}}), also for Ω^{±} with peaks, implies the representation

u(z) = u(∞) + Re C_{Γ}φ(z) ,  z ∈ Ω^{±} ,  φ ∈ W2^{\frac{1}{2}} (Γ) .

Then u(z) is harmonic in Ω^{±} and u(z) = u(∞) + O((1 + |z|)^{−1}) as |z| → ∞ and, due to Theorem 1.8, u ∈ W_{2}^{1}(\overline{Ω^{±}}).

Proof of Theorem 1.5. The first and the second claims for s = m = 0 follows from representations (1.17), (1.18) and boundedness of the singular integral operator S_{Γ} (see (1.5)) in L_{p}(Γ, ρ) (see, e.g., [GK1, Kh1, Pr1]) and in H_{μ}^{0}(Γ, ρ) (see [Du6, Du7] and also [Du3, Du5]).

The operators

W_{Γ,0}^{(k)}φ(t) := \frac{1}{4} (S_{Γ} + Vh^{k}S_{Γ}h^{−k}V) φ(t)

= \frac{1}{4πi} ∫_{Γ} φ(τ) [\frac{dτ}{τ − t} − \frac{\overline{h^{k}(t)}}{\overline{h^{k}(τ)}} \frac{d\overline{τ}}{\overline{τ} − \overline{t}}]  (2.27)

are bounded in L_{p}(Γ, ρ) and in H_{μ}^{0}(Γ, ρ) by the same reason.

For a closed contour ∂_{t}S_{Γ}φ = S_{Γ}∂_{t}φ and we get

∂_{t}W_{Γ,0}φ := \frac{1}{4} (∂_{t}S_{Γ} + V \frac{dt}{d\overline{t}}∂_{t}S_{Γ}V) φ

= \frac{1}{4} (S_{Γ} + Vh^{2}S_{Γ}h^{−2}V) ∂_{t}φ = W_{Γ,0}^{(2)}∂_{t}φ  (2.28)

(cf. (1.17)–(1.21), (1.26)); therefore W_{Γ,0} is bounded in W_{p}^{1}(Γ, ρ) and in H_{1+μ}^{0}(Γ, ρ). By interpolation (see [Tr1]) we get boundedness of W_{Γ,0} in W_{p}^{s}(Γ, ρ) for 0 ≤ s ≤ 1.

Since the operator W_{Γ,0}^{∗} is adjoint to W_{Γ,0}, it is automatically bounded in adjoint space W_{p}^{s}(Γ, ρ) (see, e.g., [Tr1]) to W_{p′}^{−s}(Γ, ρ) for −1 ≤ s ≤ 0 and p′ := p/(p − 1).

Let us prove the last claim.

V_{Γ} has a weak singular kernel and, therefore,

∥V_{Γ}φ|L_{p}(Γ, ρ)∥ ≤ C_{1}∥φ|L_{p}(Γ, ρ)∥;

on the other hand, due to (1.19),

∥∂_{t}V_{Γ}φ|L_{p}(Γ, ρ)∥ = ∥(S_{Γ} + VS_{Γ}V)φ|L_{p}(Γ, ρ)∥ ≤ C_{2}∥φ|L_{p}(Γ, ρ)∥ .

and we get the final estimate

∥V_{Γ}φ|W_{p}^{1}(Γ, ρ)∥ = ∥V_{Γ}φ|L_{p}(Γ, ρ)∥ + ∥∂_{t}V_{Γ}φ|L_{p}(Γ, ρ)∥ .

Similarly for the Hölder spaces H_{μ}^{0}(Γ, ρ) → H_{1+μ}^{0}(Γ, ρ).

Proof of Theorem 1.6. It suffices to show that W_{Γ,0}^{(k)} are bounded in

P C(Γ, ρ) for even k = 0, 2, … and W_{Γ,0}^{(0)} = W_{Γ,0} is bounded in C(Γ, ρ). In fact, h^{±}I are bounded in P C(Γ, ρ) and boundedness of W_{Γ,0}^{∗} in P C(Γ, ρ) follows since

W_{Γ,0}^{∗} = −hW_{Γ,0}^{(2)}\overline{h}I

(cf. (1.18), (2.27)). By virtue of (1.22) we get

∥W_{Γ,0}φ|P C^{1}(Γ, ρ)∥ = ∥W_{Γ,0}φ|C(Γ, ρ)∥ + ∥∂_{t}W_{Γ,0}φ|P C(Γ, ρ)∥ ,
= ∥W_{Γ,0}φ|C(Γ, ρ)∥ + ∥W_{Γ,0}^{(2)}∂_{t}φ|P C(Γ, ρ)∥ .

which means boundedness of W_{Γ,0} in P C^{1}(Γ, ρ).

Integral operator K with a weak singular kernel

|k(t, τ)| ≤ C|t − τ|^{ν−1} , 0 < ν ≤ 1 , t, τ ∈ Γ,  (2.29)

is bounded (moreover, is compact) in spaces C(Γ, ρ) and in P C(Γ, ρ).

In fact, this is easy to ascertain for ρ(t) ≡ 1. For ρ(t) ≢ 1 we have to prove that K_{1} := ρKρ^{−1}I − K is compact in C(Γ) and in P C(Γ).

The kernel k_{1}(t, τ) of K_{1} has the following estimate

|k_{1}(t, τ)| = |ρ(t) − ρ(τ)|\frac{k(t, τ)}{ρ(τ)} ≤ C \frac{g_{ρ}(t, τ)}{ρ(τ)}|t − τ|^{ν−1} ;

here g_{ρ}(t, τ) = |t − τ|^{δ_{j}} when both t and τ are close to the knot t_{j}, j = 1, …, n and g_{ρ}(t, τ) = |t − τ| otherwise. Thus, k_{1}(t, τ) is weak singular and compactness (in C(Γ) and in P C(Γ)) follows.

Let Γ′ be another Ljapunov contour and ω : Γ → Γ′ be a diffeomorphism with analytic continuation in some neighbourhood of cuspidal wedge U_{j} ⊂ Ω^{+} (outward peak of Ω^{+}) of cusps c_{j} with γ_{j} = 0. Then the operator

K_{ω} := ω_{∗}^{−1}S_{Γ}ω_{∗} − S_{Γ} , ω_{∗}φ(t) = φ(ω(t)) , t ∈ Γ ,  (2.30)

where ω^{−1} : Γ → Γ′ is the inverse diffeomorphism, has a weak singular kernel (2.29) (see [DLS1, § 3.5] and [Kh1, GK1]).

Due to representations (1.17)–(1.19), (2.27) and to boundedness of operator K_{ω} in C(Γ, ρ) and in P C(Γ, ρ) (see (2.29) and further) the contour Γ can be replaced by another one Γ′ for which we can find a diffeomorphism ω : Γ → Γ′ with local analytic continuation in the vicinity of cusps.

[figure: πγ_{j}; t_{j}; Γ_{j}^{+}; Γ_{j}^{−}; Γ_{j}^{+}; Γ_{j}^{+}; t_{j}; t_{j}; Γ_{j}^{−}; Γ_{j}^{−}; 0 < γ_{j} < 2; γ_{j} = 0; γ_{j} = 2; Fig. 2]

Due to this we can suppose Γ_{j} has rectilinear parts Γ_{j}^{+} and Γ_{j+1}^{−} in some neighbourhoods of the endpoints t_{j} and t_{j+1} except cusps; for a cusp γ_{j} = 0, 2 the right neighbourhood Γ_{j}^{+} ⊂ Γ_{j} is rectilinear in the vicinity of t_{j}, while the left neighbourhood Γ_{j}^{−} ⊂ Γ_{j−1} is not (we remind, that {t_{j}} = Γ_{j−1} ⋃ Γ_{j}; see Fig. 2). Let

Γ_{j}^{0} = Γ_{j}^{−} ⋃ Γ_{j}^{+} , Γ^{0} = ⋃_{j=1}^{n} Γ_{j}^{0} , Γ_{0} = Γ \ Γ^{0} .  (2.31)

Let v_{0} ∈ C^{1}(Γ) be a cut-off function with supp v_{0} ⊂ Γ^{0} and v_{0}(t) = 1 in some neighbourhoods of all knots t_{1}, …, t_{n}. Then

W_{Γ,0}^{(k)} = (1 − v_{0})W_{Γ,0}^{(k)} + v_{0}W_{Γ_{0},0}^{(k)} + v_{0}W_{Γ^{0},0}^{(k)} .  (2.32)

Γ_{0} is free of knots t_{1}, …, t_{n} and operators (1 − v_{0})W_{Γ,0}^{(k)}, v_{0}W_{Γ_{0},0}^{(k)} have weak singular kernels. In fact, kernels of these operators read

k_{2}(t, τ) = [1 − v_{0}(t)]k_{0}(t, τ) ,
k_{3}(t, τ) = v_{0}(t)χ_{0}(t)k_{0}(t, τ) , t, τ ∈ Γ ,

where χ_{0}(t) is the characteristic function of Γ_{0} and

k_{0}(t, τ) = \frac{1}{πi} [\frac{1}{τ − t} − \frac{\overline{h^{k}(t)}}{\overline{h^{k}(τ)}} \frac{1}{\overline{τ} − \overline{t}} \frac{d\overline{τ}}{dτ} ]

= \frac{1}{πi} [\frac{1}{τ − t} − \frac{1}{\overline{τ} − \overline{t}} \frac{d\overline{τ}}{dτ} ] − \frac{\overline{h^{k}(t)} − \overline{h^{k}(τ)}}{\overline{h^{k}(τ)}(\overline{τ} − \overline{t})} \frac{d\overline{τ}}{dτ} .  (2.33)

k_{0}(t, τ) = k_{1}(t, τ) = 0 if t, τ ∉ Γ_{0}; therefore we can suppose t, τ ∈ Γ_{0} because otherwise k_{2}(t, τ) and k_{2}(t, τ) are bounded. Γ_{0} consists of n disjoint smooth arcs and k_{0}(t, τ) is the kernel of W_{Γ,0}^{(k)} = S_{Γ} −Vh^{k}S_{Γ}h^{−k}V; therefore we can apply a diffeomorphism ω : Γ_{0} → Γ_{ℝ} ⊂ ℝ which transforms Γ_{0} to the finite union of intervals on the real axes. Since ω_{∗}^{−1}W_{Γ_{0},0}^{(k)}ω_{∗} differs from

W_{Γ_{ℝ},0}^{(k)} by a compact operator with weak singular kernel, we can consider

W_{Γ_{ℝ},0}^{(k)}. But the first summand in representation (2.33) of the kernel of

operator W_{Γ_{ℝ},0}^{(k)} vanishes

\frac{1}{τ − t} − \frac{1}{\overline{τ} − \overline{t}} \frac{d\overline{τ}}{dτ} = 0 , t, τ ∈ Γ_{ℝ} ⊂ ℝ ,

while the second summand is weak singular, because the function h^{k}(ω^{−1}(t)) is C^{1+ν}-continuous.

Thus, we have to consider only operator v_{0}W_{Γ^{0},0}^{(k)} in (2.32). This can be

simplified further and we need to treat only operators W_{Γ_{j}^{0},0}^{(k)}, because the difference

T_{0} = v_{0} [W_{Γ^{0},0}^{(k)} − ∑_{j=1}^{n} W_{Γ_{j}^{0},0}^{(k)}]

is compact (has a bounded kernel).

Let 0 < γ_{j} < 2. Without loss of generality we can suppose that

Γ_{j}^{0} = Γ_{j}^{−} ⋃ Γ_{j}^{+} , Γ_{j}^{+} = (0, 1] , Γ_{j}^{−} = {e^{iγ_{j}}x : 0 ≤ x ≤ 1} .

Consider the transformation

Z_{γ_{j},δ_{j}}φ(x) := [ \begin{matrix}e^{−δ_{j}x}φ(e^{−x}) \\ e^{−δ_{j}xi}φ(e^{πγ_{j}−x}) \end{matrix} ] , x ∈ ℝ^{+},  (2.34)

and its inverse

Z_{γ_{j},δ_{j}}^{−1} [ \begin{matrix}ψ_{1} \\ ψ_{2} \end{matrix} ] (t) = χ_{+}^{0}(t)t^{−δ_{j}}ψ_{1}(− log t)

+χ_{−}^{0}(t)e^{πγ_{j}δ_{j}i}t^{−δ_{j}}ψ_{2}(πγ_{j} − log t) , t ∈ Γ_{j}^{0} ,

where χ_{+}^{0} and χ_{−}^{0} are the characteristic functions of Γ_{j}^{+} and Γ_{j}^{−}, respectively.

Z_{γ_{j},δ_{j}} arranges the isomorphism of the space P C(Γ_{j}^{0}, t^{δ_{j}}) = P C(Γ_{j}^{0}, ρ) with the vector-space [C(ℝ̇^{+})]^{2} = C(ℝ̇^{+}) × C(ℝ̇^{+}) (see § 1.1). The transformed operator acquires the form

Z_{γ_{j},δ_{j}}W_{Γ_{j}^{0},0}^{(k)}Z_{γ_{j},δ_{j}}^{−1} = [ \begin{matrix}0 & W_{a_{γ_{j} ,δ_{j} ,k}^{+}} \\ W_{a_{γ_{j} ,δ_{j} ,k}^{−}} & 0 \end{matrix} ] ,

where

W_{a_{γ_{j} ,δ_{j} ,k}^{±}}φ(x) = ∫_{0}^{∞}k_{γ_{j},δ_{j},k}^{±}(x − y)φ(y)dy ,
a_{γ_{j},δ_{j},k}^{±}(λ) := F_{t→λ} [k_{γ_{j},δ_{j},k}^{±}(t)] , λ, x ∈ ℝ ,
k_{γ_{j},δ_{j},k}^{±}(t) = −\frac{e^{±πγ_{j}k−δ_{j}t} sin πγ_{j} + e^{−(δ_{j}+1)t} sin πγ_{j}}{2π(1 − 2e^{−t} cos πγ_{j} + e^{−2t})}
:= \frac{e^{−δ_{j}t}}{4πi} [\frac{1}{1 − e^{−πγ_{j}i−t}} − \frac{e^{±πγ_{j}ki}}{1 − e^{πγ_{j}i−t}} ] .

Obviously,

k_{γ_{j},δ_{j},k}^{±} ∈ L_{1}(ℝ) iff 0 < δ_{j} < 1  (2.35)

and, due to Lemma 2.2, the transformed operator Z_{γ_{j},δ_{j}}W_{Γ_{j}^{0},0}^{(k)}Z_{γ_{j},δ_{j}}^{−1} is bounded in [C(ℝ̇^{+})]^{2} because 0 < γ_{j} < 2 , 0 < δ_{j} < 1.

Now let γ_{j} = 0, 2. We can suppose without loss of generality that t_{k} = 0 and

Γ_{j}^{+} = J = (0, 1] ⊂ ℝ^{+} ,
Γ_{j}^{−} = {z_{j}(x) = x + ig_{j}(x) : 0 ≤ x ≤ 1} ,
g_{j} ∈ C^{1+ν}(J) , g_{j}(0) = g′_{j}(0) = 0 , g_{j}(x) ≥ 0 ,
h(z_{j}(x)) = 1 + ig_{j}(x) , h(x) = 1 , x ∈ J (see (2.21)) .

The transformation

B_{j}φ(x) = [ \begin{matrix}φ(x) \\ φ(z_{j}(x)) \end{matrix} ] , z_{j}(x) = x + ig_{j}(x) , x ∈ J ,  (2.36)

arranges the isomorphism

B : P C(Γ_{j}^{0}, |t|^{δ_{j}}) ⟶ [C(J , |t|^{δ_{j}})]^{2}

and

B_{j}W_{Γ_{j},0}^{(k)}B_{j}^{−1} = [ \begin{matrix}0 & −Ṽ_{ig_{j}} \\ V_{ig_{j}} & 0 \end{matrix} ] + [ \begin{matrix}0 & T_{12} \\ T_{21} & T_{22} \end{matrix} ] ,

where T_{12} = Ñ_{−ig_{j}}[(1 − ig′_{j})^{−k} − 1] , T_{12} = [(1 − ig′_{j})^{k} − 1]N_{−ig_{j}} , T_{21} =−K_{z_{j}} −(1 − ig′_{j})^{k}K_{z_{j}}(1 − ig′_{j})^{−k}I + (1 − ig′_{j})^{k}S_{J} (1 − ig′_{j})^{−k}I −S_{J} ,

V_{ig_{j}}N_{ig_{j}} − N_{−ig_{j}} ,  Ṽ_{ig_{j}} = Ñ_{ig_{j}}(1 + ig′_{j})I − Ñ_{−ig_{j}}(1 − ig′_{j})I ,

N_{±ig_{j}} \frac{1}{πi} ∫_{0}^{1} \frac{φ(y)dy}{y − x ± ig_{j}(x)} ,  Ñ_{±ig_{j}} = \frac{1}{πi} ∫_{0}^{1} \frac{φ(y)dy}{y − x ± ig_{j}(y)}

and K_{z_{j}} is defined in (2.30) (z_{j}(x) see in (2.36)). Operators T_{12}, T_{21}, T_{22} all have weak singular kernels and there is left to prove boundedness of operators V_{ig_{j}} and Ṽ_{ig_{j}} only.

It is easy to ascertain that

v(x) := V_{ig_{j}}1(x) = −\frac{2g_{j}(x)}{π} ∫_{0}^{1} \frac{dy}{(y − x)^{2} + g_{j}^{2}(x)}

= \frac{1}{πi} log \frac{[x + ig_{j}(x)][1 − x + ig_{j}(x)]}{[x − ig_{j}(x)][1 − x − ig_{j}(x)]} ,

ṽ(x) := Ṽ_{ig_{j}}1(x) = \frac{1}{πi} \frac{1 − x + ig_{j}(x)}{1 − x − ig_{j}(x)} ,  x ∈ J ,

and v, ṽ ∈ C(J). Functions

V_{ig_{j}}φ(x) = V_{ig_{j}}[φ(y) − φ(x)] + φ(x)v(x) ,
Ṽ_{ig_{j}}φ(x) = Ṽ_{ig_{j}}[φ(y) − φ(x)] + φ(x)ṽ(x)

are continuous provided φ ∈ C^{1}(J). On the other hand we get

|V_{ig_{j}}φ(x)| = \frac{2g_{j}(x)}{π} |∫_{0}^{1} \frac{φ(y)dy}{(y − x)^{2} + g_{j}^{2}(x)}|

≤ \frac{∥φ|C(J)∥}{π} |∫_{0}^{1} [\frac{1}{y − x + ig_{j}(x)} − \frac{1}{y − x − ig_{j}(x)}] dy|

= \frac{∥φ|C(J)∥}{π}|v(x)| ,

|Ṽ_{ig_{j}}φ(x)| = \frac{2}{π} |∫_{0}^{1} \frac{(y − x)g′_{j}(y) − g_{j}(y)}{(y − x)^{2} + g_{j}^{2}(y)}φ(y)dy|

≤ \frac{2}{π} ∥φ|C(J)∥\{| ∫_{0}^{1} \frac{[(y − x)g′_{j}(y) − g_{j}(y)][g_{j}^{2}(x) − g_{j}^{2}(y)]}{[(y − x)^{2} + g_{j}^{2}(y)][(y − x)^{2} + g_{j}^{2}(x)]}dy

+g_{j}^{2}(x) ∫_{0}^{1} \frac{(y − x)g′_{j}(y) − g_{j}(y)}{(y − x)^{2} + g_{j}^{2}(x)}dy|\}

≤ ∥φ|C(J)∥[(1 + ∥g′_{j}|C(J)∥)^{2}∥g′_{j}|C(J)∥

+∥g_{j}|C(J)∥(∥g′_{j}|C(J)∥ + ∥g′_{j}|C(J)∥)] |\frac{1}{π} ∫_{0}^{1} \frac{2ig_{j}(x)dy}{(y − x)^{2} + g_{j}^{2}(x)}|

= C_{g_{j}}∥φ|C(J)∥|v(x)| ,

C_{g_{j}} = [(1 + ∥g′_{j}|C(J)∥)^{2}∥g′_{j}|C(J)∥

+∥g_{j}|C(J)∥(∥g′_{j}|C(J)∥ + ∥g′_{j}|C(J)∥)] .

Obtained inequalities prove that V_{ig_{j}} and Ṽ_{ig_{j}} can be extended as continuous operators from C^{1}(J) ⟶ C(J) to C(J) ⟶ C(J).

3 Equations with non-elliptic symbols

3.1 Convolutions on ℝ^{+}

Let α, γ and p be as in (2.7) and the symbol a ∈ P C_{p}(ℝ) be non-elliptic (vanishing at 0):

a(λ) = \frac{λ}{λ − i}a^{(−)}(λ) , a^{(−)} ∈ P C_{p}(ℝ) ,  inf_{λ∈ℝ} | det a^{(−)}(λ)| > 0 . (3.1)

Then equation (2.19) is not Fredholm in L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α}) due to Theorem 2.6. Namely the image of the operator Im W_{a} is not closed in L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α}) (see [Du4, § 4]).

In the present section, similarly to [Pr1, § 5.2], we define the spaces L^{→}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α}) and L^{←}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α}) such that the operators

W_{a} : L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α}) ⟶L^{→}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α}) ,  (3.2)
W_{a} : L^{←}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α}) ⟶ L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α})  (3.3)

would be Fredholm.

Let

U_{0}φ(x) := ∫_{0}^{x} φ(y)dy ,  V_{∞}φ(x) := (c)∫_{x}^{∞}φ(y)dy ,  (3.4)

where (c)∫_{x}^{∞} denotes the Cesaro mean value

(c)∫_{x}^{∞}φ(y)dy := lim_{t→∞} \frac{1}{t − x} ∫_{x}^{t} dτ ∫_{x}^{τ} φ(y)dy = lim_{t→∞} ∫_{x}^{t} \frac{t − y}{t − x}φ(y)dy (3.5)

which coincides with the usual Lebesgue (or the Riemann) integral if the latter exists. The operator V_{∞} in (3.5) is equivalent to the Cesaro-type operators Ṽ_{ζ_{j}} in (1.90) and Ṽ_{t_{j}} in (1.75) modulo isomorphism of spaces (see Lemmata 1.27 and 3.8).

Let

L^{→}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α}) := {φ : φ, V_{∞}φ ∈ L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α})} , (3.6)

L^{←}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α}) := {ψ + U_{0}φ : φ, ψ ∈ L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α})} .

On defining the norms

∥φ| L^{→}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α})∥ := ∥φ|L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α})∥

+∥V_{∞}φ|L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α})∥ ,

∥φ + U_{0}ψ| L^{←}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α})∥ := ∥φ|L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α})∥

+∥ψ|L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α})∥

we make L^{→}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α}) and L^{←}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α}) into Banach spaces.

The embedding

C_{0}^{∞}(ℝ^{+}) ⊂L^{→}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α}) ⊂ L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α})

⊂L^{←}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α})  (3.7)

are dense and follow from definitions.

Let

G_{±} := W_{g_{±}} ,  g_{±} := \frac{λ}{λ ± i} .  (3.8)

Then

G_{−}φ(x) = φ(x) − ∫_{x}^{∞}e^{x−y}φ(y)dy , G_{+}φ(x) = φ(x) − ∫_{0}^{x} e^{y−x}φ(y)dy (3.9)

and we can give equivalent description of spaces (3.6) in form of the following lemma (see [Pr1, § 5.2] for a similar assertion).

Lemma 3.1 The following definitions of spaces are equivalent:

L^{→}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α}) := {G_{−}φ : φ ∈ L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α})}

= Im_{L_{p}(ℝ^{+},x^{α}(1+x)^{γ−α})}G_{−} ,

L^{←}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α}) := {φ : G_{+}φ ∈ L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α})}

= Im_{L_{p}(ℝ^{+},x^{α}(1+x)^{γ−α})}G_{+}^{−1} .

Proof. It suffices to prove that

∥ψ| L^{→}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α})∥^{0} := ∥G_{−}^{−1}ψ|L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α})∥ , (3.10)

∥φ| L^{←}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α})∥^{0} := ∥G_{+}φ|L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α})∥ (3.11)

define equivalent norms. To check this let us prove that the operators

W_{g_{−}^{−1}}ψ = G_{−}^{−1}ψ = ψ + V_{∞}ψ ,  W_{g_{+}^{−1}}ψ = G_{+}^{−1}ψ = ψ + U_{0}ψ (3.12)

represent inverses to G_{−} and to G_{+}, respectively. Let us check G_{−}(I + V_{∞})φ = φ, because all other cases are similar.

Due to the density of embedding (3.7) we have to check the claimed equality only for φ ∈ C_{0}^{∞}(ℝ^{+}). Then

V_{∞}φ(x) = ∫_{x}^{∞}φ(y)dy

and integrating by parts we find

G_{−}(I + V_{∞})φ(x) = φ(x) + ∫_{x}^{∞}φ(s)ds − e^{−x} ∫_{x}^{∞}e^{−y}φ(y)dy

−e^{−x} ∫_{x}^{∞}e^{−y}dy ∫_{0}^{∞}φ(s)ds = φ(x) .

By the definition of φ ∈L^{←}_{p} (ℝ^{+}, x^{α}(1+x)^{γ−α}) we get G_{−}^{−1}φ = φ+V_{∞}φ ∈ L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α}); therefore the mappings

G_{−} : L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α}) ⟶L^{→}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α}) ,

G_{−}^{−1} : L^{→}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α}) ⟶ L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α})

are one-to-one and continuous. Equivalence of the norm in L^{→}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α}) and of the norm in (3.10) follows from the Banach theorem.

As we already know

G_{+}(φ + U_{0}φ) = G_{+}G_{+}^{−1}φ = φ;

on the other hand φ ∈ L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α}) implies G_{+}φ ∈ L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α}) (see (3.8), (3.9)) and therefore G_{+}U_{0}φ = G_{+}(φ + U_{0}φ) − G_{+}φ = φ − G_{+}φ ∈ L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α}). Thus, the mappings

G_{+} : L^{→}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α}) ⟶ L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α}) ,

G_{+}^{−1} : L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α}) ⟶L^{→}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α})

are one-to-one and continuous. Equivalence of the norms in (3.11) and of this in L^{←}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α}) follows from the Banach theorem.

Corollary 3.2 The spaces L^{→}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α}) and L^{←}_{p′} (ℝ^{+}, x^{−α}(1 + x)^{−γ+α}), where p′ = \frac{p}{p − 1}, are dual.

Proof. The operators

\begin{matrix}W_{g_{−}} = G_{−} : L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α}) ⟶L^{→}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α}) , \\ W_{g_{+}} = G_{+} : L^{←}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α}) ⟶ L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α}) \end{matrix} (3.13)

define isomorphisms (see Lemma 3.1) and they are dual (conjugate) W_{g_{±}}^{∗} = W_{g_{∓}}. The claimed result follows since the spaces L_{p}(ℝ^{+}, x^{α}(1+x)^{γ−α}) and L_{p′}(ℝ^{+}, x^{−α}(1 + x)^{−γ+α}) are dual as well.

Lemma 3.3 The embedding

L_{p}(ℝ^{+}, x^{α}(1 + x)^{1+γ−α}) ⊂L^{→}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α}) ⊂ L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α})

⊂L^{←}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α}) ⊂ L_{p}(ℝ^{+}, x^{α}(1 + x)^{−1+γ−α})  (3.14)

are continuous and dense.

Proof (see [Pr1, Ch. 5, Theorem 2.3]). We have to prove only the first and the last embedding (see (3.7)).

Density of embedding follow from the density of C_{0}^{∞}(ℝ^{+}) in all these spaces.

First we check the embedding in (3.14). Obviously,

L_{p}(ℝ^{+}, x^{α}(1 + x)^{1+γ−α}) = L_{p}([0, 1], x^{α}) +^{·} L_{p}([1, ∞), (1 + x)^{1+γ}) ,

L^{→}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α}) = L_{p}([0, 1], x^{α}) +^{·} L_{p}([1, ∞), (1 + x)^{γ})

and it suffices to prove the embedding

L_{p}([1, ∞), (1 + x)^{1+γ}) ⊂L^{→}_{p} ([1, ∞), (1 + x)^{γ}) .  (3.15)

If we prove the inequality

∥V_{∞}φ|L_{p}([1, ∞), (1 + x)^{γ})∥ ≤ c_{1}∥φ|L_{p}([1, ∞), (1 + x)^{1+γ}) ,  (3.16)

due to the norm definition in L^{→}_{p} ([1, ∞), (1 + x)^{γ}) (see (3.6)) there will follow the embedding (3.15).

Invoking the Hölder inequality we proceed as follows

|V_{∞}φ(x)| = |∫_{x}^{∞}φ(y)dy| ≤ (∫_{x}^{∞}y^{−(1+γ)p′}dy)^{\frac{1}{p′}} (∫_{x}^{∞}|y^{1+γ}φ(y)|^{p} dy)^{\frac{1}{p}}

≤ \frac{1}{(1 + γ)p′ − 1}∥φ|L_{p}([1, ∞), (1 + x)^{1+γ})∥ ,

since −p′(1 + γ) < −1 (see (2.7)). Thus, V_{∞}φ(x) exists as an ordinary Lebesgue integral for arbitrary φ ∈ L_{p}([1, ∞), (1 + x)^{1+γ}).

For the function

f(s, t) := t |φ(st)| , s, t ∈ [1, ∞),

we have

∫_{1}^{∞}f(s, t)ds = ∫_{t}^{∞}|φ(y)dy| ≥ |V_{∞}φ(x)| ,

\{∫_{1}^{∞}[t^{γ}f(s, t)]^{p}dt\}^{\frac{1}{p}} = t^{\frac{1}{p} −γ−1} \{∫_{t}^{∞}|y^{1+γ}φ(y)|^{p}\}^{\frac{1}{p}} dy .

The latter equalities, inserted in the following well-known inequality

\{∫_{1}^{∞}[∫_{1}^{∞}t^{γ}f(s, t)ds]^{p} dt\}^{\frac{1}{p}} ≤ ∫_{1}^{∞}\{∫_{1}^{∞}[t^{γ}f(s, t)]^{p} dt\}^{\frac{1}{p}} ds

(see [HLP1, Theorem 202]) yield

∥V_{∞}φ|L_{p}([1, ∞), (1 + x)^{γ})∥ ≤ 2^{γ} ∫_{1}^{∞}s^{− \frac{1}{p} −γ−1} \{∫_{1}^{∞}|y^{1+γ}φ(y)|^{p} dy\}^{\frac{1}{p}} dt

≤ \frac{2^{γ}}{\frac{1}{p} + γ} ∥φ|L_{p}([1, ∞), (1 + x)^{1+γ})∥

since −\frac{1}{p} − γ < 0 (see (2.7)).

Thus, (3.16) is proved and implies continuity of the first embedding in (3.14).

The second embedding in (3.14) follows by density. In fact, as we already proved the embedding

L_{p′}(ℝ^{+}, x^{−α}(1 + x)^{1−γ+α}) ⊂L^{→}_{p′} (ℝ^{+}, x^{−α}(1 + x)^{−γ+α})

is continuous and dense. The spaces are reflexive and the embedding of the dual spaces

L^{←}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α}) ⊂ L_{p}(ℝ^{+}, x^{α}(1 + x)^{−1+γ−α})

are continuous and dense as well.

Corollary 3.4 Let a ∈ C(ℝ̇^{+}); then

aI ∈ L (L^{→}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α})) , aI ∈ L (L^{←}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α})) ,

provided

|a(x) − a(∞)| ≤ M(1 + x))^{−1} , x ∈ ℝ^{+} M < ∞ .

Proof. It suffices to represent

aφ = [a − a(∞)]φ + a(∞)φ

and apply Lemma 3.3 to the first summand, because the second summand, multiplication by a constant, is obviously continuous operator.

Theorem 3.5 Let a(λ) be given by (3.1) and (1.4) hold. Then operators (3.2) and (3.3) are continuous.

Operators (3.2) and (3.3) are Fredholm or are invertible if and only if the corresponding operators

W_{a^{(−)}} : L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α}) ⟶ L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α}) , (3.17)
W_{a^{(+)}} : L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α}) ⟶ L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α}) , (3.18)

a^{(+)}(λ) := \frac{λ + i}{λ}a(λ)

are Fredholm or are invertible, respectively.

The pairs of operators (3.2) and (3.17), (3.3) and (3.18) have the kernels and cokernels of equal dimension and equal indices.

Proof (see [Pr1, § 5.2.3] for a similar proof). Let b, d ∈ V_{1}(ℝ) and either b(λ) has a bounded analytic continuation b(λ − iσ) in the lower half-plane σ > 0 or d(λ) has a bounded analytic continuation d(λ + iσ) in the upper half-plane σ > 0; then

W_{bd} = W_{b}W_{d}  (3.19)

(see[Du1, GF1]). Since

a(λ) = \frac{λ}{λ − i}a^{(−)}(λ) = \frac{λ}{λ + i}a^{(+)}(λ)

(see (3.1), (3.17) and (3.18)), we get

W_{a} = G_{−}W_{a^{(−)}} = G_{+}W_{a^{(+)}}  (3.20)

(see (3.8) and (3.19)).

All claimed assertions follow from (3.20) since the operators

G_{−} : L_{p}(ℝ^{+}, x^{α}(1 + x)^{γ−α}) ⟶L^{→}_{p} (ℝ^{+}, (1 + x)^{γ−al}) ,
G_{+} : L^{←}_{p} (ℝ^{+}, x^{α}(1 + x)^{γ−α}) ⟶ L_{p}(ℝ^{+}, (1 + x)^{γ−al})

establish isomorphism (see Lemma (3.1)).

Remark 3.6 Defining the spaces

C^{→} (ℝ̇^{+}, (1 + x)^{β}) := {φ : φ, V_{∞}φ ∈ C(ℝ̇^{+}, (1 + x)^{β})} ,

C^{←} (ℝ̇^{+}, (1 + x)^{β}) := {ψ + U_{0}φ : φ, ψ ∈ C(ℝ̇^{+}, (1 + x)^{β})} ,

for β ∈ ℝ and taking in (3.1) a, a^{(−)} ∈ W_{β}(ℝ), full analogies of Lemma 3.1 and of Theorem 3.5 can be proved for the convolution operators

W_{a} : C(ℝ̇^{+}, (1 + x)^{β}) ⟶C^{→} (ℝ̇^{+}, (1 + x)^{β}) ,

W_{a} : C^{←} (ℝ̇^{+}, (1 + x)^{β}) ⟶ C(ℝ̇^{+}, (1 + x)^{β}) .

As for the analogies of Lemma 3.3 and Corollary 3.4 we easily find that continuity

[g − g(∞)]I : C(ℝ̇^{+}, (1 + x)^{β}) ⟶C^{→} (ℝ̇^{+}, (1 + x)^{β}) ,

[g − g(∞)]I : C^{←} (ℝ̇^{+}, (1 + x)^{β}) ⟶ C(ℝ̇^{+}, (1 + x)^{β}) ,

gI ∈ L (C^{→} (ℝ̇^{+}, (1 + x)^{β})) , gI ∈ L (C^{←} (ℝ̇^{+}, (1 + x)^{β}))

follows if

g ∈ C(ℝ̇) , |g(x) − g(∞)| ≤ M(1 + x))^{−1−ε} , x ∈ ℝ^{+} , M < ∞ .

3.2 Convolutions on ℝ

Let a ∈ P C_{p}(ℝ) be non-elliptic, namely, as in (3.1). Then operator W_{a}^{0} is not Fredholm in L_{p}(ℝ) and, moreover, has non-closed image Im W_{a}^{0} (see [Du4, § 4]).

Let us consider the operators

V_{∞}φ(x) := (c)∫_{x}^{∞}φ(y)dy = lim_{t→∞} \frac{1}{t − x} ∫_{x}^{t}dτ ∫_{x}^{τ} φ(z)dz = lim_{t→∞} ∫_{x}^{t} \frac{t − y}{t − x}φ(y)dy,

V_{−∞}φ(t) := (c) ∫_{−∞}^{t} φ(y)dy , F_{0}φ := ∫_{−∞}^{∞} φ(τ)dτ , t ∈ ℝ ,  (3.21)

where the integrals with prefix (c) are understood in the Cesaro mean value sense and they convert into an usual Lebesgue (or a Riemann) integral if the latter exist. We define the space

L^{→}_{p} (ℝ) := {φ ∈ L_{p}(ℝ) : V_{∞} ∈ L_{p}(ℝ)}

= {φ ∈ L_{p}(ℝ) : V_{±∞} ∈ L_{p}(ℝ), F_{0}φ = 0}  (3.22)

and endow it with the norm

∥φ| L^{→}_{p} (ℝ)∥ := ∥φ | L_{p}(ℝ)∥ + ∥V_{∞}φ | L_{p}(ℝ)∥ .

To justify the second definition in (3.22) let us prove that the conditions

V_{−∞}φ ∈ L_{p}(ℝ) , F_{0}φ = 0

follow from the principal condition V_{∞}φ ∈ L_{p}(ℝ). In fact, the inclusion V_{−∞}φ ∈ L_{p}(ℝ) follows from the principal condition and from F_{0}φ = 0, since

V_{−∞}φ(t) = F_{0}φ − V_{∞}φ(t) = −V_{∞}φ(t) .

Thus, we have to prove only F_{0}φ = 0. Since

F_{0}φ = lim_{t→−∞} V_{∞}φ(t) ,

V_{∞}φ ∈ L_{p}(ℝ) is absolutely continuous with derivative (V_{∞}φ) = φ ∈ L_{p}(ℝ), we get the result.

The embedding

{φ ∈ C_{0}^{∞}(ℝ) : F_{0}φ = 0} ⊂L^{→}_{p} (ℝ)

is dense.

Let us prove that the convolution operator

G_{−} := W_{g_{−}}^{0} : L_{p}(ℝ) ⟶L^{→}_{p} (ℝ) , g_{−}(λ) := \frac{λ}{λ − i} .  (3.23)

with vanishing symbol is bounded and, moreover, defines an isomorphism with the inverse operator written as follows

(W_{g_{−}}^{0})^{−1} = W_{g_{−}^{−1}} = I + V_{∞} : L^{→}_{p} (ℝ) ⟶ L_{p}(ℝ)  (3.24)

(cf. Lemma 3.1). In fact, by definition (3.21) operator (3.24) is bounded and, due to obvious equality W_{g}^{0}W_{h}^{0} = W_{gh}^{0} (see (2.1)) W_{g_{−}^{−1}} is the inverse from the right to W_{g_{−}}^{0}:

W_{g_{−}}^{0}W_{g_{−}^{−1}} = W_{g_{−}g_{−}^{−1}}^{0} = I .

Let us prove that operator (3.23) is bounded. According to the definition (3.22) it suffices to prove that

V_{∞}W_{g_{−}}^{0}φ ∈ L_{p}(ℝ) provided φ ∈ L_{p}(ℝ) .

Since

W_{g_{−}}^{0}φ(t) = φ(t) − ∫_{x}^{∞}e^{t−τ} φ(τ)dτ ,

we proceed as follows

V_{∞}W_{g_{−}}^{0}φ(t) = V_{∞}φ(t) − (c) ∫_{t}^{∞}dτ ∫_{τ}^{∞}e^{τ−y}φ(y)dy

= V_{∞}φ(t)−(c) ∫_{t}^{∞}φ(y)dy ∫_{t}^{y} e^{τ−y}dy =V_{∞}φ(t)−(c) ∫_{t}^{∞}(1 − e^{τ−y})φ(y)dy

= (c) ∫_{t}^{∞}e^{τ−y}φ(y)dy = φ(t) − W_{g_{−}}^{0}φ(t)  (3.25)

and get the inclusion V_{∞}W_{g_{−}}^{0}φ ∈ L_{p}(ℝ) because φ, W_{g_{−}}^{0}φ ∈ L_{p}(ℝ). Moreover, (3.25) can also be written as follows

(I + V_{∞})W_{g_{−}}^{0}φ = φ ,

which means, due to (3.24), W_{g_{−}^{−1}}^{0}W_{g_{−}}^{0} = I and W_{g_{−}}^{0} is invertible from the

left.

Similarly to (3.14) is proved that the embeddings

{φ ∈ L_{p}(ℝ, 1 + |x|) : F_{0}φ = 0} ⊂L^{→}_{p} (ℝ) ⊂ L_{p}(ℝ, (1 + |x|)^{−1})

are continuous and dense.

If g ∈ L_{∞}(ℝ̇) has the estimate

|g(x) − g(∞)| ≤ M(1 + |x|))^{−1} , x ∈ ℝ , M < ∞ ,  (3.26)

the following multiplication operators are bounded

[g − g(∞)I : L_{p}(ℝ) ⟶L^{→}_{p} (ℝ) ,  gI : L^{→}_{p} (ℝ) ⟶L^{→}_{p} (ℝ) . (3.27)

Theorem 3.7 Let a(λ) be as in (3.1). Operator W_{a}^{0} : L_{p}(ℝ) ⟶L^{→}_{p} (ℝ) is Fredholm if and only if W_{a^{(−)}}^{0} is Fredholm in the space L_{p}(ℝ), which reads

inf_{λ∈ℝ} |a^{(−)}(λ)| > 0  (3.28)

(see (2.5)). If (3.28) holds, (a^{(−)})^{−1} ∈ P C_{p}^{N×N}(ℝ̇) and the inverse is

(W_{a}^{0})^{−1} := W_{(a^{(−)})^{−1}}^{0}(I + V_{∞}) : L^{→}_{p} (ℝ) ⟶ L_{p}(ℝ) .  (3.29)

Proof. Due to (3.24), (3.24) the proof can immediately be reduced to the investigation of the operator W_{a^{(−)}}^{0} in the space L_{p}(ℝ). In this case the Fredholm criteria is known (see (2.5)).

3.3 Cesaro-type operators

We remind that Γ_{1} := {ζ ∈ ℂ : |ζ| = 1} is the unit circumference, Ξ := {ζ_{1}, …, ζ_{n}} ⊂ Γ_{1} is the conformal image of all knots of Γ and Ξ_{ow} is the subset of Ξ (conformal image of all outward peaks of Γ; see (1.74)– (1.92)); Γ_{1ζ_{j}} = Γ_{1ζ_{j}}^{−} ∪Γ_{1ζ_{j}}^{+} is a fixed neighbourhood of ζ_{j} (see (1.74)–(1.92)). We use L_{p}(Γ_{1}, {ζ_{j}}) for the space L_{p}(Γ_{1}, Ξ_{ow}) when Ξ_{ow} = {ζ_{j}} consists of a single knot.

For a Banach space 𝕏 by 𝕏^{n} we denote the spaces of vector–elements Ψ = (ψ_{1}, …, ψ_{n}) with components ψ_{j} ∈ 𝕏. Let

L_{p}^{2}(ℝ, {∞}) := {Φ = (φ_{1}, φ_{2}) ∈ L_{p}^{2}(ℝ) : Ṽ_{∞}Φ ∈ L_{p}(ℝ^{+})} , (3.30)

Ṽ_{∞}Φ := [ \begin{matrix}e^{\frac{2π}{p} i}V_{∞} & −V_{∞} \\ 0 & 0 \end{matrix} ] [ \begin{matrix}φ_{1} \\ φ_{2} \end{matrix} ] = V_{∞}[e^{− \frac{π}{p} i}φ_{1} − φ_{2}]

(see (3.4) for V_{∞}) denote the subset of L_{p}^{2}(ℝ) with the appropriate norm

∥Φ|L_{p}^{2}(ℝ, {∞})∥ := ∥Φ|L_{p}(ℝ)∥ + ∥Ṽ_{∞}Φ|L_{p}(ℝ^{+})∥ .

Lemma 3.8 There exists an isomorphism of spaces

\begin{aligned}Z_{pζ_{j}} & : & L_{p}(Γ_{1}) ⟶ L_{p}^{2}(ℝ) , \\ Z_{pζ_{j}} & : & L_{p}(Γ_{1}, \{ζ_{j}\}) ⟶ L_{p}^{2}(ℝ, \{∞\}) , \end{aligned}  (3.31)

such that the Cesaro-type operators Ṽ_{ζ_{j}} in (1.90) and Ṽ_{∞} in (3.4) are equivalent

Z_{pζ_{j}} Ṽ_{ζ_{j}}Z_{pζ_{j}}^{−1} = g_{j}Ṽ_{∞}h_{j}I = g_{j}Ṽ_{∞} + R_{j} ,  (3.32)

where the functions g_{j}^{±1}, h_{j}^{±1} ∈ C^{∞}(ℝ̇) are non-vanishing

g_{j}(x) := (\frac{1 − ie^{−x}}{1 + e^{−2x}} )^{\frac{1}{p}} , h_{j}(x) := \frac{(1 + e^{−2x})^{\frac{1}{p}}}{(1 − ie^{−x})^{1+ \frac{1}{p}}}

and the operator R_{j} : L_{p}(ℝ) → L_{p}(ℝ) is bounded.

Proof. The transformations

\begin{matrix}Z_{ζ_{j}}φ(x) := |κ′_{ζ_{j}}(x)|^{\frac{1}{p}} φ(κ_{ζ_{j}}(x)) = 2^{\frac{1}{p}} (x^{2} + 1)^{\frac{1}{p}} φ (−ζ_{j} \frac{x − i}{x + i}) , \\ Z_{p}ψ(λ) := (e^{− \frac{λ}{p}} ψ(e^{−λ}), e^{− \frac{λ}{p}} ψ(−e^{−λ})) , x, λ ∈ ℝ, \end{matrix} (3.33)

where the first one is based on the Kelly transformation

κ_{ζ_{j}}(x) := −ζ_{j} \frac{x − i}{x + i} : ℝ ⟶ Γ_{1} , κ_{ζ_{j}}(0) = ζ_{j} ,

establish isometric isomorphisms

\begin{aligned}Z_{ζ_{j}} & : & L_{p}(Γ_{1}) → L_{p}(ℝ) , ∥Z_{ζ_{j}}φ|L_{p}(ℝ)∥ = ∥φ|L_{p}(Γ_{1})∥ , \\ Z_{p} & : & L_{p}(ℝ) → L_{p}^{2}(ℝ) , ∥Z_{p}ψ|L_{p}^{2}(ℝ)∥ = ∥ψ|L_{p}(ℝ)∥ \end{aligned}  (3.34)

and have the following inverses

Z_{ζ_{j}}^{−1}ψ(ζ) := |(κ_{ζ_{j}}^{−1})′(x)|^{\frac{1}{p}} ψ(κ_{ζ_{j}}^{−1}(x)) = |ζ + ζ_{j}|^{− \frac{2}{p}} ψ (−i\frac{ζ − ζ_{j}}{ζ + ζ_{j}} ) , (3.35)

Z_{p}^{−1}Φ(x) := χ_{−}(x)(−x)^{− \frac{1}{p}} φ_{2}(− log(−x)) + χ_{+}(x)x^{− \frac{1}{p}} φ_{1}(− log x) ,

where Φ = (φ_{1}, φ_{2})^{⊤} and χ_{±}(x) are the characteristic functions of ℝ^{±} ⊂ ℝ. The transformation

Z_{pζ_{j}} := Z_{p}Z_{ζ_{j}}  (3.36)

establishes the first of claimed isomorphisms in (3.31).

To prove that Z_{pζ_{j}} arranges the second isomorphisms in (3.31) as well, let us consider the following intermediate space

L_{p}^{2}(ℝ, {0}) := {ψ ∈ L_{p}(ℝ) : Ṽ_{0}ψ ∈ L_{p}(ℝ^{+})} ,  (3.37)
Ṽ_{0}ψ(x) := V_{0}[e^{− \frac{π}{p} i}ψ(x) − ψ(−x)] ,

where the operator V_{0} is defined by the Cesaro-type mean value integral (cf. (3.5)):

V_{0}ψ(x) := (c) ∫_{0}^{x} (\frac{y}{x})^{\frac{1}{p}} ψ(y)\frac{dy}{y} = lim_{z→0} [log \frac{x}{z}]^{−1} ∫_{z}^{x} \frac{dτ}{τ} ∫_{τ}^{x} (\frac{y}{x})^{\frac{1}{p}} ψ(y)\frac{dy}{y}

= lim_{z→0} ∫_{z}^{x} \begin{matrix}log & \frac{y}{z} \\ log & \frac{x}{z} \end{matrix} (\frac{y}{x})^{\frac{1}{p}} ψ(y)\frac{dy}{y} .  (3.38)

It is easy to verify directly the following connection

Z_{p}Ṽ_{0}Z_{p}^{−1} = Ṽ_{∞}  (3.39)

(see (3.34)). Moreover, Z_{p} establishes isometric isomorphisms

\begin{matrix}Z_{p} : L_{p}(ℝ, \{0\}) → L_{p}^{2}(ℝ, \{∞\}) , \\ ∥Z_{p}ψ|L_{p}^{2}(ℝ, \{∞\})∥ = ∥ψ|L_{p}(ℝ, \{0\})∥ . \end{matrix}  (3.40)

Therefore, to justify the second isomorphism in (3.31) we just need to verify

Z_{ζ_{j}}V_{ζ_{j}}Z_{ζ_{j}}^{−1} = g_{0}V_{0}h_{0}I ,  (3.41)

where g_{0}^{±1}, h_{0}^{±1} ∈ C^{∞}(ℝ) are non-vanishing functions

g_{0}(x) := (\frac{1 − ix}{1 + x^{2}} )^{\frac{1}{p}} , h_{0}(x) := \frac{(1 + x^{2})^{\frac{1}{p}}}{(1 − ix)^{1+ \frac{1}{p}}}

because applying equivalence (3.39) to equality (3.41) we immeadiately get (3.32).

Let us consider the following operators

V_{ζ_{j}}^{1}φ(ζ) := lim_{η→ζ_{j}} [log \frac{(ζ − ζ_{j})(η + ζ_{j})}{(η − ζ_{j})(ζ + ζ_{j})}]^{−1} ∫_{η ζ^{⌣}} \frac{dλ}{λ − ζ_{j}} ∫_{λζ^{⌣}} (\frac{τ − ζ_{j}}{ζ − ζ_{j}} )^{\frac{1}{p}}

×φ(τ)\frac{dτ}{τ − ζ_{j}} = lim_{η→ζ_{j}} ∫_{η ζ^{⌣}} \begin{matrix}log & \frac{τ−ζ_{j}}{η−ζ_{j}} \\ log & \frac{(ζ−ζ_{j})(η+ζ_{j})}{(η−ζ_{j})(ζ+ζ_{j})} \end{matrix} (\frac{τ − ζ_{j}}{ζ − ζ_{j}} )^{\frac{1}{p}} φ(τ)\frac{dτ}{τ − ζ_{j}} ,

V_{ζ_{j}}^{2}φ(ζ) := lim_{η→ζ_{j}} ∫_{η ζ^{⌣}} \begin{matrix}log & \frac{(τ−ζ_{j})(η+ζ_{j})}{(η−ζ_{j})(τ+ζ_{j})} \\ log & \frac{(ζ−ζ_{j})(η+ζ_{j})}{(η−ζ_{j})(ζ+ζ_{j})} \end{matrix} (\frac{τ − ζ_{j}}{ζ − ζ_{j}} )^{\frac{1}{p}} φ(τ)\frac{dτ}{τ − ζ_{j}}

and prove that

V_{ζ_{j}} = V_{ζ_{j}}^{1} = V_{ζ_{j}}^{2} .  (3.42)

In fact,

(V_{ζ_{j}} − V_{ζ_{j}}^{1})φ(ζ) := lim_{η→ζ_{j}} \frac{log \frac{η+ζ_{j}}{ζ+ζ_{j}}}{log \frac{ζ−ζ_{j}}{η−ζ_{j}} + log \frac{η+ζ_{j}}{ζ+ζ_{j}}} V_{ζ_{j}}φ(ζ) = 0 ,

because, for a fixed ζ ∈ Γ_{1ζ_{j}},

|log \frac{η + ζ_{j}}{ζ + ζ_{j}} | ≤ M_{0} < ∞ and lim_{η→ζ_{j}} \frac{log \frac{η+ζ_{j}}{ζ+ζ_{j}}}{log \frac{ζ−ζ_{j}}{η−ζ_{j}} + log \frac{η+ζ_{j}}{ζ+ζ_{j}}} = 0 .

For the difference V_{ζ_{j}}^{2} − V_{ζ_{j}}^{1} we have

(V_{ζ_{j}}^{2} − V_{ζ_{j}}^{1})φ(ζ) = lim_{η→ζ_{j}} ∫_{η ζ^{⌣}} \begin{matrix}log & \frac{η+ζ_{j}}{τ+ζ_{j}} \\ log & \frac{(ζ−ζ_{j})(η+ζ_{j})}{(η−ζ_{j})(ζ+ζ_{j})} \end{matrix} (\frac{τ − ζ_{j}}{ζ − ζ_{j}} )^{\frac{1}{p}} φ(τ)\frac{dτ}{τ − ζ_{j}} . (3.43)

If log^{2}(τ − ζ_{j})φ(τ) belongs to L_{p}(Γ_{1ζ_{j}}) integrand in (3.43) is absolutely integrable and we can drug the limit inside; on the other hand

lim_{η→ζ_{j}} \begin{matrix}log & \frac{η+ζ_{j}}{τ+ζ_{j}} \\ log & \frac{(ζ−ζ_{j})(η+ζ_{j})}{(η−ζ_{j})(ζ+ζ_{j})} \end{matrix} = 0

for all fixed τ ∈ ) ; η ζ . Therefore, with above constraints on φ(τ) we get

(V_{ζ_{j}}^{2} − V_{ζ_{j}}^{1})φ(ζ) = lim_{η→ζ_{j}} V_{ζ_{j}}^{1}v_{j}φ(ζ) = 0 , v_{j}(τ, η) := \begin{matrix}log & \frac{η+ζ_{j}}{τ+ζ_{j}} \\ log & \frac{η−ζ_{j}}{τ−ζ_{j}} \end{matrix} .

Since the above taken functions are dense in the space L_{p}(Γ_{1j}, {ζ_{j}}), equality V_{ζ_{j}}^{2}φ = V_{ζ_{j}}^{1}φ holds for all φ ∈ L_{p}(Γ_{1j}, {ζ_{j}}).

Due to (3.42) all three operators V_{ζ_{j}}, V_{ζ_{j}}^{1} and V_{ζ_{j}}^{2} define the same space L_{p}(Γ_{1}, {ζ_{j}}) (cf. (1.92)). Therefore, to prove the second isomorphism in (3.31) we use the operator V_{ζ_{j}}^{2} instead of V_{ζ_{j}}. We proceed as follows:

(Z_{ζ_{j}}χ_{ζ_{j}}V_{ζ_{j}}^{2}Z_{ζ_{j}}^{−1}ψ)(x)= lim_{κ_{ζ_{j}} (z)→ζ_{j}} |κ′_{ζ_{j}}(x)|^{\frac{1}{p}} ∫ ; )_{κ_{ζ_{j}} (z)κ_{ζ_{j}} (x)}\begin{matrix}log & \frac{(τ−ζ_{j})(κ_{ζ_{j}} (z)+ζ_{j})}{(κ_{ζ_{j}} (z)−ζ_{j})(τ+ζ_{j})} \\ log & \frac{(κ_{ζ_{j}} (x)−ζ_{j})(κ_{ζ_{j}} (z)+ζ_{j})}{(κ_{ζ_{j}} (z)−ζ_{j})(κ_{ζ_{j}} (x)+ζ_{j})} \end{matrix}

× [\frac{τ − ζ_{j}}{κ_{ζ_{j}}(x) − ζ_{j}} ]^{\frac{1}{p}} |(κ_{ζ_{j}}^{−1})′(τ)|^{\frac{1}{p}} ψ(κ_{ζ_{j}}^{−1}(τ))\frac{dτ}{τ − ζ_{j}} ;

inserting τ = κ_{ζ_{j}}(y), dτ = κ′_{ζ_{j}}(y)dy and taking into account the equalities

(κ_{ζ_{j}}^{−1})′(κ_{ζ_{j}}(y)) = [κ′_{ζ_{j}}(y)]^{−1} , \frac{κ_{ζ_{j}}(x) − ζ_{j}}{κ_{ζ_{j}}(x) + ζ_{j}} = ix ,

κ_{ζ_{j}}(x) − ζ_{j} = \frac{−2ζ_{j}x}{x + i} , κ′_{ζ_{j}}(x) = \frac{−2iζ_{j}}{(x + i)^{2}} ,  (3.44)

we continue as follows

(Z_{ζ_{j}}χ_{ζ_{j}}V_{ζ_{j}}^{2}Z_{ζ_{j}}^{−1}ψ)(x) = lim_{z→0} ∫_{z}^{x} \begin{matrix}log & \frac{(κ_{ζ_{j}} (y)−ζ_{j})(κ_{ζ_{j}} (z)+ζ_{j})}{(κ_{ζ_{j}} (z)−ζ_{j})(κ_{ζ_{j}} (y)+ζ_{j})} \\ log & \frac{(κ_{ζ_{j}} (x)−ζ_{j})(κ_{ζ_{j}} (z)+ζ_{j})}{(κ_{ζ_{j}} (z)−ζ_{j})(κ_{ζ_{j}} (x)+ζ_{j})} \end{matrix} [\frac{κ_{ζ_{j}}(y) − ζ_{j}}{κ_{ζ_{j}}(x) − ζ_{j}} ]^{\frac{1}{p}}

× |\frac{κ′_{ζ_{j}}(x)}{κ′_{ζ_{j}}(y)}|^{\frac{1}{p}} ψ(y)\frac{κ′_{ζ_{j}}(y)dy}{κ_{ζ_{j}}(y) − ζ_{j}} = lim_{z→0} g_{0}(x) ∫_{z}^{x} \begin{matrix}log & \frac{y}{z} \\ log & \frac{x}{z} \end{matrix} (\frac{y}{x})^{\frac{1}{p}} h_{0}(y)ψ(y)\frac{dy}{y}

= g_{0}V_{0}h_{0}ψ(x) , g_{0}(x) := (\frac{1 − ix}{1 + x^{2}} )^{\frac{1}{p}} , h_{0}(x) := \frac{(1 + x^{2})^{\frac{1}{p}}}{(1 − ix)^{1+ \frac{1}{p}}}

and we get (3.41).

Boundedness of

R_{j} := g_{j}Ṽ_{∞}[h_{j} − 1]I : L^{2}(ℝ) ⟶ L^{2}(ℝ)

follows since h_{j}(x) − 1 = h_{j}(x) − h_{j}(+∞) = O (e^{− \frac{x}{p}} ) as x → +∞ which

yields the boundedness [h_{j} − 1]I : L^{2}(ℝ) →L^{→} ^{2}(ℝ) (see (3.27)).

Proof of Lemma 1.27. Let us apply the isomorphism Z_{pζ_{j}}, defined in (3.31), (3.33). Then φ, log(ζ − ζ_{j})φ ∈ L_{p}(Γ_{1ζ_{j}}) for all ζ_{j} ∈ Ξ_{ow} imply

Z_{pζ_{j}}φ ∈ L_{p}^{2}(ℝ^{+}), (Z_{pζ_{j}} log(ζ − ζ_{j})φ)(x)=log (−ζ_{j} \frac{e^{−x}}{e^{−x} + i}) (Z_{pζ_{j}}φ)(x)

= (−x + log \frac{−ζ_{j}}{e^{−x} + i}) (Z_{pζ_{j}}φ)(x) ∈ L_{p}^{2}(ℝ^{+})

(see (3.33)); due to Lemma 3.3 Z_{pζ_{j}}φ ∈L^{→} _{p}^{2}(ℝ^{+}). Applying the inverse iso-

morphism Z_{pζ_{j}}^{−1} (see (3.31), (3.35)) we find φ = Z_{pζ_{j}}^{−1}Z_{pζ_{j}}φ ∈L^{→} _{p}^{2}(Γ_{1ζ_{j}}, {ζ_{j}}).

The remainder claims of the Lemma (see (1.98)) follow from the proved part as Corollary 3.4 from Lemma 3.3.

Remark 3.9 Due to the above established isomorphism (3.40) and to Corollary 3.4 if a function g(x) has the property

g ∈ C(J), g(x) − g(0) = O((1 − log x)^{−1}) , J := (−c, c) ⊂ ℝ (3.45)

the following multiplication operators (see (3.37))

\begin{aligned}gI & : & L_{p}(J , \{0\}) ⟶ L_{p}(J , \{0\}) , \\ [g − g(0)]I & : & L_{p}(J) ⟶ L_{p}(J , \{0\}) \end{aligned}

are bounded.

3.4 Equations on the circumference (example)

Let Γ_{1}, Ξ := {ζ_{1}, …, ζ_{n}} ⊂ Γ_{1}, Ξ_{ow} ⊂ Ξ, Γ_{1ζ_{j}} and Γ_{1ζ_{j}}^{±} be the same as in § 3.3.

We us consider, as an example, the following operator with fixed singularities at Ξ in the kernel

A_{Ξ}φ(ζ) = φ(ζ)
+ ∑_{j=1}^{n} χ_{ζ_{j}}^{+}(ζ)\frac{μ_{j}ζ_{j}}{π} ∫_{Γ_{1ζ_{j}}^{+}} (\frac{ζ−ζ_{j}}{τ−ζ_{j}} )^{γ_{j}} \frac{φ(τ)dτ}{ζ_{j}^{2}−τt} , ζ ∈ Γ_{1} ,  (3.46)

where χ_{ζ_{j}}^{+}(t) is the characteristic function of the arc Γ_{1ζ_{j}}^{+} ⊂ Γ_{1ζ_{j}} ⊂ Γ_{1} and

μ_{j} = \{ \begin{matrix}sin π (\frac{1}{p} + γ_{j}) for & ξ ∈ Ξ_{ow} , \\ sin π (\frac{1}{p} + γ′_{j}) for & ξ ∉ Ξ_{ow} , \end{matrix}  (3.47)

−\frac{1}{p} < γ_{j} < 1 − \frac{1}{p} , ξ ∈ Ξ_{ow} ,

−\frac{1}{p} < γ′_{k} ≠ γ_{k} < 1 − \frac{1}{p} , ξ ∉ Ξ_{ow} .  (3.48)

Theorem 3.10 Let conditions (3.47) and (3.48) hold. Then the operator

A_{Ξ} : L_{p}(Γ_{1}) → L_{p}(Γ_{1}, Ξ_{ow}) , 1 < p < ∞  (3.49)

is Fredholm provided

\frac{1}{p} + γ_{j} ≠ \frac{1}{2} for all j = 1, …, n  (3.50)

and then

dim Ker A_{Ξ} = ∑_{σ_{j}>0} σ_{j} , dim Coker A_{Ξ} = − ∑_{σ_{j}<0} σ_{j} ,  (3.51)

where

σ_{j} = \{ \begin{matrix}0 & for & \frac{1}{2} < \frac{1}{p} + γ_{j} < 1 , & ζ_{j} ∈ Ξ_{ow} , \\ 0 & for & γ_{j} ∈ (γ′_{j}, 1 − γ′_{j}) , & ζ_{j} ∉ Ξ_{ow} , \\ −1 & for & 0 < \frac{1}{p} + γ_{j} < \frac{1}{2} , & ζ_{j} ∈ Ξ_{ow} , \\ 1 & for & γ_{j} > max\{γ′_{j}, 1 − γ′_{j}\} , & ζ_{j} ∉ Ξ_{ow} , \\ −1 & for & γ_{j} < min\{γ′_{j}, 1 − γ′_{j}\} , & j = m + 1, …, n . \end{matrix} (3.52)

In particular, A_{Ξ} in (3.49) is invertible provided

\frac{1}{2} < \frac{1}{p} + γ_{j} < 1 for all ξ ∈ Ξ_{ow}

and γ_{j} ∈ (γ′_{j}, 1 − γ′_{j}) for all ξ ∉ Ξ_{ow} .  (3.53)

Proof. Note that since Γ_{1ζ_{k}}^{+} ∩ Γ_{1ζ_{j}}^{+} = ∅ for k ≠ j, we have

A_{Ξ} = ∏_{j=1}^{n} A_{ζ_{j}} ,

A_{ζ_{j}} := φ(ζ) + χ_{ζ_{j}}^{+}(ζ)\frac{μ_{j}ζ_{j}}{π} ∫_{Γ_{1ζ_{j}}^{+}} (\frac{ζ − ζ_{j}}{τ − ζ_{j}} )^{γ_{j}} \frac{φ(τ)dτ}{ζ_{j}^{2} − τt} , ζ ∈ Γ_{1} . (3.54)

Therefore it suffices to prove the Theorem for a single knot Ξ = {ζ_{j}}. We will apply the isomorphisms of spaces

\begin{aligned}Z_{ζ_{j}} & : & L_{p}(Γ_{1ζ_{j}}^{+}) ⟶ L_{p}(I) , \\ Z̃_{p} & : & L_{p}(I) ⟶ L_{p}(ℝ^{+}) , \end{aligned}  (3.55)

where I = [0, 1] and Z_{ζ_{j}} is defined in (3.33), while

Z̃_{p}φ(x) := e^{− \frac{x}{p}} φ(e^{−x}) .  (3.56)

We have assumed, without loss of generality, that

Γ_{1ζ_{j}}^{+} = {e^{iϑ}ζ_{j} : 0 < ϑ < π}

is the half-circumference; otherwise we will use another Kelly transformation

κ_{ζ_{j}}(x) := −ζ_{j} \frac{x − i cot \frac{ϑ_{j}}{2}}{x + i cot \frac{ϑ_{j}}{2}} : I = [0, 1] ⟶ Γ_{1ζ_{j}}^{+} = ) ; ζ_{j}(ζ_{j}e^{iϑ_{j}})⊂ Γ_{1}

while defining the isomorphism Z_{ζ_{j}} in (3.33). The operators Z_{ζ_{j}} and Z̃_{p}, besides (3.55) and similarly to (3.31), (3.40), establish the following isomorphisms

\begin{aligned}Z_{ζ_{j}} & : & L_{p}(Γ_{1ζ_{j}}^{+}, \{ζ_{j}\}) ⟶ L_{p}(I, \{0\}) , \\ Z̃_{p} & : & L_{p}(I, \{0\}) ⟶L^{→} _{p}(ℝ^{+}) . \end{aligned}  (3.57)

Lifting the operator (3.54) to the equivalent operator first by the isomorphism Z_{ζ_{j}}, we get, by applying (3.44),

B̃_{ζ_{j}}ψ(x) := (Z_{ζ_{j}}A_{ζ_{j}}Z_{ζ_{j}}^{−1}ψ)(x)=ψ(x)+ \frac{μ_{j}ζ_{j} |κ′_{ζ_{j}}(x)|^{\frac{1}{p}}}{π} ∫_{Γ_{1ζ_{j}}} (\frac{κ_{ζ_{j}}(x) − ζ_{j}}{τ − ζ_{j}} )^{γ_{j}}

× |(κ_{ζ_{j}}^{−1})′(τ)|^{\frac{1}{p}} ψ(κ_{ζ_{j}}^{−1}(τ))\frac{dτ}{ζ_{j}^{2} − τκ_{ζ_{j}}(x)}

= ψ(x) + \frac{μ_{j}ζ_{j}}{π} ∫_{0}^{1} |\frac{κ′_{ζ_{j}}(x)}{κ′_{ζ_{j}}(y)}|^{\frac{1}{p}} (\frac{κ_{ζ_{j}}(x) − ζ_{j}}{κ_{ζ_{j}}(y) − ζ_{j}} )^{γ_{j}} \frac{κ′_{ζ_{j}}(y)ψ(y)dy}{ζ_{j}^{2} − κ_{ζ_{j}}(y)κ_{ζ_{j}}(x)}

= ψ(x) − \frac{μ_{j}}{π} ∫_{0}^{1}[\frac{x}{y}]^{γ_{j}} [\frac{x + i}{y + i}]^{1−γ_{j}} [\frac{x^{2} + 1}{y^{2} + 1}]^{\frac{1}{p}} \frac{ψ(y)dy}{y + x} =g_{j}B_{ζ_{j}}g_{j}^{−1}ψ(x)

for x ∈ I, where

g_{j}(x) := \frac{(x + i)^{1−γ_{j}}}{(1 + x^{2})^{\frac{1}{p}}} , B_{ζ_{j}}ψ(x) := ψ(x) − \frac{μ_{j}}{π} ∫_{0}^{1} [\frac{x}{y}]^{γ_{j}} \frac{ψ(y)dy}{y + x} (3.58)

and g_{j}^{±} ∈ C^{∞}(I) satisfy condition (3.45). Therefore we can detach invertible operators g_{j}^{±1}I and study the equivalent operators

B_{ζ_{j}} : L_{p}(I) ⟶ L_{p}(I, {0}) for ζ_{j} ∈ Ξ_{ow} ,
B_{ζ_{j}} : L_{p}(I) ⟶ L_{p}(I) for ζ_{j} ∉ Ξ_{ow} .

The operator B_{ζ_{j}} can be lifted further, now by Z_{p}, to the following equivalent operator

W_{B_{ζ_{j}}} = Z_{p}B_{ζ_{j}}Z_{p}^{−1} : L_{p}(ℝ^{+}) ⟶ L⃗_{p}(ℝ^{+}) for ζ_{j} ∈ Ξ_{ow} ,

W_{B_{ζ_{j}}} = Z_{1}B_{ζ_{j}}Z_{1}^{−1} : L_{p}(ℝ^{+}) ⟶ L_{p}(ℝ^{+}) for ζ_{j} ∉ Ξ_{ow} (3.59)

(see (3.56)), which turn out to be convolutions. In fact,

(Z_{1}B_{ζ_{j}}Z_{1}^{−1}φ)(x) = φ(x) − \frac{μ_{j}}{π} ∫_{0}^{1} e^{− \frac{x}{p}} [\frac{e^{−x}}{y} ]^{γ_{j}} \frac{z^{− \frac{1}{p}} φ(− log z)dz}{z + e^{−x}}

= φ(x) − \frac{μ_{j}}{π} ∫_{0}^{∞}e^{(y−x)(\frac{1}{p} +γ_{j})} \frac{φ(y)dy}{1 + e^{y−x}} = W_{B_{ζ_{j}}} φ(x) ,

where

B_{ζ_{j}}(λ) := 1 − \frac{μ_{j}}{π} F_{t→λ} [\frac{e^{−(\frac{1}{p} +γ_{j})t}}{1 + e^{−t}} ] = 1 − \frac{μ_{j}i}{sinh π [(\frac{1}{p} + γ_{j}) i + λ]}

= 1 − \frac{μ_{j}}{sin π (\frac{1}{p} + γ_{j} − iλ)} , λ ∈ ℝ , j = 0, …, n  (3.60)

(see [Du1, Ch. II, § 1]).

First let ζ_{j} ∉ Ξ_{ow}; then (see (3.47))

B_{ζ_{j}}(λ) := 1 − \frac{sin π (\frac{1}{p} + γ′_{j})}{sin π (\frac{1}{p} + γ_{j} − iλ)} , λ ∈ ℝ , γ′_{j} ≠ γ_{j} .

From the property \overline{B_{ζ_{j}}(λ)} = B_{ζ_{j}}(−λ) we easily conclude that B_{ζ_{j}}(λ) = 0 implies λ = 0 and, due to conditions (3.48),

inf_{λ∈ℝ} |B_{ζ_{j}}(λ)| > 0 for ζ_{j} ∉ Ξ_{ow} .

Since B_{ζ_{j}}(λ) depends continuously on the parameter β_{j} := \frac{1}{p} + γ_{j}, 0 < β_{j} < 1, the index ind B_{ζ_{j}} might have at most 3 different values. For γ_{j} ∈ (γ′_{j}, 1 − γ′_{j}) we apply the homotopy

B_{j,μ}(λ) := 1 − μ\frac{sin π (\frac{1}{p} + γ′_{j})}{sin π (\frac{1}{p} + γ_{j} − iλ)} ≠ 0 for λ ∈ ℝ̇ , 0 ≤ μ ≤ 1 ,

since B_{j,μ}(λ) ≠ 0 for all λ ∈ ℝ̇ and μ ∈ [0, 1] we conclude ind B_{ζ_{j}} = ind B_{j,1} = ind B_{j,0} = 0.

For γ_{j} < min{γ′_{j}, 1 − γ′_{j}} and for γ_{j} > max{γ′_{j}, 1 − γ′_{j}} it is sufficient to calculate the index only for one value of parameters in each case. The images of the test functions on the complex plane are plotted on Fig. 5 in the Appendix with the arrows showing the orientation of the image when the argument λ ranges through ℝ from −∞ to ∞.

Finally we get

ind B_{ζ_{j}} = \{ \begin{matrix}1 & if & γ_{j} < min\{γ′_{j}, 1 − γ′_{j}\} , \\ 0 & if & γ_{j} ∈ (γ′_{j}, 1 − γ′_{j}) , \\ −1 & if & γ_{j} > max\{γ′_{j}, 1 − γ′_{j}\} , \end{matrix}  (3.61)

for ζ_{j} ∉ Ξ_{ow} (cf. [Du1, Du3]).

Next let ζ_{j} ∈ Ξ_{ow}. Then (see (3.47))

B_{ζ_{j}}(λ) := 1 − \frac{sin π (\frac{1}{p} + γ_{j})}{sin π (\frac{1}{p} + γ_{j} − iλ)} = \frac{λ}{λ − i}B_{j}^{0}(λ)

and

B_{j}^{0}(0) = lim_{λ→0} \frac{λ − i}{λ}B_{j}^{0}(λ) = −iB′_{j}(0)

= −π cot π (\frac{1}{p} + γ_{j}) ≠ 0 iff \frac{1}{p} + γ_{j} ≠ \frac{1}{2} .

Therefore,

inf_{λ∈ℝ} |B_{j}^{0}(λ)| > 0 iff \frac{1}{p} + γ_{j} ≠ \frac{1}{2} , ζ_{j} ∈ Ξ_{ow} .  (3.62)

Further we find easily that ind B_{j}^{0} might have at most two different values,

B_{j}^{0}(±∞) = 1, ±B_{j}^{0}(0) > 0 for ± (\frac{1}{p} + γ_{j} − \frac{1}{2}) > 0 and ± Im B_{j}^{0}(λ) > 0 for ±λ > 0. The images of the test functions on the complex plane are plotted on Fig. 6 in Appendix with the arrows showing the orientation of the image when the argument λ ranges through ℝ from −∞ to ∞. These tests show that

ind B_{j}^{0} = \{ \begin{matrix}1 & if & 0 < \frac{1}{p} + γ_{j} < \frac{1}{2} , \\ 0 & if & \frac{1}{2} < \frac{1}{p} + γ_{j} < 1 for ζ_{j} ∈ Ξ_{ow} . \end{matrix}

According to Theorems 2.5 and 3.5 we get: the operator W_{B_{ζ_{j}}} in (3.59) is Fredholm iff conditions (3.48) and (3.50) hold (see (3.51) and (3.62)) and Ind W_{B_{ζ_{j}}} = − ind B_{j}^{0} = σ_{j} for ζ_{j} ∈ Ξ_{ow} (see (3.61)), Ind W_{B_{ζ_{j}}} = − ind B_{ζ_{j}} = σ_{j} for ζ_{j} ∉ Ξ_{ow} (see (3.61)), where σ_{j} is defined in (3.52).

4 Elliptic boundary integral equations

Let Γ be as in § 1.1, the weight function ρ(t) be defined in (1.2).

For our purposes we need to define the order of cusp: σ_{j} > 0 is called the order of a cusp t_{j} ∈ Γ if there exists q_{j} ≠ 0 such that

arg \frac{τ_{−}(t_{j}, r) − t_{j}}{τ_{+}(t_{j}, r) − t_{j}} = q_{j}r^{σ_{j}} + o (r^{σ_{j}}) as r → 0,

where τ_{−}(t_{j}, r) ∈ Γ_{j−1} and τ_{+}(t_{j}, r) ∈ Γ_{j} are equidistant points |τ_{±}(t_{j}, r) − t_{j}| = r (see Fig. 3).

[figure: x_{+}; x_{−}; x_{+}; x_{−}; x; x; t_{j}; t_{j}; τ_{+}; τ_{−}; τ_{−}; τ_{+}; Γ_{j−1}; Γ_{j}; Γ_{j−1}; Γ_{j}; r; r]

Fig. 3 The obvious equivalent condition is

τ_{−}(t_{j}, r) − τ_{+}(t_{j}, r) = q_{j}r^{1+σ_{j}} + o (r^{1+σ_{j}}) as r → 0 .

Further equivalent definitions of the order can be found in [DLS1].

Throughout this section we assume the orders of cusps are all equal 1

if γ_{j} = 0 or γ_{j} = 2 , then σ_{j} = σ(t_{j}) = 1  (4.1)
for all j = 1, …, n

(see (3.2)) and will investigate the following integral equations:

A_{0}φ = a_{0}φ + a_{1}S_{Γ} + a_{2}W_{Γ,0}φ + a_{3}W_{Γ,0}^{∗}φ + a_{4}∂_{t}V_{Γ}φ = f  (4.2)

with N × N matrix coefficients a_{0}, a_{1}, a_{2}, a_{3}, a_{4} ∈ P C^{N×N}(Γ) (a_{0}, a_{1}, a_{2}, a_{3}, a_{4} ∈ P H_{μ}^{N×N}(Γ)) in the vector space L_{p}^{N}(Γ, ρ) (in the vector space (H_{μ}^{0})^{N}(Γ, ρ), respectively, provided Γ has no cusps 0 < γ_{j} < 2, j = 1, …, n)

A_{1}φ = a_{0}φ + a_{1}W_{Γ,0}φ + a_{2}W_{Γ,0}^{∗}φ = f , a_{0}, a_{1}, a_{2} ∈ P C^{N×N}(Γ) (4.3)

in the vector spaces L_{p}^{N}(Γ, ρ) and P C^{N}(Γ, ρ),

B_{0}φ = b_{0}φ + b_{1}W_{Γ,0}φ = g ,  b_{0}, b_{1} ∈ (P C^{1})^{N×N}(Γ) ⊂ C^{N×N}(Γ) (4.4)

in the vector spaces (W^{1})_{p}^{N}(Γ, ρ), C^{N}(Γ, ρ), (P C^{1})^{N}(Γ, ρ) and in (H^{0})_{μ+1}^{N}(Γ, ρ) (in the latter case cusps are absent and coefficients belong to P H_{μ}^{N×N}(Γ)).

Due to Theorems 1.5 and 1.6 respective conditions in (1.4) ensure boundedness of operators A_{0}, A_{1}, B_{0} in spaces listed above.

4.1 Equation (4.2) in the spaces L_{p}^{N}(Γ, ρ) and H_{μ}^{0}(Γ, ρ)

Let 𝕏(Γ) denote the space L_{p}^{N}(Γ, ρ) or, if cusps are absent, the space H_{μ}^{0}(Γ, ρ) and appropriate condition in (1.4) hold. Symbol of equation (4.2) in the space 𝕏(Γ) reads as follows

(A_{0})_{𝕏(Γ)} := ã_{0} + ã_{1}S_{𝕏(Γ)} + ã_{2}W_{𝕏(Γ)} + ã_{3}W_{𝕏(Γ)}^{∗} + ã_{4}(∂_{t}V)_{𝕏(Γ)} , (4.5)

where

ã := [ \begin{matrix}a(t + 0) & 0 \\ 0 & a(t − 0) \end{matrix} ] , a ∈ P C^{N×N}(Γ) , t ∈ Γ ,

W_{𝕏(Γ)}(t, λ, ξ) := \frac{1}{4} [S_{𝕏(Γ)}(t, λ, ξ) + \overline{S_{𝕏(Γ)}(t, −λ, −ξ)}] , λ, ξ ∈ ℝ ,

(∂_{t}V)_{𝕏(Γ)}(t, λ, ξ) := \frac{i}{4} [S_{𝕏(Γ)}(t, λ, ξ) − \overline{S_{𝕏(Γ)}(t, −λ, −ξ)}] ,

W_{𝕏(Γ)}^{∗}(t, λ, ξ) := −\frac{1}{4}[h̃^{−1}(t)S_{𝕏(Γ)}(t, λ, ξ)h̃(t) + h̃(t)\overline{S_{𝕏(Γ)}(t, −λ, −ξ)}h̃^{−1}(t)]

h̃(t) := \{ \begin{matrix}[ \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ] & if & t ≠ t_{1}, …, t_{n} , \\ [ \begin{matrix}1 & 0 \\ 0 & e^{π(γ_{j}−1)i} \end{matrix} ] & if & t = t_{j}, j = 1 …, n , \end{matrix}

S_{𝕏(Γ)}(t, λ, ξ) := \{ \begin{matrix}[ & coth π(iβ_{t} + λ) −\frac{e^{π(γ_{t}−1)(iβ_{t}+λ)}}{sinh π(iβ_{t} + λ)} \frac{e^{π(1−γ_{t})(iβ_{t}+λ)}}{sinh π(iβ_{t} + λ)} − coth π(iβ_{t} + λ) & ] \\ if 0 < γ_{t} < 2 , \\ (γ_{j} − 1) [ \begin{matrix}− sign λ & 2χ_{−}(λ)e^{2λ} \\ 2χ_{+}(λ)e^{−2λ} & sign λ \end{matrix} ] \\ if t = t_{j}, γ_{t} = γ_{j} = 0, 2 , λ ≠ 0 , \\ (γ_{j} −1) [ \begin{matrix}− coth π(iβ_{t} + ξ) & 1+coth π(iβ_{t} + ξ) \\ 1−coth π(iβ_{t} + ξ) & − coth π(iβ_{t} + ξ) \end{matrix} ] \\ if t = t_{j}, γ_{t} = γ_{j} = 0, 2, λ = 0 , \end{matrix} (4.6)

β_{t} := \{ \begin{matrix}\frac{1}{p} & if t ≠ t_{1}, …, t_{n} , & 𝕏(Γ) = L_{p}(Γ, ρ) , \\ \frac{1}{2} & if t ≠ t_{1}, …, t_{n} , & 𝕏(Γ) = H_{μ}^{0}(Γ, ρ) , \\ \frac{1}{p} + α_{t} & if t = t_{j} , & 𝕏(Γ) = L_{p}(Γ, ρ) , \\ α_{j} − μ & if t = t_{j} , & 𝕏(Γ) = H_{μ}^{0}(Γ, ρ) , \end{matrix}

γ_{t} := \{ \begin{matrix}1 & if & t ≠ t_{1}, …, t_{n} , \\ γ_{j} & if & t = t_{j} , \end{matrix} χ_{±}(λ) := \frac{1}{2}(1 + sign λ) .

Due to constraints (1.4) 0 < β_{t} < 1 for all t ∈ Γ and the symbol (A_{0})_{𝕏(Γ)}(t, λ, ξ) represents piecewise-continuous uniformly bounded function of all variables.

Although h(t_{j} − 0) = h(t_{j} + 0)e^{π(γ_{j}−1)i} (see (1.20)), we have dropped the factor h(t_{j} + 0) for t = t_{j} and the factor h(t) for t ≠ t_{1}, …, t_{n} in the definition of the symbol matrix h̃(t) above since it cancels out in the combined symbol (A_{0})_{𝕏(Γ)}(t, λ, ξ). In fact, h̃(t) and h̃^{−1}(t) enter the symbol (A_{0})_{𝕏(Γ)}(t, λ, ξ) only as the combination h̃^{−1}(t)S_{𝕏(Γ)}h̃(t) and the constant factors h^{−1}(t_{j} + 0), h(t_{j} + 0) cancel out.

Theorem 4.1 Let 𝕏(Γ) = L_{p}^{N}(Γ, ρ) or, if Γ has no cusps, 𝕏(Γ) = (H_{μ}^{0})^{N}(Γ, ρ)). Equation (4.2) is Fredholm in the space 𝕏(Γ) if and only if

inf_{t∈Γ, λ,ξ∈ℝ} | det (A_{0})_{𝕏(Γ)}(t, λ, ξ)| > 0 .  (4.7)

If condition (4.7) holds, then

Ind A_{0} = −\frac{1}{2π} [arg det (A_{0})_{𝕏(Γ)}(t, +∞, 0)]_{Γ}

− ∑_{j=1}^{n} \frac{1}{2π} \{ [arg det (A_{0})_{𝕏(Γ)}(t_{j}, λ, 0)]_{ℝ\\{0\}}

+ [arg det (A_{0})_{𝕏(Γ)}(t_{j}, 0, ξ)]_{ℝ} } .  (4.8)

Proof. Due to Lemma 3.1

A_{0} = a_{0}I + a_{1}S_{Γ} + \frac{a_{2}}{4} (S_{Γ} + VS_{Γ}V) + \frac{a_{3}}{4} (S_{Γ}^{∗} + VS_{Γ}^{∗}V) + \frac{a_{4}}{4} (S_{Γ} − VS_{Γ}V)

and the claimed result follows from [DLS1, Theorem 1.1] for the case 𝕏(Γ) = L_{p}^{N}(Γ, ρ) and from [Du6, Du7] for the case 𝕏(Γ) = (H_{μ}^{0})^{N}(Γ, ρ) (when cusps are absent) if we take into account the following:

I. The symbol of operator A_{0} defined in [DLS1] and in [Du8] (see also [Du3, Du5]) has a block-diagonal form

[ \begin{matrix}(A_{0})_{𝕏(Γ)}(t, λ, ξ) & 0 \\ 0 & \overline{(A_{0})_{𝕏(Γ)}(t, −λ, −ξ)} \end{matrix} ]

and it suffices to consider only the first block as a symbol of A_{0}. Due

to this change we should multiply the index formula by factor \frac{1}{2}.

Let us note that symbol would be a full matrix-function if the corresponding operator contains terms VS_{Γ}, VaI, aV or S_{Γ}V.

II. The dual operator W_{Γ,0}^{∗} to W_{Γ,0} is defined in (3.9) and the symbol for it is composed according to the usual rule (see (4.5)) with h̃(t) denoting the symbol of hI (see (3.7) for h(t)).

III. If B(t, λ, ξ) is the symbol of B, the symbol of B^{0} = VBV reads as

follows

B^{0}(t, λ, ξ) = \overline{B(t, −λ, −ξ)}  (4.9)

(see [DLS1, § 1]).

Corollary 4.2 For the operator

A_{0} = a_{0}I+a_{1}S_{Γ} = (a_{o}+a_{1})(P_{+}+GP_{−}), P_{±} := \frac{1}{2}(I±S_{Γ}), G := \frac{a_{o} − a_{1}}{a_{o} + a_{1}} ,

following conditions are equivalent to (4.7):

(i) inf_{t∈Γ} |a_{0}(t) ± a_{1}(t)| > 0;

(ii′) −2πβ_{t_{j}} < arg \frac{G(t_{j} − 0)}{G(t_{j} + 0)} < 2π(1 − β_{t_{j}}), j = 1, …, n, where β_{t_{j}} is

defined in (4.6);

(ii′′) (equivalent to ii’) G(t) has the representation

G(t) = G_{0}(t) ∏_{j=1}^{n}(t − z_{0})_{t_{j}}^{ν_{j}} , G_{o} ∈ C(Γ_{1}) ,

z_{0} ∈ Ω^{+} , −β_{t_{j}} < ν_{j} < 1 − β_{t_{j}} , j = 1, …, n

and (t − z_{0})_{t_{j}}^{ν_{j}} has the jump only at the point t_{j} ∈ Γ.

If conditions (i) and (ii’) (or (i) and (ii”)) hold,

Ind A = ind G_{0} .

4.2 Equation (4.3) in the spaces L_{p}^{N}(Γ, ρ) and P C^{N}(Γ, ρ)

Although equation (4.3) is a particular case of equation (4.2), in this case we can define substantially simpler symbol and consider equations also in the space P C^{N}(Γ, ρ).

Let 𝕏(Γ) denote either L_{p}^{N}(Γ, ρ) or P C^{N}(Γ, ρ) and (1.4) hold.

Symbol of equation (4.3) in the space 𝕏(Γ) reads as follows

(A_{1})_{𝕏(Γ)}(t, λ) := [ \begin{matrix}a_{0}(t + 0) & A_{+}(t, λ) \\ A_{−}(t, λ) & a_{0}(t − 0) \end{matrix} ] ,  (4.10)

where A_{±}(t, λ) := a_{1}(t ± 0)w_{𝕏(Γ)}(t, λ) + a_{2}(t ± 0)w_{𝕏(Γ)}^{∗}(t, λ) , t ∈ Γ, λ ∈ ℝ ,

w_{𝕏(Γ)}(t, λ) = \{ \begin{matrix}0 & if & t ≠ t_{1}, …, t_{n} , \\ \frac{sinh π(1 − γ_{j})(iβ_{j} + λ)}{sinh π(iβ_{j} + λ)} & if & t = t_{j}, 0 < γ_{j} < 2 , \\ \frac{γ_{j} − 1}{2}e^{−|λ|} & if & t = t_{j}, γ_{j} = 0, 2 , \end{matrix}

w_{𝕏(Γ)}^{∗}(t, λ) = \{ \begin{matrix}0 & if & t ≠ t_{1}, …, t_{n} , \\ \frac{sinh π(1 − γ_{j})[(β_{j} + 1)i + λ]}{sinh π(iβ_{j} + λ)} & if & t = t_{j}, 0 < γ_{j} < 2 , \\ \frac{γ_{j} − 1}{2}e^{−|λ|} & if & t = t_{j}, γ_{j} = 0, 2 , \end{matrix}

β_{j} := \{ \begin{matrix}\frac{1}{p} + α_{j} & if & 𝕏(Γ) = L_{p}^{N}(Γ, ρ) , \\ α_{j} & if & 𝕏(Γ) = P C^{N}(Γ, ρ) . \end{matrix}

Since 0 < β_{j} < 1, j = 1, …, n (see (1.4)) the symbol (A_{1})_{𝕏(Γ)}(t, λ) is a correctly defined 2N × 2N matrix-function, is continuous and

(A_{1})_{𝕏(Γ)}(t_{j}, −∞) = (A_{1})_{𝕏(Γ)}(t_{j}, +∞) = diag {a_{0}(t_{j} − 0), a_{0}(t_{j} + 0)} . Theorem 4.3 Let 𝕏(Γ) denote either L_{p}^{N}(Γ, ρ) or P C^{N}(Γ, ρ) and (1.4) hold.

Equation (4.3) is Fredholm in 𝕏(Γ) if and only if

inf_{t∈Γ, λ∈ℝ} | det (A_{1})_{𝕏(Γ)}(t, λ)| > 0 .  (4.11)

If condition (4.11) holds, then

Ind A_{1} = ∑_{j=1}^{n} \frac{1}{2π} [arg det (A_{1})_{𝕏(Γ)}(t_{j}, λ)]_{ℝ} .  (4.12)

Remark 4.4 It is easy to ascertain that condition (4.11) for a cusp t_{j} (with γ_{j} = 0, 2) reads as follows a_{0}(t_{j}−0)a_{0}(t_{j}+0)−[a_{1}(t_{j}−0)+a_{2}(t_{j}−0)][a_{1}(t_{j}+0)+a_{2}(t_{j}+0)]e^{−λ} ≠ 0, λ∈ℝ, or, equivalently,

[\frac{a_{0}(t_{j} − 0)a_{0}(t_{j} + 0)}{[a_{1}(t_{j} − 0) + a_{2}(t_{j} − 0)][a_{1}(t_{j} + 0) + a_{2}(t_{j} + 0)]}] > 1 .

Proof of Theorem 4.3. For 𝕏(Γ) = L_{p}^{N}(Γ, ρ) the proof can be derived from Theorem 4.1 (see (4.36) how to get symbol (4.10) from (4.6)). We expose independent proof to cover the case 𝕏(Γ) = P C^{N}(Γ, ρ) which is not covered by Theorem 4.1.

We suppose, as in the proof of Theorem 1.6 in § 2.3, that Γ has rectilinear parts Γ_{j}^{−}, Γ_{j}^{+} in some neighbourhood of all knots t_{1}, …, t_{n} except cusps; for a cusp γ_{j} = 0, 2 the right neighbourhood Γ_{j}^{+} is rectilinear, while the left one Γ_{j}^{−} is not (cf. (2.31) and Fig. 2). Such changes of the contour Γ cause a compact perturbation of equation (4.3) and does not influence the Fredholm properties as well as the index of equation (see [DLS1]).

Next we notice that operators W_{Γ_{0},0} and W_{Γ_{0},0}^{∗} are compact due to Corollary (1.6) since Γ_{0} has no angular points and cusps.

Applying the “macro localization”, described in [DLS1, Theorem 1.1, § 3.2], we find that A_{1} is Fredholm in 𝕏(Γ) iff det a_{0}(t) ≠ 0 for t ∈ Γ \ {t_{1}, …, t_{n}} and operators

A_{1,Γ_{j}^{0}} = a_{0,j}I + a_{1,j}W_{Γ_{j}^{0},0} + a_{2,j}W_{Γ_{j}^{0},0}^{∗} , Γ_{j}^{0} = Γ_{j}^{−} ∪ Γ_{j}^{+} ,  (4.13)

a_{k,j}(t) := \{ \begin{matrix}a_{k}(t_{j} − 0) & if & t ∈ Γ_{j}^{−} , \\ a_{k}(t_{j} + 0) & if & t ∈ Γ_{j}^{+} , k = 0, 1, 2 \end{matrix}

are Fredholm in X(Γ_{j}^{0}) for all j = 1, …, n; for the index we have

Ind A_{1} = ∑_{j=1}^{n} Ind A_{1,Γ_{j}^{0}} .  (4.14)

First let us consider the space 𝕏(Γ) = L_{p}^{N}(Γ, ρ) and 0 < γ_{j} < 2; without loss of generality t_{j} = 0.

The transformation Z_{γ_{j},β_{j}} with β_{j} := \frac{1}{p} + α_{j} has the inverse Z_{γ_{j},β_{j}}^{−1} (see (2.34)) and arranges an isomorphism

Z_{γ_{j},β_{j}} : L_{p}^{N}(Γ_{j}^{0}, |t|^{α_{j}}) ⟶ L_{p}^{2N}(ℝ^{+}) .  (4.15)

Obviously,

Z_{γ_{j},β_{j}}A_{1,Γ_{j}^{0}}Z_{γ_{j},β_{j}}^{−1} = [ \begin{matrix}a_{0}(t_{j} + 0) & 0 \\ 0 & a_{0}(t_{j} − 0) \end{matrix} ]

+\frac{1}{2} [ \begin{matrix}a_{1}(t_{j} + 0) & 0 \\ 0 & a_{1}(t_{j} − 0) \end{matrix} ] [ \begin{matrix}0 & N_{γ_{j}}^{0} − N_{−γ_{j}}^{0} \\ N_{γ_{j}}^{0} − N_{−γ_{j}}^{0} & 0 \end{matrix} ]

+\frac{1}{2} [ \begin{matrix}a_{2}(t_{j} + 0) & 0 \\ 0 & a_{2}(t_{j} − 0) \end{matrix} ]

× [ \begin{matrix}0 & e^{−π(γ_{j}−1)i}N_{γ_{j}}^{0} −e^{π(γ_{j}−1)i}N_{−γ_{j}}^{0} \\ e^{−π(γ_{j}−1)i}N_{γ_{j}}^{0} −e^{π(γ_{j}−1)i}N_{−γ_{j}}^{0} & 0 \end{matrix} ] ,

where

N_{±γ_{j}}^{0}φ(x) := \frac{1}{2πi} ∫_{0}^{∞} \frac{e^{−(x−y)β_{j}}φ(y)dy}{1 − e^{−(x−y)±πγ_{j}i}}  (4.16)

(see (4.6) where the symbols of hI, S_{Γ} and of VS_{Γ}V is possible to pick up). Thus, we get a convolution operator

Z_{γ_{j},β_{j}}A_{1,Γ_{j}^{0}}Z_{γ_{j},β_{j}}^{−1} = W_{(A_{1})_{𝕏(Γ)}(t_{j},·)} : L_{p}^{2N}(ℝ^{+}) ⟶ L_{p}^{2N}(ℝ^{+}) (4.17)

(cf. (2.6)) with the symbol (A_{1})_{𝕏(Γ)}(t_{j}, λ) defined in (4.10). In fact, N_{±γ_{j}}^{0} in (4.16) are convolutions with the symbols

N_{±γ_{j}}^{0}(λ) := \frac{1}{2πi} ∫_{−∞}^{∞} \frac{e^{iλy−β_{j}y}dy}{1 − e^{−y±πγ_{j}i}} = \frac{1}{2πi} ∫_{0}^{∞} \frac{t^{β_{j}−iλ−1}dt}{1 − e^{±πγ_{j}i}t}  (4.18)

= \frac{e^{±π(1−γ_{j})(β_{j}−iλ)}}{sinh π(iβ_{j} + λ)} = \frac{e^{±π(1−γ_{j})Λ_{t_{j}} i}}{sinh π(iβ_{j} + λ)} , β_{j} = \frac{1}{p} + α_{j} , λ ∈ ℝ ,

since −π < π − πγ_{j} < π (see [GR1, 3.194.4]). Thus, N_{±γ_{j}}^{0} = W_{N_{±γ_{j}}^{0}} and from (4.18) we get (4.17).

From (4.17) and from Theorem 2.6 follows: A_{1,Γ_{j}} is Fredholm iff

inf | det (A_{1})_{𝕏(Γ)}(t_{j}, λ)| > 0 , λ ∈ ℝ  (4.19)

and, for 0 < γ_{j} < 2

Ind A_{1,Γ_{j}} = − ind det (A_{1})_{𝕏(Γ)}(t_{j}, ·) .  (4.20)

Now let γ_{j} = 0 or γ_{j} = 2. Then Γ_{j}^{+} = [0, 1] and, due to condition (4.1) Γ_{j}^{−} can be taken as the quarter part of the circumference centered at z_{0} = \frac{1−γ_{j}}{2} i, starting at z_{1} = \frac{i}{2} + \frac{1−γ_{j}}{2} i and terminating at z_{2} = 0 (see Fig. 4).

[figure: 0; 0; \frac{1}{2} + \frac{i}{2}; \frac{1}{2} − \frac{i}{2}; z_{0}; z_{0}; Γ_{j}^{−}; Γ_{j}^{−}; Γ_{j}^{+}; Γ_{j}^{+}; γ_{j} = 2; γ_{j} = 0]

Fig. 4

The transformations

Z_{0}φ(x) := [ \begin{matrix}\frac{1}{x + 1}φ (\frac{1}{x + 1}) \\ \frac{1}{x − i + 1}φ (\frac{1}{x − i + 1}) \end{matrix} ] if γ_{j} = 0 ,

Z_{2}φ(x) := [ \begin{matrix}\frac{1}{x + 1}φ (\frac{1}{x + 1}) \\ \frac{1}{x + i + 1}φ (\frac{1}{x + i + 1}) \end{matrix} ] if γ_{j} = 2 , x ∈ ℝ^{+}, (4.21)

define isomorphisms

Z_{γ_{j}} : L_{p}^{N}(Γ_{j}^{0}, |t|^{α_{j}}) ⟶ L_{p}^{2N}(ℝ^{+}, (1 + x)α̃_{j}) , α̃_{j} := p − α_{j} − 2 (4.22)

and their inverses read

Z_{0}^{−1} [ \begin{matrix}ψ_{1} \\ ψ_{2} \end{matrix} ] (t) = χ_{+}^{0}(t)\frac{1}{t}ψ_{1}(t − 1) + χ_{−}^{0}(t)\frac{1}{t}ψ_{2}(\frac{1}{t} + i − 1) ,

Z_{2}^{−1} [ \begin{matrix}ψ_{1} \\ ψ_{2} \end{matrix} ] (t) = χ_{+}^{0}(t)\frac{1}{t}ψ_{1}(t − 1) + χ_{−}^{0}(t)\frac{1}{t}ψ_{2}(\frac{1}{t} − i − 1) ,

where χ_{+}^{0} and χ_{−}^{0} are the characteristic functions of Γ_{j}^{+} and Γ_{j}^{−}, respectively. Obviously 1 < α̃_{j} < p − 1 and

Z_{γ_{j}}A_{1,Γ_{j}^{0}}Z_{γ_{j}}^{−1} = [ \begin{matrix}a_{0}(t_{j} + 0) & 0 \\ 0 & a_{0}(t_{j} − 0) \end{matrix} ]

+\frac{γ_{j} − 1}{2} [ \begin{matrix}a_{1}(t_{j} + 0) + a_{2}(t_{j} + 0) & 0 \\ 0 & a_{1}(t_{j} − 0) + a_{2}(t_{j} − 0) \end{matrix} ]

× [ \begin{matrix}0 & N_{i} − N_{−i} \\ N_{i} − N_{−i} & 0 \end{matrix} ] ,

where

N_{±i}φ(x) := \frac{1}{2πi} ∫_{0}^{∞} \frac{φ(y)dy}{y − x ± i} = W_{N_{±i}}φ(x)  (4.23)

are convolutions with the symbols

N_{±i}(λ) := \frac{1}{2πi} ∫_{−∞}^{∞} \frac{e^{iλy}dy}{±i − y} = ∓χ_{±}(λ)e^{∓λ} ,  (4.24)

χ_{±}(λ) := \frac{1}{2}(1 ± sign λ) , λ ∈ ℝ .

Therefore,

Z_{γ_{j}}A_{1,Γ_{j}}Z_{γ_{j}}^{−1} = [ \begin{matrix}a_{0}(t_{j} + 0) & 0 \\ 0 & a_{0}(t_{j} − 0) \end{matrix} ]

+\frac{γ_{j} − 1}{2} [ \begin{matrix}a_{1}(t_{j} + 0) + a_{2}(t_{j} + 0) & 0 \\ 0 & a_{1}(t_{j} − 0) + a_{2}(t_{j} − 0) \end{matrix} ]

× [ \begin{matrix}0 & W_{N_{i}−N_{−i}} \\ W_{N_{i}−N_{−i}} & 0 \end{matrix} ] = W_{(A_{1})_{𝕏(Γ)}(t_{j},·)}

and, due to Theorem 2.6, A_{1,γ_{j}} is Fredholm iff (4.19) holds; the index formula (4.20) remains valid for γ_{j} = 0, 2.

Now let 𝕏(Γ) = P C^{N}(Γ, ρ).

For 0 < γ_{j} < 2 we consider the transformation Z_{γ_{j},δ_{j}}, defined in (3.20). Similarly to (4.15)–(4.18) we find that

Z_{γ_{j},δ_{j}} : C^{N}(Γ_{j}^{0}, |t|^{δ_{j}}) ⟶ C^{2N}(ℝ̇^{+})

defines an isomorphism and

Z_{γ_{j},δ_{j}}A_{1,Γ_{j}^{0}}Z_{γ_{j},δ_{j}}^{−1} = W_{(A_{1})_{𝕏(Γ)}(t_{j},·)}

is a Fredholm operator in the space C^{2N}(ℝ̇^{+}) iff

inf | det (A_{1})_{P C(Γ,ρ)}(t_{j}, λ)| > 0 , λ ∈ ℝ  (4.25)

and

Ind A_{1,Γ_{j}^{0}} = − [arg det (A_{1})_{P C(Γ,ρ)}(t_{j}, λ)]_{ℝ} ,  (4.26)

provided 0 < γ_{j} < 2.

For γ_{j} = 0 and γ_{j} = 2 (see Fig. 4) the transformation

Z_{γ_{j}} : P C^{N}(Γ_{j}^{0}, |t|^{δ_{j}}) ⟶ C^{2N}(ℝ̇^{+}, (1 + x)^{−δ_{j}+1}) , Γ_{j}^{0} = Γ_{j}^{−} ∪ Γ_{j}^{+} ,

defined in (4.21), arranges an isomorphism and

Z_{γ_{j}}A_{1,Γ_{j}^{0}}Z_{γ_{j}}^{−1} = W_{(A_{1})_{P C(Γ,ρ)}(t_{j},·)}

is Fredholm in the space P C^{2N}(ℝ̇^{+}, (1+x)^{−δ_{j}+1}) iff condition (4.25) holds (see Theorem 2.6); again the index is defined by (4.26).

Remark 4.5 If S_{𝕏}(Γ)(t, λ) is the symbol of S_{Γ} (see (4.5), (4.6), (4.10)), the symbol of VS_{Γ}V is \overline{S_{𝕏(Γ)}(t, −λ)}. We know the symbol of aI for a ∈ P C^{N×N}(Γ) (𝕏(Γ) = L_{p}^{N}(Γ, ρ) or 𝕏(Γ) = P C^{N}(Γ, ρ)). Therefore we can compose the symbol of equation

a_{0}φ + a_{1}W_{Γ,0}φ + a_{2}W_{Γ,0}^{∗}φ + ∑_{k=1}^{M} a_{2+k}W_{Γ,0}^{(k)}φ = f ,  (4.27)

a_{0}, …, a_{2+M} ∈ P C^{N×N}(Γ)

and prove Theorem 4.3 for equation (4.27).

4.3 Equation (4.4) in the spaces (W_{p}^{1})^{N}(Γ, ρ), (H_{μ+1}^{0})^{N}(Γ, ρ), C^{N}(Γ, ρ) and (P C^{1})^{N}(Γ, ρ)

Let 𝕏(Γ) denote one of the spaces mentioned in the headline.

To equation (4.4) in the space 𝕏(Γ) with smooth matrix coefficients we assign the symbol

(B_{0})_{𝕏(Γ)}(t, λ) := [ \begin{matrix}b_{0}(t) & b_{1}(t)e^{−π(1−γ_{j})i}w_{𝕏(Γ)}(t, λ) \\ b_{1}(t)e^{π(1−γ_{j})i}w_{𝕏(Γ)}(t, λ) & b_{0}(t) \end{matrix} ] , (4.28)

where

w_{𝕏(Γ)}(t, λ) = \{ \begin{matrix}0 & if & t ≠ t_{1}, …, t_{n} , \\ \frac{γ_{j} − 1}{2}e^{−|λ|} & if & t = t_{j}, γ_{j} = 0, 2 \end{matrix}

and w_{𝕏(Γ)}(t_{j}, λ) has following values for the different spaces 𝕏(Γ):

w_{W_{p}^{1}(Γ,ρ)}(t_{j}, λ) = \frac{sinh π(1 − γ_{j}) (\frac{i}{p} + α_{j}i − i + λ)}{2 sinh π (\frac{i}{p} + α_{j}i + λ)} ,

w_{H_{μ+1}^{0}(Γ,ρ)}(t_{j}, λ) = \frac{sinh π(1 − γ_{j}) (α_{j}i − μi − i + λ)}{2 sinh π (\frac{i}{p} + α_{j}i + λ)} ,

w_{P C^{1}(Γ,ρ)}(t_{j}, λ) = \frac{sinh π(1 − γ_{j})(α_{j}i − i + λ)}{2 sinh π(α_{j}i + λ)} ,

w_{C(Γ,ρ)}(t_{j}, λ) = \frac{sinh π(1 − γ_{j})(α_{j}i + λ)}{2 sinh π(α_{j}i + λ)} .

Due to conditions (1.4) the symbol (B_{0})_{𝕏(Γ)}(t, λ) is correctly defined, i.e., is a piecewise-continuous and uniformly bounded function of all variables.

Theorem 4.6 Let 𝕏(Γ) denote one of the following spaces (W_{p}^{1})^{N}(Γ, ρ), (H_{μ+1}^{0})^{N}(Γ, ρ) (if cusps are absent), (P C^{1})^{N}(Γ, ρ) or C^{N}(Γ, ρ) and conditions (1.4) hold.

Equation (4.4) is Fredholm in the space 𝕏(Γ) if and only if ^{9)}

inf_{t∈Γ, λ∈ℝ} | det (B_{0})_{𝕏(Γ)}(t, λ)| > 0 .  (4.29)

If condition (4.29) holds, then (cf. (4.12))

Ind B_{0} = − ∑_{j=1}^{n} \frac{1}{2π} [arg det (B_{0})_{𝕏(Γ)}(t_{j}, λ)]_{ℝ} .  (4.30)

Proof. For the space 𝕏(Γ) = C^{N}(Γ, ρ) the proof is verbatim the case 𝕏(Γ) = P C^{N}(Γ, ρ), exposed in Theorem 4.3.

Let

g(s) : [0, ℓ] ⟶ Γ , r(t) := g^{−1}(t) : Γ ⟶ [0, ℓ] , g(r(t)) ≡ t

be some parametrisation of Γ and the inverse to the parametrisation. The operator

Λ_{Γ}^{1}φ(t) := ∂_{t}φ(t) + r′(t)\frac{2πi}{ℓ}[φ(t) − φ(t_{n})] + φ(t_{n})e^{− \frac{2πi}{ℓ} r(t)}

= ∂_{s}φ_{0}(s) + \frac{2πi}{ℓ}[φ_{0}(s) − φ_{0}(0)] + φ_{0}(0)e^{− \frac{2πi}{ℓ} s} ,  (4.31)
s = r(t) , φ_{0}(s) = φ(g(s)) , 0 ≤ s ≤ ℓ , t ∈ Γ

9)An equivalent condition for a cusp see in Remark 4.4.

(see [Du3, § 2.2]) defines an isomorphism of spaces

Λ_{Γ}^{1} : (W_{p}^{1})^{N}(Γ, ρ) ⟶ L_{p}^{N}(Γ, ρ) .  (4.32)

and the inverse operator reads

Λ_{Γ}^{−1}ψ(t) := e^{− \frac{2πi}{ℓ} r(t)} ∫_{t_{n}t} e^{\frac{2πi}{ℓ} r(τ)}ψ(τ)dτ

+\frac{1}{ℓ} [1 − r(t)e^{− \frac{2πi}{ℓ} r(t)}] ∫_{Γ} e^{\frac{2πi}{ℓ} r(τ)}ψ(τ)dτ .  (4.33)

Namely,

Λ_{Γ}^{−1}Λ_{Γ}^{1}ψ = ψ , ψ ∈ L_{p}^{N}(Γ, ρ) , Λ_{Γ}^{1}Λ_{Γ}^{−1}φ = φ , φ ∈ (W_{p}^{1})^{N}(Γ, ρ) and

Λ_{Γ}^{1} = ∂_{t} + R , ∂_{t}, R : (W_{p}^{1})^{N}(Γ, ρ) ⟶ L_{p}^{N}(Γ, ρ) ,

where R is a compact operator.

Then the equation

B_{1}ψ := Λ_{Γ}^{1}B_{0}Λ_{Γ}^{−1}ψ = u ,  (4.34)
u, ψ ∈ L_{p}^{N}(Γ, ρ) , ψ := Λ_{Γ}^{1}φ , u = Λ_{Γ}^{1}f

is equivalent to (4.3). Since

∂_{t}Λ_{Γ}^{−1} = I + K , K : L_{p}^{N}(Γ, ρ) ⟶ L_{p}^{N}(Γ, ρ)

where K is a compact operator, applying (2.27). we get

B_{1} = (∂_{t} + R)(a_{0}I + a_{1}W_{Γ,0})Λ_{Γ}^{−1} = a_{0}I + a_{1}W_{Γ,0}^{(2)} + T

= a_{0}I + a_{1}[S_{Γ} + h^{−2}VS_{Γ}Vh^{2}I] + T ,  (4.35)
T = (a′_{0}I + a′_{1}W_{Γ,0})Λ_{Γ}^{−1} + R(a_{0}I + a_{1}W_{Γ,0})Λ_{Γ}^{−1}
+(a_{0}I + a_{1}W_{Γ,0}^{(2)})K : L_{p}^{N}(Γ, ρ) ⟶ L_{p}^{N}(Γ, ρ) .

T is a compact operator because Λ_{Γ}^{−1}, R and K are compact in L_{p}^{N}(Γ, ρ).

Symbol of the operator B_{1} in L_{p}^{N}(Γ, ρ), according to (4.6) and to Remark 4.5, reads

(B_{1})_{L_{p}(Γ,ρ)}(t, λ) := [ \begin{matrix}b_{0}(t) & 0 \\ 0 & b_{0}(t) \end{matrix} ] if t ≠ t_{1}, …, t_{n} ,

while for the knots t = t_{j} we get

(B_{1})_{L_{p}(Γ,ρ)}(t_{j}, λ) = [ \begin{matrix}b_{0}(t_{j}) & 0 \\ 0 & b_{0}(t_{j}) \end{matrix} ] + [ \begin{matrix}b_{1}(t_{j}) & 0 \\ 0 & b_{1}(t_{j}) \end{matrix} ]

× \{[ \begin{matrix}coth π(iβ_{j} + λ) & −\frac{e^{−π(1−γ_{j})(iβ_{j}+λ)}}{sinh π(iβ_{j} + λ)} \\ \frac{e^{π(1−γ_{j})(iβ_{j}+λ)}}{sinh π(iβ_{j} + λ)} & − coth π(iβ_{j} + λ) \end{matrix} ] + [ \begin{matrix}1 & 0 \\ 0 & e^{2π(1−γ_{j})i} \end{matrix} ]

× [ \begin{matrix}− coth π(iβ_{j} + λ) & \frac{e^{π(1−γ_{j})(iβ_{j}+λ)}}{sinh π(iβ_{j} + λ)} \\ −\frac{e^{−π(1−γ_{j})(iβ_{j}+λ)}}{sinh π(iβ_{j} + λ)} & coth π(iβ_{j} + λ) \end{matrix} ] [ \begin{matrix}1 & 0 \\ 0 & e^{−2π(1−γ_{j})i} \end{matrix} ]\} (4.36)

= [ \begin{matrix}b_{0}(t) & b_{1}(t)e^{−π(1−γ_{j})i}w_{W_{p}^{1}(Γ,ρ)}(t_{j}, λ) \\ b_{1}(t)e^{π(1−γ_{j})i}w_{W_{p}^{1}(Γ,ρ)}(t_{j}, λ) & b_{0}(t) \end{matrix} ] ,

w_{W_{p}^{1}(Γ,ρ)}(t_{j}, λ) := \frac{sinh π(1 − γ_{j})[i(β_{j} − 1) + λ)}{sinh π(iβ_{j} + λ)} ,

where β_{j} is defined in (1.79). Thus, we get the symbol defined in (4.28).

As proved above, the operator B_{1} (see (4.35)) in the space L_{p}^{N}(Γ, ρ) is equivalent (as a Fredholm operator) with B_{0} (see (4.4)) in the space (W_{p}^{1})^{N}(Γ, ρ) and their indices are equal Ind B_{0} = Ind B_{1} (see (4.34)). Thus, the symbol (B_{0})_{L_{p}(Γ,ρ)}(t_{j}, λ) := (B_{1})_{L_{p}(Γ,ρ)}(t, λ) defined in (4.36) is responsible for the Fredholm properties and the index of B_{0} in the space (W_{p}^{1})^{N}(Γ, ρ). Now the assertion follows from Theorem 4.1 (and from Theorem 4.3).

In the cases 𝕏(Γ) = (H_{μ+1}^{0})^{N}(Γ, ρ) and 𝕏(Γ) = (P C^{1})^{N}(Γ, ρ) the proofs follow verbatim the above exposed case 𝕏(Γ) = (W_{p}^{1})^{N}(Γ, ρ).

5 Conformal mapping and BVPs

Through this section we use the notation from § 1.1: for domains Ω^{±}, for their boundary Γ = ∂Ω^{±}, for the weight function ρ(t) (see (1.2), (1.4)), for the unit disk D_{1} and the unit circumference Γ_{1} = ∂D_{1}.

5.1 The Cisotti formula and its applications

In the present subsection we prove the Cisotti formula (5.5). It was published in 1921 (see [LS1, Ch. III, § 1, n^{o}. 44, Example 5]) and was rediscovered in [PK1] for piecewise-smooth curves by a different method (namely, by reducing the problem to the Riemann–Hilbert BVP for analytic functions). This formula has several interesting applications (see [KKP1]) and we will give some further applications below. Returning to the original method (see [LS1]) we prove the Cisotti formula for arbitrary domain bounded by a rectifiable Jordan curve.

Next Theorem is easy to ascertain if properties of conformal mapping ω : D_{1} ⟶ Ω^{+} and of the inverse to it ω^{−1} : Ω^{+} ⟶ D_{1} are taken into account: it suffices to change variables in the integrals ζ = ω(z), z = ω^{−1}(ζ). (see (1.47) and [Ev1, Ch. V, § 1]).

Theorem 5.1 The derivatives ω′(z) and (ω^{−1})′(ζ) of conformal mapping (1.46) and its inverse are both square integrable

∫_{Ω^{+}} |(ω^{−1})′(ζ)|^{2}|dζ| = π^{2} , ∫_{D_{1}} |ω′(z)|^{2}|dz| = ( mes Ω^{+})^{2} ,  (5.1)

while restricted to the boundaries they become absolutely integrable

∫_{Γ} |(ω^{−1})′(ζ)||dζ| = 2π , ∫_{Γ_{1}} |ω′(z)||dz| = mes Γ .  (5.2)

Next Theorem is a far non-trivial and subtle consequence of the foregoing theorem and we quote [Go1, p.p. 405–411] (see also [Ko1, Ch. I, II]) for rigorous proofs.

Theorem 5.2 If ω(z) in (5.1) is a conformal mapping of the unit disk D_{1} onto a simply connected domain Ω^{+} with the rectifiable Jordan boundary, then:

i. ω ∈ W_{1}^{1}(\overline{D_{1}}) (see § 1.1).

ii. ω(z) is absolutely continuous on the boundary Γ_{1}.

iii. For almost all t_{0} ∈ [0, 2π] there exists an angular (i.e., non-tangential) boundary limit →^{∧} of the function ω′(z)

lim_{re^{it}→^{∧}e^{it_{0}}} ω′(re^{it}) = −ie^{−it_{0}} \frac{dω(e^{iτ})}{dτ}|_{τ=t_{0}} .  (5.3)

The limit is denoted again by ω′(e^{it_{0}}).

Theorem 5.3 The derivatives ω′(z) of the conformal mapping ω : D_{1} → Ω^{+} has the following representation

ω′(z) = ω′(0) exp [\frac{1}{π} ∫_{|ζ|=1} \frac{β(ζ)dζ}{ζ − z} − \frac{1}{π} ∫_{|ζ|=1} β(ζ)\frac{dζ}{ζ} ] , z ∈ D_{1} , (5.4)

β(e^{it}) := α(t) − t − \frac{π}{2} = ϑ(t) − t for a.a. t ∈ [−π, π] , (5.5)

where α(t) and ϑ(t) = ϑ_{ω(e^{it})} = arg ν⃗(ω(e^{it})) denote the inclinations with respect to the abscissa axes of the tangent and the outer unit normal vectors at the point ω(e^{it}), respectively (see Fig. 1).

Proof. Due to (5.3) β_{0}(t) := β(e^{it}) in (5.5) exists for almost all t ∈ (−π, π] and for those t we have

ω′(e^{it}) = −ie^{−it} \frac{dω(e^{it})}{dt} = −ie^{iβ_{0}(t)} |\frac{dω(e^{it})}{dt}| .

Since ω′(e^{it}) ≠ 0 (ω(z) is a conformal mapping!)

Re [−i log ω′(e^{it})] = Im log ω′(e^{it}) = β_{0}(t) = β(e^{it}) for a.a. t ∈ (−π, π]

and the Schwartz integral recovers the analytic function −i log ω′(z) ∈ w_{1}^{1}(\overline{D_{1}}) by its real part on the boundary

−i log ω′(z) = iC + \frac{1}{2π} ∫_{−π}^{π} \frac{e^{iτ} + z}{e^{iτ} − z} β(e^{iτ})dτ

(see [Ko1, Ch. I, II], [LS1, §. 44]); therefore

ω′(z) = exp(−C) exp [\frac{i}{2π} ∫_{−π}^{π} \frac{e^{iτ} + z}{e^{iτ} − z} β(e^{iτ})dτ]

= C_{0} exp [−\frac{i}{2π} ∫_{−π}^{π} β(e^{iτ})dτ + \frac{i}{π} ∫_{−π}^{π} \frac{β(e^{iτ})e^{iτ} dτ}{e^{iτ} − z} ]

= C_{1} exp [\frac{1}{π} ∫_{|ζ|=1} \frac{β(ζ)dζ}{ζ − z} ]

and taking z = 0 easily locate the constant C_{1}:

C_{1} = ω′(0) exp [−\frac{1}{π} ∫_{|ζ|=1} β(ζ)\frac{dζ}{ζ} ] .

It is sometimes helpful to have the Cisotti formula (5.4) in the following equivalent form

ω′(re^{it}) = ω′(0) exp [i(P_{r}β_{0})(re^{it}) − (P̃_{r}β_{0})(re^{it}) − \frac{i}{2π} ∫_{π}^{π} β_{0}(τ)dτ] , (5.6)
0 < r < 1 , −π < t ≤ π ,

where P_{r}φ(z) is the Poisson operator and P̃_{r}φ(z) defines the adjoint harmonic function to P_{r}φ(z) (|z| < 1; see [Ko1, Ch. I]):

P_{r}φ(re^{it}) := \frac{1}{2π} ∫_{−π}^{π} \frac{1 − r^{2}}{1 − 2r cos(t − τ) + r^{2}} φ(τ)dτ ,

P̃_{r}φ(re^{it}) = \frac{1}{2π} ∫_{−π}^{π} \frac{r sin(t − τ)}{1 − 2r cos(t − τ) + r^{2}} φ(τ)dτ .

(5.7)

In the next theorem we have collected properties of the Poisson operator P_{r} and its adjoint P̃_{r} from [Ko1, Ch.I] and [Ko1, Ch. V, § D.1^{o}], necessary for further investigations.

Theorem 5.4 Let φ ∈ L_{p}(Π), Π := [−π, π], 1 ≤ p < ∞. Then

i. P_{r}φ(z) is harmonic in D_{1} and

∥P_{r}φ|L_{p}(Π)∥ ≤ ∥φ|L_{p}(Π)∥ , 0 ≤ r < 1 , lim_{r→0} ∥P_{r}φ − φ|L_{p}(Π)∥ = 0

ii. If φ(t) is continuous at some t_{0} ∈ Π, then

lim P_{r}φ(z) = φ(t_{0}) as z = re^{it} → e^{it_{0}} , r < 1.  (5.8)

iii. If Im g(t) ≡ 0, |g(t)| ≤ λ\frac{π}{2} for all t ∈ Π and λ < 1, then

∫_{−π}^{π} exp [|P̃_{r}g(e^{it})|] ≤ \frac{4π}{cos \frac{π}{2} λ} .  (5.9)

In particular, if φ ∈ C(Π), φ(−π) = φ(π), then the convergence in (5.8) is uniform (including convergence across tangent paths) with respect to t_{0} ∈ Π.

Remark 5.5 Easy to check that

P_{r}φ(z) = Im C_{Γ}φ(z) − \frac{1}{2π} ∫_{π}^{π} φ(τ)dτ ,

P̃_{r}φ(z) = − Re C_{Γ}φ(z) for Im φ(t) ≡ 0

(5.10)

(see (1.3)). Therefore for P̃_{r}φ we can apply the Plemelji formulae and get

lim_{z→^{∧}e^{it}} P̃_{r}φ(z) = −\frac{1}{2}φ(t) − \frac{1}{2π} ∫_{π}^{π} cot \frac{t − τ}{2} φ(τ)dτ ,  (5.11)

where the limit is angular (see (1.25)).

Corollary 5.6 If the inclination of the tangent vector to the boundary curve Γ is continuous on the entire boundary, derivative ω′(ζ) of the conformal mapping in (1.46) belongs to the Smirnov–Lebesgue space ω′ ∈ e_{p}(\overline{D_{1}}) for all 1 < p < ∞.

Proof. Due to the asserted conditions β_{0}(t) = β(e^{it}) in (5.5) is a continuous function β_{0} ∈ C(Π) and β_{0}(π) = β_{0}(−π) (see (5.5)); then

β_{0}(t) = β_{1}(t) + β_{2}(t) , β_{1} ∈ C^{1}(Π) , β_{1}(π) = β_{1}(−π) ,

|β_{2}(t)| ≤ \frac{π}{4p} = \frac{1}{2p} \frac{π}{2} for all t ∈ Π .

From (5.6) and (5.8) we have

∫_{−π}^{π} |ω′(re^{iτ})|^{p}dτ = ∫_{−π}^{π} | exp [|pP̃_{r}β_{1}(e^{iτ}) + pP̃_{r}β_{2}(e^{iτ})|] dτ

≤ \frac{4πC_{0}}{cos \frac{π}{4}} for all 0 < r ≤ 1

(see Corollary (5.6)), where

C_{0} = ∫_{−π}^{π} exp [|pP̃_{r}β_{1}(e^{iτ})|] dτ < ∞

since β_{1} ∈ H_{1}(Γ_{1}) and P_{r}β_{1}(ζ) is uniformly bounded with respect to 0 < r ≤ 1 (see (5.10) and (5.12) below).

Let us formulate several consequences of the foregoing results. First of them is a weak form of the Lindelöf theorem; in full generality it can be found e.g. [Ko1] and deals with arbitrary domain with Jordan boundary. For a domain with the smooth boundary it is proved e.g in [Go1, ] by different method and in [KKP1, p. 141]–as here, by using the Cisotti formula, but for piecewise-smooth curves.

Theorem 5.7 Let Ω^{±} be a simply connected domain with the rectifiable Jordan boundary Γ and ω(z) be a conformal mapping of the unit disk D_{1} onto the domain Ω^{+}. If the tangent exists at some point of the boundary t_{0} ∈ Γ, then the argument arg ω′(z) of the derivative of the conformal mapping is continuous at e^{iϑ_{0}} ∈ Γ_{1} = ∂D_{1}, where t_{0} = ω(e^{iϑ_{0}}):

lim arg ω′(z) = arg ω′(e^{iϑ_{0}}) = e^{iα(ϑ_{0})} as z → e^{iϑ_{0}} and z ∈ Ω^{+}.

In particular, if the tangent exists at each point of the boundary Γ, then arg ω′(x) is a continuous function on the closed domain \overline{Ω^{+}}.

Proof. The proof follows from Theorem 5.4.ii and from the equality

arg ω′(z) = arg ω′(0) + P_{r}β_{0}(t) − \frac{1}{2π} ∫_{π}^{π} β_{0}(τ)dτ ,  (5.12)
z = re^{iϑ} , 0 < r < 1 , −π < t ≤ π

(see (5.6)), where β_{0}(t) := β(e^{it}) is defined in (5.5).

Let 0 < μ < ∞ and 𝕏 be a compact sufficiently smooth manifold (we can take 𝕏 = [0, 1], 𝕏 = \overline{Ω^{+}} or even 𝕏 = Γ if the latter is sufficiently smooth). Norm in the Zygmund space ℤ^{μ}(𝕏) is defined as follows

||φ|ℤ^{μ}(𝕏)|| = ||f|C^{[μ]^{−}}(𝕏)||

+ ∑_{|α|=[μ]^{−}} sup_{x,x±h∈𝕏} \frac{|(∂^{α}φ)(x + h) − 2(∂^{α}φ)(x) + (∂^{α}φ)(x − h)|}{|h|^{\{μ\}^{+}}},

μ = [μ]^{−} + {μ}^{+} [μ]^{+} ∈ ℕ_{0}, 0 < {μ}^{+} ≤ 1,

where

∥f|C^{m}(𝕏)∥ = ∑_{|α|≤m} sup_{x∈𝕏} |∂^{α}f(x)|.

For μ ∈ ℝ^{+} \ ℕ the space ℤ^{μ}(𝕏) coincides with the generalized Hölder space H_{μ}(𝕏) (see [St1]), where (cf. § 1.1)

||φ|H^{μ}(𝕏)|| = ||f|C^{[μ]}(𝕏)|| + ∑_{|α|=[μ]} sup_{x,y∈𝕏,x≠y} \frac{|(∂^{α}φ)(y) − (∂^{α}φ)(x)|}{|y − x|^{\{μ\}}},

μ = [μ] + {μ}, [μ] ∈ ℕ_{0}, 0 < {μ} < 1.

ℤ^{μ}(Γ) coincides with the Besov space B_{∞,∞}^{μ}(Γ) (see [Tr1]) and the next theorem represents very particular case of [Du10, Theorem 3.2] (cf. Theorem 1.8 above). the assertion can readily be derived from the Muskhelishvili–Privalov theorem (the case μ < 1), proved in [Mu1, § 21], for non-integer μ ∈ ℝ and extended to integer values μ = 1, 2, … by the interpolation of Zygmund spaces (see [St1, Tr1] for theorems on interpolation).

Theorem 5.8 Let 0 < μ < ∞ and the boundary Γ = ∂Ω^{±} be m-smooth, where m ∈ ℕ_{0}, m ≥ μ.

The potential operators

C_{Γ} : ℤ^{μ}(Γ) ⟶ ℤ^{μ}(\overline{Ω^{±}}) ,
W_{Γ} : ℤ^{μ}(Γ) ⟶ ℤ^{μ}(\overline{Ω^{±}}) ,
V_{Γ} : ℤ^{μ}(Γ) ⟶ ℤ^{μ+1}(\overline{Ω^{±}})

(5.13)

(see (1.3) and (1.16)) are bounded.

In particular, if Γ is piecewise-smooth, we should restrict 0 < μ < 1.

Kellogg proved that if the inclination of the tangent vector is a Hölder continuous function with some exponent 0 < μ < 1 (so called Ljapunov boundary), then the derivative ω′(x) of the conformal mapping ω :; D_{1} → Ω^{+} also is Hölder continuous with the same exponent μ. The simple proof of this assertion is exposed in [KKP1, p. 143] and is based on the Cisotti formula. . Next theorem generalization the Kellogg theorem for μ ≥ 1.

Theorem 5.9 Let Ω^{±} be a simply connected domain and the inclination of the tangent to the boundary Γ = ∂Ω^{±} with respect to some fixed direction belongs to the Zygmund space ℤ^{μ}([0, ℓ]) for some 0 < μ < ∞.

If ω(z) is a conformal mapping of the unit disk D_{1} onto the domain Ω^{+}, then ω ∈ ℤ^{μ+1}(\overline{Ω^{+}}).

Proof. Let us consider the natural parametrisation of the curve Γ by the arc length parameter ζ(s)[0, ℓ] → Γ, ζ(0) = ζ(ℓ) (cf. (1.21)). The derivative ζ′(s) coincides with the unit tangent vector to Γ and the condition of the theorem can be written as follows

arg ζ′(·) ∈ ℤ^{μ}([0, ℓ]) , arg ∂_{s}^{k+1}ζ(ℓ−0) = arg ∂_{s}^{k+1}ζ(0+0) , k = 0, …, [μ] .

From (5.3) we find easily that

β(e^{it(s)}) = α(t(s)) − t(s) − \frac{π}{2} ,

where t(s) : [0, ℓ] → [−π, π] is a continuous function of the arc length parameter, defined by the equality ω(e^{it(s)}) Thus, we need to prove the implication

t′(·) ∈ ℤ^{μ}([0, ℓ]) ⇒ t′(s(ω(·))) ∈ ℤ^{μ}(Γ_{1}) .

From the asserted conditions β ∈ C(Γ_{1}) and from Corollary (5.6) we get ω′ ∈ e_{2}(\overline{D_{1}}). Then

|s(ω(ζ_{2})) − s(ω(ζ_{1}))| = |∫_{ζ_{1}}^{ζ_{2}}|ω′(ζ)||dζ|| ≤ (∫_{ζ_{1}}^{ζ_{2}}|ω′(ζ)|^{2}|dζ|)^{\frac{1}{2}} (∫_{ζ_{1}}^{ζ_{2}}|dζ|)^{\frac{1}{2}}

= C_{0}|ζ_{2} − ζ_{1}|^{\frac{1}{2}} .  (5.14)

Thus, s(ω(·)) ∈ H_{\frac{1}{2}} (Γ_{1}) and we find the first crude inclusion β(s(ω(·))) ∈ ℤ^{ν_{1}}(Γ_{1}) = H_{ν_{1}}(Γ_{1}) with ν_{1} = min \{\frac{1}{2}, \frac{μ}{2}\}. Due to Theorem 5.8 and to the Cisotti formula (5.4) we get another crude result ω′ ∈ ℤ^{ν_{1}}(\overline{D_{1}}). We return to (5.14) and find

|s(ω(ζ_{2})) − s(ω(ζ_{1}))| = ∫_{ζ_{1}}^{ζ_{2}}|ω′(ζ)||dζ| ≤ C_{1}|ζ_{2} − ζ_{1}| , ζ_{1}, ζ_{2} ∈ Γ ,

where C_{1} = sup_{ζ∈Γ_{1}} |ω′(ζ)|. the obtained estimate and the inclusion β(·) ∈

ℤ^{μ}([0, ℓ]) give the second crude inclusion β(s(ω(·))) ∈ ℤ^{ν_{2}}(Γ_{1}) with ν_{2} = min{1, μ}. Due to Theorem 5.8 and to formula (5.4) this inclusion yields ω′ ∈ ℤ^{ν_{2}}(\overline{D_{1}}), which is the final result provided 0 < μ ≤ 1.

If μ > 1 we take the derivative in (5.4)

ω′′(z) = ω′(0) exp [C_{Γ_{1}}β(z) − B_{0}] C_{Γ_{1}}β′(z) ,  (5.15)

B_{0} := \frac{1}{π} ∫_{|ζ|=1} β(ζ)\frac{dζ}{ζ} , z ∈ D_{1} .

On the other hand,

(∂_{ζ}β)(s(ω(ζ))) := \frac{dβ(s(ω(ζ)))}{dζ} = β′(s(ω(ζ)))(∂_{ζ}s)(ω(ζ)) .  (5.16)

From the equality

(∂_{ζ}s)(ω(ζ))| = |ω′(ζ)| ,

(cf. (5.13)), and from the inclusion ω′ ∈ ℤ^{1}(Γ_{1}) ⊂ H_{1}(Γ_{1}) we conclude (∂_{ζ}s)(ω(·)) ∈ H_{1}(Γ_{1}). This inclusion, together with β′(·) ∈ ℤ^{μ−1}([0, ℓ]) yields ∂_{ζ}β(s(ω(·))) ∈ ℤ^{ν_{3}}(Γ_{1}) (see (5.16) with ν_{3} = min{1, μ − 1}.

Again, we derive ω′′ ∈ ℤ^{ν_{3}}(\overline{Ω^{+}}) ⇒ ω ∈ ℤ^{ν_{3}+2}(\overline{Ω^{+}}) from (5.15) and from Theorem 5.8. The final result is obtained if μ ≤ 2 which implies ν_{3} = μ.

If μ > 2 we repeat the foregoing proof, taking further derivatives in (5.15) and accomplish the proof by the mathematical induction.

Corollary 5.10 (see also [KKP1]). The inequality

0 < C_{1} ≤ |\frac{log[ω(ζ) − ω(ζ_{j})]}{log[ζ − ζ_{j}]}| ≤ C_{2} < ∞  (5.17)

holds for all |ζ| = 1 provided t_{j} = ω(ζ_{j}) is not a cusp of Γ, i.e., if 0 < γ_{j} < 2.

Proof. Invoking the Lagrange theorem and Cisotti formula (5.5) with the Plemelji formula (the last one in (1.25)) we get

log[ω(ζ) − ω(ζ_{j})] = log(ζ − ζ_{j}) + log ω′(ζ′)

= C_{0} + log(ζ − ζ_{j}) + β(ζ′) + \frac{1}{π} ∫_{|τ|=1} \frac{β(τ)dτ}{τ − ζ′} ,

where ζ′ = ζ′(ζ, ζ_{j}) ∈ζ_{j}ζ^{⌣} and

C_{0} := log ω′(0) − \frac{1}{π} ∫_{|τ|=1} β(τ)\frac{dτ}{τ} = const .

The density β(τ) in the Cauchy integral is piecewise-Hölder continuous β ∈ H_{ν}(Γ_{1ζ_{j}} \ {ζ_{j}}) by condition and has the following jump at ζ_{j} ∈ Ξ_{ow}

\frac{β(ζ_{j} + 0) − β(ζ_{j} − 0)}{π} = 1 − γ_{j} .

Applying the estimates

\frac{1}{π} ∫_{|τ|=1} \frac{β(τ)dτ}{τ − ζ′} = −\frac{β(ζ_{j} + 0) − β(ζ_{j} − 0)}{π} log(ζ_{j} − ζ′) + β_{1}(ζ′)

= (γ_{j} − 1) log(ζ′ − ζ_{j}) + β_{1}(ζ′) = (γ_{j} − 1) log(ζ − ζ_{j}) + β_{2}(ζ′)

as ζ → ζ_{j}, |ζ′ − ζ_{j}|/|ζ − ζ_{j}| ≤ 1, where β_{1}, β_{2} ∈ H_{ν}(Γ_{1ζ_{j}} \ {ζ_{j}}) (see [Mu1, § 26]) we find

log[ω(ζ) − ω(ζ_{j})] = γ_{j} log(ζ − ζ′) + β_{3}(ζ_{j}, ζ)

with uniformly bounded β_{3}(ζ_{j}, ·) ∈ H_{ν}(Γ_{1ζ_{j}} \ {ζ_{j}}) when ζ → ζ_{j} and (5.17) follows.

5.2 Proof of Lemma 1.11

Repeating verbatim the arguments exposed in the proof of Theorem 1.16 (see (1.51)–(1.56)) we find easily that the Riemann–Hilbert problem (1.35) in the space Ψ ∈ E_{p}(\overline{Ω^{±}}, ρ), g ∈ L_{p}(Γ, ρ) is equivalent to the singular integral equation (1.50) in the space L_{p}(Γ).

Let, for definiteness, consider the domain Ω^{+}. The case of outer domain differs only by angles: we should replace all γ_{j} by 2−γ_{j} (i.e., by the measure of the complementary angle).

First let us prove that G ∈ P C(Γ_{1}); namely,

\frac{G(ζ_{j} − 0)}{G(ζ_{j} + 0)} = exp [−\frac{2π}{p} i + 2π (\frac{1}{p} + α_{j}) γ_{j}i] , j = 1, …, n . (5.18)

In fact, in the vicinity of t_{j} ∈ Γ we get

ρ_{0}(ω(ζ)) = ρ_{j}(ζ) [ω(ζ) − ω(ζ_{j})]^{α_{j}} = ρ_{j}(ζ) [ω′(ζ′_{j})]^{α_{j}} (ζ − ζ_{j})^{α_{j}} , ζ → ζ_{j} ,

ζ′_{j} := λ_{j}ζ_{j} + (1 − λ_{j})ζ , 0 < λ_{j} < 1 , ρ_{j}(ζ) = ∏_{k≠j} [ω(ζ) − ω(ζ_{k})]^{α_{k}}

(see (1.46), (1.48)) and ρ_{j}(t) is continuous at t_{j}: ρ_{j}(t_{j} − 0) = ρ_{j}(t_{j} + 0). Therefore,

\frac{G(ζ_{j} − 0)}{G(ζ_{j} + 0)} = \frac{ρ_{0}(ω(ζ_{j} − 0))}{\overline{ρ_{0}(ω(ζ_{j} − 0))}} \frac{\overline{ρ_{0}(ω(ζ_{j} + 0))}}{ρ_{0}(ω(ζ_{j} + 0))} [\frac{ω′(ζ_{j} − 0)}{\overline{ω′(ζ_{j} − 0)}} \frac{\overline{ω′(ζ_{j} + 0)}}{ω′(ζ_{j} + 0)}]^{\frac{1}{p}}

= \frac{(ζ_{j} − 0 − ζ_{j})^{α_{j}}}{\overline{(ζ_{j} − 0 − ζ_{j})^{α_{j}}}} \frac{\overline{(ζ_{j} + 0 − ζ_{j})^{α_{j}}}}{(ζ_{j} + 0 − ζ_{j})^{α_{j}}} [\frac{ω′(ζ_{j} − 0)}{\overline{ω′(ζ_{j} − 0)}} \frac{\overline{ω′(ζ_{j} + 0)}}{ω′(ζ_{j} + 0)}]^{\frac{1}{p} +α_{j}}

= exp \{2πα_{j}i + 2 (\frac{1}{p} + α_{j}) [arg ω′(ζ_{j} − 0) − arg ω′(ζ_{j} + 0)]i\} .

We proceed with the help of (1.66) (see also (5.6) and (5.8))

\frac{G(ζ_{j} − 0)}{G(ζ_{j} + 0)} = exp \{2πα_{j}i + 2 (\frac{1}{p} + α_{j}) [arg β(ζ_{j} − 0) − arg β(ζ_{j} + 0)]i\}

= exp [2πα_{j}i − 2π (\frac{1}{p} + α_{j}) (1 − γ_{j})i]

= exp [−\frac{2π}{p} i + 2π (\frac{1}{p} + α_{j}) γ_{j}i] .

The function ζ_{ζ_{j}}^{−ν_{j}} with

ν_{j} := −\frac{1}{p} + (\frac{1}{p} + α_{j}) γ_{j} , j = 1, …, n,  (5.19)

has discontinuity on the unit circumference if ν_{j} ≠ 0, ±1, … and this discontinuity we fix at the point ζ_{j} ∈ Γ_{1}; then

\frac{(ζ_{j} − 0)_{ζ_{j}}^{−ν_{j}}}{(ζ_{j} + 0)_{ζ_{j}}^{−ν_{j}}} = exp(−2πν_{j}i) = exp [\frac{2π}{p} i − 2π (\frac{1}{p} + α_{j}) γ_{j}i]

and consider the function

G_{0}(ζ) := G(ζ) ∏_{j=1}^{n} ζ_{ζ_{j}}^{−ν_{j}} , ζ ∈ Γ_{1} .  (5.20)

Let us prove that

G_{0} ∈ C(Γ_{1}) , |G_{0}(ζ)| = 1 for all |ζ| = 1 and ind G_{0} = 0 . (5.21)

Continuity on Γ_{1} follows from (5.18) G_{0}(ζ_{j} − 0) = G_{0}(ζ_{j} + 0), j = 1, …, n, while from (1.51), (5.20) we find immediately that the function is unimodular |G_{0}(ζ)| = 1.

To prove the last claim ind G_{0} = 0 we rewrite (5.20) as follows

G_{0}(ζ) = G(ζ) ∏_{j=1}^{n} (−\frac{ζ − ζ_{j}}{ζ_{j} (\overline{ζ} − \overline{ζ_{j}})})_{ζ_{j}}^{−ν_{j}} = G(ζ) ∏_{j=1}^{n}(−ζ_{j})^{ν_{j}} (\frac{ζ − ζ_{j}}{\overline{ζ} − \overline{ζ_{j}}} )^{−ν_{j}}

= c_{0} \frac{ρ_{0}(ω(ζ))}{\overline{ρ_{0}(ω(ζ))}} [\frac{ω′(ζ)}{\overline{ω′(ζ)}}]^{\frac{1}{p}} (\frac{ζ − ζ_{j}}{\overline{ζ} − \overline{ζ_{j}}} )^{−ν_{j}} , ζ ∈ Γ_{1} , c_{0} := ∏_{j=1}^{n}(−ζ_{j})^{ν_{j}} .

Thus, G_{0}(ζ) has a continuous extension inside the unit disk

G_{0} ∈ C(\overline{D_{1}}) , G_{0}(z)| ≠ 0 for all z ∈ \overline{D_{1}}

and the homotopy

G_{0,r}(ζ) := G_{0}(rζ) , |ζ| = 1 , 0 ≤ r ≤ 1

is continuous, non-vanishing and connects the function G_{0} = G_{0,1} with the constant G_{0,0} = G_{0}(0), confirming ind G_{0} = 0.

Let us rewrite (5.20) in the form

G(ζ) := G_{0}(ζ) ∏_{j=1}^{n} ζ_{ζ_{j}}^{ν_{j}} , ζ ∈ Γ_{1} .  (5.22)

From (5.22), (5.21) and Corollary 4.2 we find that conditions (1.36) (1.32) are necessary and sufficient the singular integral equation (1.50) to have a solution, because under these conditions A is Fredholm in L_{p}(Γ_{1}) and has the following index

Ind A = ∑_{ν_{j}>1} 1 ,

since ind ζ_{ζ_{j}}^{ν_{j}} = 0 when ν_{j} < 1 and ind ζ_{ζ_{j}}^{ν_{j}} = 1 when ν_{j} > 1.

In conclusion it is worth mentioning that the problem has alwayes nonnegative index Ind_{L_{p}(Γ_{1})}A ≥ 0, i.e., is surjective if it is Fredholm.

5.3 Proof of Theorem 1.26

As in the proof of Lemma 1.11 in § 5.2 we treat, for definiteness, the domain Ω^{+}. In the case of outer domain we have just to replace all γ_{j} by 2 − γ_{j}.

First we suppose Ξ_{ow} = ∅. Then

G(ζ) := G̃_{0}(ζ) ∏_{j=1}^{n} ζ_{ζ_{j}}^{ν̃_{j}} , ζ ∈ Γ_{1} ,  (5.23)

ν̃_{j} := \{ \begin{matrix}ν_{j} & for & \frac{1}{p} ≤ ν_{j} , \\ ν_{j} − 1 & for & \frac{1}{p} > ν_{j} , \end{matrix} G̃_{0}(ζ) := G_{0}(ζ)ζ^{σ} , σ := ∑_{\begin{matrix}ζ_{j}∉Ξ_{ow} \\ ν_{j}>1 \end{matrix}}1

(see (5.22) and (1.93)–(1.95)). Due to Corollary 4.2 equation (1.50) is Fredholm in L_{p}(Γ_{1}) if and only if conditions (1.94) hold and then

Ind A = ind G̃_{0} = σ = ∑_{\begin{matrix}ζ_{j}∉Ξ_{ow} \\ ν_{j}>1 \end{matrix}}1

(see (5.23)). Proposition (1.95) follows because the equivalent Riemann– Hilbert BVP (1.55) has non-negative index σ ≥ 0 and has the trivial kernel dim Ker A = 0 (if the index is positive, BVP (1.55) would have the trivial cokernel dim Coker A = 0; cf. [GK1, Kh1, Mu1]).

Now let Ξ_{ow} ≠ ∅ and consider equation (1.50) for g_{0} ∈ L_{p}(Γ_{1}, Ξ_{ow}), φ ∈ L_{p}(Γ_{1}) or, what is equivalent, consider operator (1.93). We should start by proving boundedness of (1.93). First note that due to Lemma 1.25 the operator

\frac{G − 1}{2}K : L_{p}(Γ) ⟶ P C(Γ) ⊂ L_{p}(Γ, Ξ_{ow})

is bounded and since is one-dimensional influences neither the Fredholm property nor the index of the operator

A = P_{Γ_{1}}^{+} + G(ζ)P_{Γ_{1}}^{−} + \frac{G(ζ) − 1}{2}K .

Therefore, in what follows, we ignore this summand in the operator A and put

A = P_{Γ_{1}}^{+} + G(ζ)P_{Γ_{1}}^{−} .

Let Γ_{1j}^{±} := {ζ ∈ Γ_{1} : ± Im (ζ/ζ_{j}) > 0} be the semi-circles having ±ζ_{j} as endpoints and χ_{ζ_{j}}^{±}(ζ) be the corresponding characteristic functions (ζ ∈ Γ_{1}).

Boundedness of the operator in (1.93) follows from the boundedness of the restrictions

A_{ζ_{j}} := (1 − χ_{ζ_{j}})I + g_{1}χ_{ζ_{j}}Aχ_{ζ_{j}}g_{1}^{−1}I : L_{p}(Γ_{1ζ_{j}}) ⟶ L_{p}(Γ_{1ζ_{j}}, {ζ_{j}}) , (5.24)

g_{1}(ζ) := \frac{|ζ + ζ_{j}|^{\frac{2}{p}}}{ζ + ζ_{j}}

for all ζ_{j} ∈ Ξ_{ow}. Easy to ascertain that if

G_{ζ_{j}}(ζ) := G(ζ_{j} − 0)χ_{ζ_{j}}^{−}(ζ) + G(ζ_{j} + 0)χ_{ζ_{j}}^{+}(ζ) ,

then

G(ζ) − G_{ζ_{j}}(ζ) = O(|ζ − ζ_{j}|) as ζ → ζ_{j} ∈ Ξ_{ow} .  (5.25)

Due to Lemma 1.22 the operator

A_{ζ_{j}} − A_{ζ_{j}}^{0} = g_{1}[G(ζ) − G_{ζ_{j}}]P_{Γ_{1}}^{−}g_{1}^{−1}I : L_{p}(Γ_{1ζ_{j}}) ⟶ L_{p}(Γ_{1}, {ζ_{j}}) ,
A_{ζ_{j}}^{0} := g_{1}[P_{Γ_{1}}^{+} + G_{ζ_{j}}P_{Γ_{1}}^{−}]g_{1}^{−1}I  (5.26)

is bounded. Moreover, if ε > 0 and χ_{ζ_{j},ε} is the characteristic function of the neighbourhood Γ_{1ζ_{j},ε} ⊂ Γ_{1ζ_{j}}, contracting to {ζ_{j}} as ε → 0, then

∥χ_{ζ_{j},ε}(A_{ζ_{j}} − A_{ζ_{j}}^{0})|L(L_{p}(Γ_{1}), L_{p}(Γ_{1}, {ζ_{j}}))∥
≤ M_{0}∥χ_{ζ_{j},ε}(G − G_{ζ_{j}})|L_{∞}(Γ_{1})∥^{1−δ} ,

which yields

lim_{ε→0} ∥χ_{ζ_{j},ε}(A_{ζ_{j}} − A_{ζ_{j}}^{0})|L(L_{p}(Γ_{1}), L_{p}(Γ_{1}, {ζ_{j}}))∥ = 0 as ε → 0

since δ > 0 is arbitrary. Thus, boundedness of operator (1.93) follows from the boundedness of the operator

A_{ζ_{j}}^{0} : L_{p}(Γ_{1}) ⟶ L_{p}(Γ_{1}, {ζ_{j}}) .

The boundedness of A^{0}, in its turn, follows from the estimates

∥Ṽ_{ζ_{j}}A_{ζ_{j}}^{0}φ|L_{p}(Γ_{1}^{+})∥ ≤ M_{j}∥φ|L_{p}(Γ_{1})∥ , M_{j} < ∞ for all ζ_{j} ∈ Ξ_{ow}

(see (1.90), (1.92)).

We can suppose, that

G_{ζ_{j}}(ζ) = \{ \begin{matrix}e^{\frac{2π}{p} i} & for & ζ ∈ Γ_{1}^{+} , \\ 1 & for & ζ ∈ Γ_{1}^{−} \end{matrix}  (5.27)

In fact, the operator

B_{j} := P_{Γ_{1}}^{+} + G^{−1}(ζ_{j} − 0)P_{Γ_{1}}^{−}  (5.28)

has constant coefficients G(ζ_{j} − 0) =const≠ 0 and due to the following well-known properties of the singular projections

(P_{Γ_{1}}^{±})^{2} = P_{Γ_{1}}^{±} , P_{Γ_{1}}^{+}P_{Γ_{1}}^{−} = P_{Γ_{1}}^{−}P_{Γ_{1}}^{+} = 0 , P_{Γ_{1}}^{−} + P_{Γ_{1}}^{+} = I  (5.29)

is invertible B_{j}^{−1} = P_{Γ_{1}}^{+} + G(ζ_{j} − 0)P_{Γ_{1}}^{−}, B_{j}^{−1}B_{j} = B_{j}B_{j}^{−1} = I. Therefore it suffices to prove boundedness of the operator

AB_{j} = P_{Γ_{1}}^{+} + G^{−1}(ζ_{j} − 0)GP_{Γ_{1}}^{−} : L_{p}(Γ_{1}) ⟶ L_{p}(Γ_{1}, Ξ_{ow})  (5.30)

instead of (1.93). The coefficient G^{0}(ζ) := G^{−1}(ζ_{j} − 0)G(ζ) of the operator (5.30) has limits G^{0}(ζ_{j} − 0) = 1 and G^{0}(ζ_{j} + 0) = e^{\frac{2π}{p} i} and corresponding local representative G_{ζ_{j}}^{0}(ζ) has the form (5.27).

Let us apply the isomorphisms Z_{pζ_{j}} = Z_{p}Z_{ζ_{j}} defined in (3.31)–(3.36). Since

A_{0,ζ_{j}} := Z_{pζ_{j}}A_{ζ_{j}}^{0}Z_{pζ_{j}}^{−1} = \frac{1}{2}(I + Z_{pζ_{j}}g_{1}S_{Γ_{1}}g_{1}^{−1}Z_{pζ_{j}}^{−1})

+\frac{1}{2} [ \begin{matrix}e^{\frac{2π}{p} i} & 0 \\ 0 & 1 \end{matrix} ] (I − Z_{pζ_{j}}g_{1}S_{Γ_{1}}g_{1}^{−1}Z_{pζ_{j}}^{−1})  (5.31)

it suffices to find Z_{pζ_{j}}g_{1}S_{Γ_{1ζ_{j}}} g_{1}^{−1}Z_{pζ_{j}}^{−1}. Applying (3.44) we proceed as follows

Z_{ζ_{j}}g_{1}S_{Γ_{1}}g_{1}^{−1}Z_{ζ_{j}}^{−1}ψ(x) = \frac{1}{πi} ∫_{−∞}^{∞} |\frac{κ′_{ζ_{j}}(x)}{κ′_{ζ_{j}}(y)}|^{\frac{1}{p}} \frac{κ_{ζ_{j}}(y)}{κ_{ζ_{j}}(x)} \frac{κ′_{ζ_{j}}(y)φ(y)dy}{κ_{ζ_{j}}(y) − κ_{ζ_{j}}(x)}

= \frac{1}{πi} ∫_{−∞}^{∞} \frac{φ(y)dy}{y − x} = S_{ℝ}φ(x)

and further

Z_{pζ_{j}}S_{Γ_{1}}Z_{pζ_{j}}^{−1} = Z_{p}(Z_{ζ_{j}}S_{Γ_{1}}Z_{ζ_{j}}^{−1})Z_{p}^{−1} = [ \begin{matrix}S_{p} & −N_{p} \\ N_{p} & −S_{p} \end{matrix} ] ,

where

S_{p}φ(x) := \frac{1}{πi} ∫_{−∞}^{∞} \frac{e^{− \frac{x−y}{p}} φ(y)dy}{1 − e^{−(x−y)}} = W_{s_{p}}^{0} ,

s_{p}(λ) := coth π (\frac{i}{p} + λ) , λ, x ∈ ℝ,

N_{p}φ(x) := \frac{1}{πi} ∫_{−∞}^{∞} \frac{e^{− \frac{x−y}{p}} φ(y)dy}{1 + e^{−(x−y)}} = W_{n_{p}}^{0} ,

n_{p}(λ) := \frac{1}{sinh π (\frac{i}{p} + λ)} .  (5.32)

Easy to ascertain that

A_{0,ζ_{j}} := Z_{pζ_{j}}A_{ζ_{j}}^{0}Z_{pζ_{j}}^{−1} = [ \begin{matrix}\frac{1}{2}(I + S_{p}) + e^{\frac{2π}{p} i} \frac{1}{2}(I − S_{p}) & \frac{1}{2}(e^{\frac{2π}{p} i} − 1)N_{p} \\ 0 & I \end{matrix} ]

= W_{A_{0}(ζ_{j},·)}  (5.33)

(see (5.31)–(5.32)), where A_{0}(ζ_{j}, ·) is the symbol. Since

cosh z sinh w − sinh z cosh w = sinh(w − z) , z, w ∈ ℂ , we find the symbol

A_{0}(ζ_{j}, λ) := [ \begin{matrix}e^{\frac{π}{p} i} [cosh \frac{π}{p} i − sinh \frac{π}{p} i coth π (\frac{i}{p} + λ)] & e^{\frac{π}{p} i} \frac{sinh \frac{π}{p} i}{sinh π(\frac{i}{p} +λ)} \\ 0 & 1 \end{matrix} ]
= [ \begin{matrix}e^{\frac{π}{p} i}_{\frac{sinh πλ}{sinh π(\frac{i}{p} +λ)}} & e^{\frac{π}{p} i} \frac{sinh \frac{π}{p} i}{sinh π(\frac{i}{p} +λ)} \\ 0 & 1 \end{matrix} ]  (5.34)

Applying (3.30), (3.32) we get

Z_{pζ_{j}} Ṽ_{ζ_{j}}Z_{pζ_{j}}^{−1} = g_{j}Ṽ_{∞}I =g_{j} [ \begin{matrix}e^{− \frac{π}{p} i}V_{∞} & −V_{∞} \\ 0 & 0 \end{matrix} ]+R_{j} =g_{j}Wṽ_{0} + R_{j} (5.35)

ṽ_{0} := [ \begin{matrix}e^{− \frac{π}{p} i} [1 − g_{−}^{−1}(λ)] & g_{−}^{−1}(λ) − 1 \\ 0 & 0 \end{matrix} ] = [ \begin{matrix}e^{− \frac{π}{p} i} \frac{i}{λ} & −\frac{i}{λ} \\ 0 & 0 \end{matrix} ]

(see (3.8)). From (5.33) and (5.35) we have

Z_{pζ_{j}} Ṽ_{ζ_{j}}A_{ζ_{j}}^{0}φ = g_{j}Ṽ_{∞}A_{0,ζ_{j}}(ψ_{1}, ψ_{2})^{⊤} + R_{j}A_{0,ζ_{j}}(ψ_{1}, ψ_{2})^{⊤} = (V_{∞}W_{a_{1j}}^{0}ψ_{1}

+V_{∞}W_{a_{2j}}^{0}ψ_{2}, 0)^{⊤} = (W_{b_{1j}}^{0}ψ_{1} + W_{b_{2j}}ψ_{2}, 0)^{⊤} + R_{j}A_{0,ζ_{j}}(ψ_{1}, ψ_{2})^{⊤} , (5.36)

where (ψ_{1}, ψ_{2})^{⊤} := Z_{pζ_{j}}φ and

a_{1j}(λ) := \frac{sinh πλ}{sinh π (\frac{i}{p} + λ)} , b_{1j}(λ) := \frac{−i sinh πλ}{λ sinh π (\frac{i}{p} + λ)} ,  (5.37)

a_{2j}(λ) := \frac{sinh \frac{π}{p} i}{sinh π (\frac{i}{p} + λ)} , b_{2j}(λ) := \frac{i [sinh \frac{π}{p} i − sinh π (\frac{i}{p} + λ)]}{λ sinh π (\frac{i}{p} + λ)} ,

because V_{∞} = W_{g_{−}^{−1}}^{0} − I = W_{g_{−}^{−1}−1}^{0} (see (3.23)) and g_{−}^{−1}(λ) − 1 = −i/λ.

The functions b_{kj}(λ) satisfy conditions (3.4) and, therefore, b_{kj} ∈ P C_{p}(ℝ̇). This yields the estimate

∥Ṽ_{ζ_{j}}A_{ζ_{j}}^{0}φ|L_{p}(Γ_{1ζ_{j}}^{+})∥ ≤ ∥Z_{pζ_{j}}^{−1}∥∥Z_{pζ_{j}} Ṽ_{ζ_{j}}A_{ζ_{j}}^{0}φ|L_{p}(ℝ^{+})∥  (5.38)

= ∥Z_{pζ_{j}}^{−1}∥∥g_{j}Ṽ_{∞}A_{0,ζ_{j}}(ψ_{1}, ψ_{2})^{⊤}|L_{p}^{2}(ℝ)∥ + ∥R_{j}A_{0,ζ_{j}}(ψ_{1}, ψ_{2})^{⊤}|L_{p}^{2}(ℝ)∥

≤ ∥Z_{pζ_{j}}^{−1}∥ [ ∑_{k=1,2} ∥g_{j}W_{b_{kj}}^{0}ψ_{k}|L_{p}(ℝ)∥ + ∥R_{j}A_{0,ζ_{j}}(ψ_{1}, ψ_{2})^{⊤}|L_{p}^{2}(ℝ)∥]

≤ M_{j}^{0}∥(ψ_{1}, ψ_{2})^{⊤}|L_{p}^{2}(ℝ)∥ = M_{j}^{0}∥Z_{pζ_{j}}φ|L_{p}^{2}(ℝ)∥ ≤ M_{j}∥φ|L_{p}(Γ_{1ζ_{j}})∥ .

Estimates (5.30) follow and imply the boundedness in (1.93).

To prove the Fredholm criteria (1.94) we apply the localization method, due to I.Gohberg and N.Krupnik (see [GK1, RS1]) modified for operators between two different spaces (see [Du9, § 3]). We skip over exposing details of the method because they are well-known and even modified version is operating with similar objects–localization classes, local equivalence, local representatives, local invertibility etc.

We choose a standard covering system of localizing classes {M_{ζ}}_{ζ∈Γ_{1}}, where M_{ζ} consists of all multiplication operators vI by smooth functions v ∈ C^{∞}(Γ_{1}), |v(t)| ≤ 1 (t ∈ Γ_{1}) which are equal 1 in some neighbourhood of ζ. Boundedness of operators vI ∈ M_{ζ} in the space L_{p}(Γ_{1}) is trivial, while in L_{p}(Γ_{1}.Ξ_{ow}) follows from Lemma 1.22. Another essential property– compactness of commutators

[vI, A] = vA − AvI : L_{p}(Γ_{1}) ⟶ L_{p}(Γ_{1}.Ξ_{ow}) ,

which is a bounded operator already, follows from the well-known criteria of compactness in L_{p}(Γ_{1}) space modified with the help of Lemma 1.22

∫_{Γ_{1}} [∫_{Γ_{1}} | log(ζ − ζ_{j})k(ζ, τ)|^{p′}|dτ|]^{\frac{p}{p′}} |dζ| < ∞ ,

since the kernel k(ζ, τ) of the commutator [vI, A] is a uniformly bounded function.

As a local representative of A at a regular point ζ_{0} ≠ ζ_{1}, … ζ_{n} we choose the following operator

A ∼^{M_{ζ_{0}}} A_{ζ_{0}} := P_{Γ_{1}}^{+} + G(ζ_{0})P_{Γ_{1}}^{−} , A_{ζ_{0}} : L_{p}(Γ_{1}) ⟶ L_{p}(Γ_{1})  (5.39)

with the constant (“frozen” at ζ_{0}) coefficient. This operator is invertible A_{ζ_{0}}^{−1} := P_{Γ_{1}}^{+} + G^{−1}(ζ_{0})P_{Γ_{1}}^{−} (see (5.28), (5.29)).

Before localizing at the point ζ_{j}, where the coefficient has discontinuity G(ζ_{j}+0) ≠ G_{j}(ζ_{j}−0) ≠ 0 let us simplify the operator by taking composition with the invertible operator B_{j} in (5.28). The composition AB_{j} has the same image Im A_{j} = Im A and due to invertibility of B_{j} we can consider the composition

A_{j} := P_{Γ_{1}}^{+} + G^{−1}(ζ_{j} − 0)GP_{Γ_{1}}^{−} : L_{p}(Γ_{1}) ⟶ L_{p}(Γ_{1}, Ξ_{ow})  (5.40)

instead of (1.93). The local representative of the operator (5.40) at the point ζ_{j} ∈ Γ_{1} is chosen as follows

A_{j} ∼^{M_{ζ_{j}}} A_{γ_{j},ζ_{j}}^{0} := g_{1}[P_{Γ_{1}}^{+} + G_{ζ_{j}}P_{Γ_{1}}^{−}]g_{1}^{−1}I , G_{ζ_{j}}(t) := e^{2πν_{j}i}χ_{j}^{+} + χ_{j}^{−} ,(5.41)

ν_{j} = \frac{1}{p} − (\frac{1}{p} + α_{j}) γ_{j} ,

since G^{−1}(ζ_{j} −0)G(ζ_{j} +0) = e^{2πν_{j}i} (see (5.18) and note that in (5.24)–(5.27) we have taken the outward peak which means γ_{j} = 0); χ_{j}^{±} in (5.41) are the characteristic functions of the semi-circumference ± Im (ζ/ζ_{j}) ≥ 0.

The localized operator A_{γ_{j},ζ_{j}}^{0} should be considered in the appropriate local spaces:

A_{γ_{j},ζ_{j}}^{0} : L_{p}(Γ_{1}) ⟶ L_{p}(Γ_{1})  if 0 < γ_{j} ≤ 2 ,
A_{0,ζ_{j}}^{0} : L_{p}(Γ_{1}) ⟶ L_{p}(Γ_{1}, {ζ_{j}})  if γ_{j} = 0, (i.e., ζ_{j} ∈ Ξ_{ow}) .

(5.42)

The lifted operators (cf. (5.33))

\begin{matrix}A_{γ_{j},ζ_{j}} := Z_{pζ_{j}}A_{γ_{j},ζ_{j}}^{0}Z_{pζ_{j}}^{−1} =: L_{p}^{2}(ℝ) ⟶ L_{p}^{2}(ℝ) & if 0 < γ_{j} ≤ 2 , \\ A_{0,ζ_{j}} := Z_{pζ_{j}}A_{0,ζ_{j}}^{0}Z_{pζ_{j}}^{−1} =: L_{p}^{2}(ℝ) ⟶ L_{p}^{2}(ℝ, \{∞\}) & if γ_{j} = 0 \end{matrix} (5.43)

are convolutions

A_{γ_{j},ζ_{j}} = W_{A_{γ_{j}} (ζ_{j},·)}^{0}  (5.44)

(cf. (5.33)) with the symbols

A_{γ_{j}}(ζ_{j}, λ) := [ \begin{matrix}\frac{1}{2}(I + s_{p}(λ)) + e^{2πν_{j}i} \frac{1}{2}(I − s_{p}(λ)) & \frac{1}{2}(e^{2πν_{j}i} − 1)n_{p}(λ) \\ 0 & I \end{matrix} ]

= [ \begin{matrix}e^{πν_{j}i} & 0 \\ 0 & 1 \end{matrix} ] [ 0  1^{\frac{sinh π(\frac{i}{p} −ν_{j}i+λ)}{sinh π(\frac{i}{p} +λ)} \frac{sinh πν_{j}i}{sinh π(\frac{i}{p} +λ)}} ]
= [ \begin{matrix}e^{πν_{j}i} & 0 \\ 0 & 1 \end{matrix} ] [ 0  1^{\frac{sinh π(λ−(\frac{1}{p} +α_{j})γ_{j}i)}{sinh π(\frac{i}{p} +λ)} \frac{sinh π[\frac{i}{p} −(\frac{1}{p} +α_{j})γ_{j}]}{sinh π(\frac{i}{p} +λ)}} ] . (5.45)

The operator A_{γ_{j},ζ_{j}} = W_{A_{γ_{j}} (ζ_{j},·)}^{0} for γ_{j} ≠ 0 is invertible in L_{p}^{2}(ℝ) iff

A_{0}(ζ_{j}, λ)=e^{πν_{j}i} \frac{sinh π (λ−(\frac{1}{p} + α_{j}) γ_{j}i)}{sinh π (\frac{i}{p} + λ)} ≠ 0⟹(\frac{1}{p} +α_{j}) γ_{j} ≠ 1, (5.46)

as it follows from (5.45) and (2.5). Condition (1.94) is justified.

Now let γ_{j} = 0; then ν_{j} = \frac{1}{p} and (see (5.45))

A_{0}(ζ_{j}, λ) := [ \begin{matrix}e^{\frac{π}{p} i}_{\frac{sinh πλ}{sinh π(\frac{i}{p} +λ)}} & e^{\frac{π}{p} i} \frac{sinh \frac{π}{p} i}{sinh π(\frac{i}{p} +λ)} \\ 0 & 1 \end{matrix} ]  (5.47)

(cf. (5.36), (5.37)). The operator

Ṽ_{∞} := [ \begin{matrix}I + V_{∞} & 0 \\ 0 & I \end{matrix} ] [ \begin{matrix}e^{− \frac{π}{p} i}I & −I \\ 0 & I \end{matrix} ]=[ \begin{matrix}e^{− \frac{π}{p} i}(I + V_{∞}) & −(I + V_{∞}) \\ 0 & I \end{matrix} ]

= W_{v_{0}}^{0} : L_{p}^{2}(ℝ, \{∞\}) ⟶ L_{p}^{2}(ℝ) , v_{0}(λ) := [ \begin{matrix}e^{− \frac{π}{p} i} \frac{λ−i}{λ} & −\frac{λ−i}{λ} \\ 0 & 1 \end{matrix} ] (5.48)

arranges an isomorphism (see (1.92) and (5.35)). Therefore, the operator A_{0,ζ_{j}} in (5.43), (5.44) (the case γ_{j} = 0) is equivalent to the operator

Ṽ_{∞}W_{A_{0}(ζ_{j},·)}^{0} = W_{v_{0}}^{0}W_{A_{0}(ζ_{j},·)}^{0} = W_{A_{0}^{1}(ζ_{j},·)}^{0} : L_{p}^{2}(ℝ) ⟶ L_{p}^{2}(ℝ) , (5.49)

where

A_{0}^{1}(ζ_{j}, λ) := v_{0}(λ)A_{0}(ζ_{j}, λ) := [ 0  1^{\frac{(λ−i) sinh πλ}{λ sinh π(\frac{i}{p} +λ)} \frac{(λ−i)[sinh \frac{π}{p} i−sinh π(\frac{i}{p} +λ)]}{λ sinh π(\frac{i}{p} +λ)}} ]

Obviously, A_{0}^{1}(ζ_{j}, ·) ∈ P C_{p}^{2×2}(ℝ) (see (3.4)) and

det A_{0}^{1}(ζ_{j}, λ) = \frac{(λ − i) sinh πλ}{λ sinh π (\frac{i}{p} + λ)} ≠ 0 for all λ ∈ ℝ̇ .

W_{A_{0}^{1}(ζ_{j},·)} is invertible in L_{p}^{2}(ℝ) and yields invertibility of the local representatives A_{ζ_{j}} in (5.42) for all ζ_{j} ∈ Ξ_{ow}.

Thus, under conditions (1.94), all local representatives of the operator (1.93) are invertible, which implies that (1.93) is Fredholm.

To prove the index formula (1.95) we recall the representation (5.23) and arrange a homotopy sending the function G(ζ) to

g(ζ) := g_{0}(ζ) ∏_{ζ_{j}∈Ξ_{ow}} g_{j}(ζ) , ζ ∈ Γ_{1} ,  (5.50)

where the functions g_{0}(ζ) and g_{j}(ζ) have the same images (accept the same

values) as G̃_{0}(ζ) and ζ_{j}^{−1}ζζ_{j}^{\frac{1}{p}}, respectively, when ζ ranges over Γ_{1} (we remind that ν̃_{j} = ν_{j} = \frac{1}{p} as soon as γ_{j} = 0). More of this, supports of g_{0} − 1 and of g_{j} − 1 are “squeezed” and belong to Γ_{10} and Γ_{ζ_{j}}, respectively. Therefore,

supp (g_{k} − 1) ∩ (ζ) supp (g_{j} − 1) := ∅ for all k ≠ j ,

g_{0} ∈ C^{1}(Γ_{1}) , ind g_{0} = ind G̃_{0} = σ .

g_{j} ∈ C^{1}(Γ_{1} \ \{ζ_{j}\}) , \frac{g_{j}}{ζζ_{j}^{\frac{1}{p}}} ∈ C^{1}(Γ_{1}) , ind \frac{g_{j}}{ζζ_{j}^{\frac{1}{p}}} = 0 .

(5.51)

To arrange such homotopy we just define

G_{ϑ}(ζ) := G̃_{0}(ζ) [\frac{g_{0}(ζ)}{G̃_{0}(ζ)}]^{ϑ} ∏_{ζ_{j}∈Ξ_{ow}} [\frac{g_{j}(ζ)}{ζζ_{j}^{\frac{1}{p}}} ]^{ϑ} ζζ_{j}^{\frac{1}{p}} ∏_{ζ_{j}∉Ξ_{ow}} ζ_{ζ_{j}}^{(1−ϑ)ν_{j}} (5.52)

for 0 ≤ ϑ ≤ 1. Since the functions [g_{0}(ζ)/G̃_{0}(ζ)]^{ϑ} and [g_{j}(ζ)/ζζ_{j}^{\frac{1}{p}}]^{ϑ} are continuous for all 0 ≤ ϑ ≤ 1 (see (5.51)) and the exponents (1 − ϑ)ν̃_{j} continue to satisfy conditions (1.94) when ζ_{j} ∉ Ξ_{ow}, we get the operators

A_{ϑ} := P_{Γ_{1}}^{+} + G_{ϑ}P_{Γ_{1}}^{−} : L_{p}(Γ_{1}) ⟶ L_{p}(Γ_{1}, Ξ_{ow})

which are Fredholm for all 0 ≤ ϑ ≤ 1. Then these operators maintain the index

Ind A = Ind A_{0} = Ind A_{1} = Ind (P_{Γ_{1}}^{+} + G_{1}P_{Γ_{1}}^{−}) .  (5.53)

Due to the disjoint supports of g_{j} − 1 (see (5.51)) we get

A_{1} = P_{Γ_{1}}^{+} + G_{1}P_{Γ_{1}}^{−} = D_{0} ∏_{ζ_{j}∈Ξ_{ow}} D_{j} ,

D_{0} := P_{Γ_{1}}^{+} + g_{0}P_{Γ_{1}}^{−} : L_{p}(Γ_{1}) ⟶ L_{p}(Γ_{1}) ,  (5.54)
D_{j} := P_{Γ_{1}}^{+} + g_{j}P_{Γ_{1}}^{−} : L_{p}(Γ_{1}) ⟶ L_{p}(Γ_{1}, {ζ_{j}})

and the operators commute D_{j}D_{k} = D_{k}D_{j}. Therefore

Ind A_{1} = Ind D_{0} + ∑_{ζ_{j}∈Ξ_{ow}} Ind D_{j} = ind G̃_{0} + ∑_{ζ_{j}∈Ξ_{ow}} Ind D_{j} (5.55)

and to justify the index formula (1.95) we just have to show that

Ind D_{j} = 0 for all ζ_{j} ∈ Ξ_{ow} .  (5.56)

By the condition the image of g_{j}(ζ) coincides with the image of ζ_{j}^{−1}ζζ_{j}^{\frac{1}{p}} which means that

|g_{j}(ζ)| = 1 , g_{j}(ζ_{j} − 0) = e^{\frac{2π}{p} i} , g_{j}(ζ_{j} + 0) = +1 .  (5.57)

Let us consider the operator

H_{ζ_{j}} = I + Z_{pζ_{j}}^{−1}W_{H}^{0}Z_{pζ_{j}} = Z_{pζ_{j}}^{−1}W_{1+H}^{0}Z_{pζ_{j}} : L_{p}(Γ_{1}) ⟶ L_{p}(Γ_{1}, {ζ_{j}}) ,

1 + H(λ) = [ \begin{matrix}e^{\frac{π}{p} i} \frac{λ}{λ−i} & e^{\frac{π}{p} i} \\ 0 & 1 \end{matrix} ] .  (5.58)

The lifted operator (see (3.37)–(3.40))

Z_{pζ_{j}}H_{ζ_{j}}Z_{pζ_{j}}^{−1} = W_{1+H}^{0} : L_{p}^{2}(ℝ) ⟶ L_{p}^{2}(ℝ, {∞})  (5.59)

is invertible. In fact,

[1 + H(λ)]^{−1} = [ \begin{matrix}e^{− \frac{π}{p} i} \frac{λ−i}{λ} & −\frac{λ−i}{λ} \\ 0 & 1 \end{matrix} ] = v_{0}(λ)

(cf. (5.48)) and therefore Ṽ_{∞} in (5.48) is the inverse operator to (5.58)

Ṽ_{∞}W_{1+H}^{0} = W_{v_{0}(1+H)}^{0} = I  (5.60)

(see (3.23)).

For the parameter-dependent operator

R_{ϑ} := (1−ϑ)B_{j}−ϑe^{μi}H_{ζ_{j}} : L_{p}(Γ_{1}) ⟶ L_{p}(Γ_{1}, {ζ_{j}}) , 0 ≤ ϑ ≤ 1 , (5.61)

where \frac{2π}{p} ≤ μ ≤ 2π will be chosen later, the local representatives for ζ_{0} ∉ Ξ_{ow} read

R_{ϑ} ∼^{M_{ζ_{0}}} R_{ϑ,ζ_{0}} = g_{1}^{−1}[(1 − ϑ)P_{Γ_{1}}^{+} + (1 − ϑ)g_{j}(ζ_{0})P_{Γ_{1}}^{−}]g_{1}I − ϑe^{μi}I

=]g_{1}^{−1}[(1 − ϑ − ϑe^{μi})P_{Γ_{1}}^{+} + [(1 − ϑ)g_{j}(ζ_{0})]]g_{1} − ϑe^{μi}]P_{Γ_{1}}^{−}
: L_{p}(Γ_{1}) ⟶ L_{p}(Γ_{1}) , (5.62)

while for ζ_{j} ∈ Ξ_{ow} we get

R_{ϑ} ∼^{M_{ζ_{j}}} R_{ϑ,ζ_{j}} =]g_{1}^{−1}[(1 − ϑ)P_{Γ_{1}}^{+} + (1 − ϑ)G_{ζ_{j}}(ζ_{0})P_{Γ_{1}}^{−}]]g_{1}I

−ϑe^{μi}Z_{pζ_{j}}^{−1}W_{1+H}Z_{pζ_{j}}
= Z_{pζ_{j}}^{−1}W_{R_{ϑ}(ζ_{j},·)}Z_{pζ_{j}} : L_{p}(Γ_{1}) ⟶ L_{p}(Γ_{1}, {ζ_{j}})  (5.63)

(cf. (5.39), (5.41), (5.42)–(5.47)), where G_{ζ_{j}}(ζ) = +1 for Im (ζζ_{j}^{−1}) > 0 and G_{ζ_{j}}(ζ) = e^{\frac{2π}{p} i} for Im (ζζ_{j}^{−1}) < 0 (cf. (5.27), (5.41)) and

R_{ϑ}(ζ_{j}, λ) = (1 − ϑ)A(ζ_{j}, λ) − ϑe^{μi}[1 + H(λ)] .  (5.64)

The operators R_{ϑ,ζ_{0}} in (5.62) are invertible having constant non-vani-shing coefficients

1 − ϑ(1 + e^{μi}) ≠ 0 , (1 − ϑ)g_{j}(ζ_{0}) − ϑe^{μi} ≠ 0 for all 0 ≤ ϑ ≤ 1 , ζ_{0} ≠ ζ_{j}

provided μ > π (we remind that g_{j}(ζ_{0}) = e^{μi} with \frac{2π}{p} ≤ μ ≤ 2π is impossible since ζ_{0} ≠ ζ_{j}). The inverse operator is written as in (5.28)–(5.30).

The operators R_{ϑ,ζ_{j}} in (5.63) are also invertible because the lifted operators

W_{R_{ϑ}(ζ_{j},·)}^{0} = Z_{pζ_{j}}R_{ϑ,ζ_{j}}Z_{pζ_{j}}^{−1} : L_{p}^{2}(ℝ) ⟶ L_{p}^{2}(ℝ, {∞})  (5.65)

are invertible. To verify this we should apply the isomorphism Ṽ_{∞} from (5.48)

Ṽ_{∞}W_{R_{ϑ}(ζ_{j},·)}^{0} = W_{v_{0}R_{ϑ}(ζ_{j},·)}^{0} : L_{p}^{2}(ℝ) ⟶ L_{p}^{2}(ℝ)  (5.66)

(see (3.23)), where

v_{0}(λ)R_{ϑ}(ζ_{j}, λ) = (1 − ϑ)A_{0}^{1}(ζ_{j}, λ) − ϑe^{μi}v_{0}(λ)[1 + H(λ)] = (1 − ϑ)A_{0}^{1}(ζ_{j}, λ) − ϑe^{μi}I

= [ \begin{matrix}(1 − ϑ)\frac{(λ−i) sinh πλ}{λ sinh π(\frac{i}{p} +λ)} − ϑe^{μi} & (1 − ϑ)(λ − i)\frac{sinh \frac{π}{p} i−sinh π(\frac{i}{p} +λ)}{λ sinh π(\frac{i}{p} +λ)} \\ 0 & 1 − ϑ(1 + e^{μi}) \end{matrix} ]

(see (5.49), (5.60)). The image of the function

h_{p}(λ) := \frac{(λ − i) sinh πλ}{λ sinh π (\frac{i}{p} + λ)} = \frac{sinh πλ}{λ} [\frac{λ sinh πλ cos \frac{π}{p} − cosh πλ sin \frac{π}{p}}{|sinh π (\frac{i}{p} + λ)|^{2}}

−i \frac{sinh πλ cos \frac{π}{p} + λ cosh πλ sin \frac{π}{p}}{|sinh π (\frac{i}{p} + λ)|^{2}} ] , \overline{h_{p}(λ)} = h_{p}(−λ)

on the complex plane ℂ when λ ranges through ℝ is a continuous curve connecting points h_{p}(±∞) = e^{± \frac{π}{p} i} on the unit circumference and passing through h_{p}(0) = −\frac{π}{sin \frac{π}{p}} < 0 on the negative semi-axes. Easy to ascertain, that \frac{π}{p} ≤ arg h_{p}(λ) ≤ 2π − \frac{π}{p} and the constraints

max \{π, 2π − \frac{π}{p} , \frac{2π}{p} \} < μ ≤ 2π

on the parameter μ ensure the ellipticity

det v_{0}(λ)R_{ϑ}(ζ_{j}, λ) = [(1 − ϑ)h_{p}(λ) − ϑe^{μi}][1 − ϑ(1 + e^{μi})] ≠ 0
for all 0 ≤ ϑ ≤ 1 , λ ∈ ℝ̇,

which yields invertibility of the operator in (5.66) (see (2.5)).

Thus, the operator R_{ϑ} in (5.61) depends on the parameter ϑ ∈ [0, 1] continuously and connects the operator B_{j} with the invertible one −e^{μi}H_{ζ_{j}} in the group of Fredholm operators, which yields equality of indices

Ind B_{j} = Ind R_{0} = Ind R_{1} = Ind H_{ζ_{j}} = 0 .

5.4 Proof of Theorem 1.23

First suppose Γ has no peaks T_{pk} = ∅.

Let us write the symbols of equations (1.39) and (1.40) in the spaces 𝕏^{m}(Γ, ρ) = W_{p}^{m}(Γ, ρ), H_{μ+m}^{0}(Γ, ρ), C(Γ, ρ), P C^{m}(Γ, ρ) according to (4.6), (4.10) and (4.28)

(A_{±})_{𝕏^{m}(Γ,ρ)}(t, λ) := \{ \begin{matrix}\frac{1}{p} [ \begin{matrix}±1 & 0 \\ 0 & ±1 \end{matrix} ] & if t ≠ t_{1}, …, t_{n} , \\ \frac{1}{p} [ \begin{matrix}±1 & H_{m,j}(λ) \\ H_{m,j}(λ) & ±1 \end{matrix} ] & if t = t_{j} , \end{matrix}

where m = 0, 1, β_{j} is defined in (1.79) and

H_{m,j}(λ) := \frac{sinh π(1 − γ_{j})(iβ_{j} − mi + λ)}{sinh π(iβ_{j} + λ)} .  (5.67)

According to Theorems 4.1, 4.3 and 4.6 equations (1.39) and (1.40) are Fredholm in 𝕏^{m}(Γ, ρ) if and only if

inf_{λ∈ℝ} | det (A_{±})_{𝕏^{m}(Γ,ρ)}(t, λ)| = \frac{1}{4} inf_{λ∈ℝ} |1 − H_{m,j}^{2}(λ)| ≠ 0 .  (5.68)

Invoking the formulae

sinh^{2} a − sinh^{2} b = sinh(a − b) sinh(a + b) ,
sinh(a + 2πk) = sinh a , a, b ∈ ℂ , k = 0, ±1, ….

we find easily

\frac{1}{4}(1 − H_{m,j}^{2}(λ)) = \frac{sinh^{2} π(iβ_{j} + λ) − sinh^{2} π(1 − γ_{j})(iβ_{j} − mi + λ)}{4 sinh^{2} π(iβ_{j} + λ)}

= −\frac{sinh π[(2 − γ_{j})(iβ_{j} + λ) − mi + 2γ_{j}i] sinh πγ_{j}(iβ_{j} + λ − mi)}{sinh^{2} π(iβ_{j} + λ)}

= −\frac{sinh π(2 − γ_{j})(iβ_{j} + λ − mi) sinh πγ_{j}(iβ_{j} + λ − mi)}{sinh^{2} π(iβ_{j} + λ)} . (5.69)

Due to (5.69) condition (5.68) holds if and only if

(2 − γ_{j})(iβ_{j} + λ − mi) ≠ 0, ±i, …, γ_{j}(iβ_{j} + λ − mi) ≠ 0, ±i, ….

Since 0 < β_{j} < 1, m = 0, 1 the latter conditions can be written as follows

β_{j} ≠ \{ \begin{matrix}γ_{j}^{0} & if m = 0 , \\ 1 − γ_{j}^{0} & if m = 1 . \end{matrix}  (5.70)

and the condition of the theorem is justified.

On the other hand due to (5.70) the group of non-degenerate symbols (5.68) is divided in four homotopy groups (two for each m = 0, 1); the symbols inside each group have equal indices and it suffices to find the value for one representative of the group. Since

det (A_{±})_{𝕏^{m}(Γ,ρ)}(t, λ) = \frac{1}{4}[1 − H_{m,j}^{2}(λ)] = \frac{1}{4}[1 − H_{m,j}(λ)][1 + H_{m,j}(λ)]

it is sufficient to investigate simpler functions 1 ± H_{m,j}(λ). Images on the complex plane of representatives

γ_{j} = \frac{1}{p} , β_{j} = \frac{1}{4}, \frac{3}{4} , m = 0, 1

are plotted on Fig. 7–Fig.10 in Appendix. The result can be summarized as follows:

ind det (A_{±})_{𝕏^{m}(Γ,ρ)}(t_{j}, ·) = \{ \begin{matrix}0 & for & β_{j} < 1 − γ_{j}^{0} & and m = 0 , \\ −1 & for & β_{j} > 1 − γ_{j}^{0} & and m = 0 , \\ 1 & for & β_{j} < γ_{j}^{0} & and m = −1 , \\ 0 & for & β_{j} > γ_{j}^{0} & and m = −1 . \end{matrix}

From Theorems 4.1, 4.3 and 4.6 we get the index formula (we remind that T_{pk} = ∅)

Ind A_{±} = − ∑_{j=1}^{n} ind det (A_{±})_{𝕏^{m}(Γ,ρ)}(t_{j}, ·)=\{ \begin{matrix}∑_{\begin{matrix}t_{j}∉T_{pk} \\ β_{j}>γ_{j}^{0} \end{matrix}} 1 & for m=0, \\ − ∑_{\begin{matrix}t_{j}∈T \\ β_{j}<1−γ_{j}^{0} \end{matrix}} 1 & for m=1 \end{matrix}

(see (1.81)).

Now we need information about the kernels dim Ker A_{±} to derive the remainder equalities in (1.81).

Solvability results follow from from (1.81) provided (1.82) or (1.83) hold. First of all note, that due to Lemma 1.21 it suffices to establish values

of dim Ker_{𝕏}A_{±} and dim Coker_{𝕏}A_{±} only for one space among those where operators A_{±} have equal indices.

Equalities dim Ker A_{±} = ε_{±}, dim Coker A_{−} = ε_{±} under condition (1.82) and, in general, equalities in (1.81) can be derived from the equivalence of BVPs and our BIEs stated in Theorem 1.12 by invoking Remark 1.10, Lemma 1.15 and equivalence of BVPs with the Riemann–Hilbert problem, stated in Theorem 1.16, because either the kernel or the cokernel of the Riemann–Hilbert problem (and of characteristic singular integral equation) are trivial (see [Du1, GK1, Kh1]).

If one of conditions of the theorem is missing we can apply above mentioned equivalence with the Riemann–Hilbert problem to find that our BIEs are not Fredholm. Moreover, since in all cases the kernels and cokernels are finite dimensional dim Ker A_{±} ≤ n+1 and dim Coker A_{±}^{∗} ≤ n+1, the images Im A_{±} can not be closed.

Now suppose Γ has peaks T_{pk} ≠ ∅.

Localization method applied in §,5.3, can be applied in the present situation as well. Due to Corollary 1.7 local representatives of operators A_{±} in (1.39) at t_{0} ∉ T_{pk} are

A_{±} ∼^{M_{t_{0}}} ±\frac{1}{2}I

and are invertible in L_{p}(Γ).

At the inward peak t_{l} ∉ T_{iw} we should localize the operator A_{±} to the same one, but replace the curve Γ by a new one L_{j} which coincides with Γ in the vicinity of t_{j} and has t_{j} as a single outward peak. Therefore we can suppose, without restricting generality, that Γ has a single knot T = {t_{1}}, which is either an angular point or an outward peak.

WARNING! While changing from the inward peak to outward, we change the orientation of the curve. Then operators A_{±} and B_{±} are replaced by ∓A_{∓} and ∓B_{∓}, respectively. We should also interchange one-side neighbourhoods Γ_{t_{j}}^{−} and Γ_{t_{j}}^{+} which leads, due to non-equal rights of these neighbourhoods in the definition of the space L_{p}(Γ, ρ, T_{iw}) (see (1.76)) to differences, which should be taken into consideration.

Due to Lemma 1.13 the Riemann–Hilbert problem is surjective and we can enjoy equivalent reduction of (1.39) and of (1.44) to the corresponding BVPs (1.6)–(1.8) for the domain Ω^{+} justified in Theorems 1.12 and 1.14. Due to equivalence established in Theorems 1.16 and 1.17 we find that equation (1.39) is equivalent to (1.50) while (1.44)–to (1.60). By applying Theorem 1.26 and 1.29 we accomplish the proof of Fredholm properties.

The same equivalence can be used to prove the index formulae for the case of one knot. In case of multiple knots we can use exactly the same approac as in (5.51)–(5.54) and reduce the proof to the case of one knot.

For equations (1.40) and (1.45) we make conclusions as for dual equations to (1.39) and to (1.44), respectively.

As for dim Ker A_{±} and dim Ker B_{±} in (1.39)–(1.40) and in (1.44)– (1.45), the formulae can be derived from the index formulae and above mentioned results on kernels in L_{p}(Γ) spaces (see Remark 1.10).

Remark 5.11 Due to Lemma 1.21 any integrable solutions φ_{±} ∈ L_{p}(Γ, ρ) of integral equations (1.39) and (1.40) are continuous (are Hölder continuous with the exponent 0 < μ < 1 or even belong to the Zygmund space ℤ^{ν}(Γ) for 0 < ν < ∞) provided the right-hand sides are continuous (belong to H^{μ}(Γ) or to ℤ^{ν}(Γ), respectively and, in the latter cases, Γ sufficiently smooth).

Moreover, invoking Theorem 5.8 we find that the solution u(x) to the Dirichlet BVP (1.6), (1.7) is continuous on \overline{Ω^{±}} (is Hölder continuous with the exponent 0 < μ < 1 or even belongs to the Zygmund space ℤ^{ν}(Γ) for 0 < ν < ∞) provided the same condition holds for the date g(t) on Γ.

Similar assertions for L_{p}-spaces and continuous solutions can be found in [Mi2, § 14] and in [Ma1, Ch. I, Theorems 3 and 5].

Remark 5.12 Non-equal rights of curves Γ_{t_{j}}^{±} in the definition of the space L_{p}(Γ, ρ, T_{pk}) in (1.76) originates in the behavior of the convolution operator with 2 × 2 matrix symbol which is a local representative of the boundary integral operator and can easily be traced in the proof of Theorem 1.26 in § 5.3 (see (5.42)–(5.47)). Difference of conditions on the function φ ∈ L_{p}(Γ, ρ, T_{pk}) at outward and inward peaks in the definition (1.76) reflected in ε_{j} = ±1, is due to the above-mentioned non-equal rights of curves Γ_{t_{j}}^{±} and can be explained by the change of domain Ω^{+} to some outer domain by localization to make an inward peak outward (see the proof of Theorem 1.23 above).

Acknowledgements

The first author was supported by the grants: a) DFG 436 GEO 17/3/99;

b) Academy of Sciences of Georgia 1.2.97 and the second one was supported by DFG 436 GEO 17/3/99.

Appendix

[figure: (0, 0); (0, 0); B_{j}(λ) = 1 − \frac{sin \frac{π}{4}}{sin π(\frac{6}{25} −iλ)}; B_{j}(λ) = 1 − \frac{sin \frac{π}{4}}{sin π(\frac{19}{25} −iλ)}; ind B_{j} = 1; ind B_{j} = −1]

Fig. 5

[figure: (0, 0); (1, 0); B_{j}^{0}(λ) = \frac{λ−i}{λ} [1 − \frac{sin \frac{π}{8}}{sin π(\frac{1}{8} −iλ)}]; B_{j}^{0}(λ) = \frac{λ−i}{λ} [1 − \frac{sin \frac{9π}{10}}{sin π(\frac{9}{10} −iλ)}]; ind B_{j}^{0} = 1; ind B_{j}^{0} = 0]

Fig. 6

[figure: (0.44, 0); (1, 0); 1 − H_{j}(λ) = 1 − sinh \frac{π}{4} (\frac{i}{4} +λ) ; sinh \frac{π}{4} (\frac{i}{4} +λ); 1 + H_{j}(λ) = 1 + sinh \frac{π}{4} (\frac{i}{4} +λ) ; sinh \frac{π}{4} (\frac{i}{4} +λ); ind (1 − H_{j}^{2}) = 0]

Fig. 7

[figure: (0, 0); (1, 0); 1 − H_{j}(λ) = 1 − sinh \frac{π}{4} (\frac{3i}{4} +λ) ; sinh \frac{π}{4} (\frac{3i}{4} +λ); 1 + H_{j}(λ) = 1 + sinh \frac{π}{4} (\frac{3i}{4} +λ) ; sinh \frac{π}{4} (\frac{3i}{4} +λ); ind (1 − H_{j}^{2}) = −1]

Fig. 8

[figure: (0, 0); (1, 0); 1 − H_{j}(λ) = 1 − sinh \frac{π}{4} (\frac{3i}{4} −λ) ; sinh \frac{π}{4} (\frac{i}{4} +λ); 1 + H_{j}(λ) = 1 + sinh \frac{π}{4} (\frac{3i}{4} −λ) ; sinh \frac{π}{4} (\frac{i}{4} +λ); ind (1 − H_{j}^{2}) = 1]

Fig. 9

[figure: (0.46, 0); (1, 0); 1 − H_{j}(λ) = 1 − sinh \frac{π}{4} (\frac{i}{4} −λ) ; sinh \frac{π}{4} (\frac{3i}{4} +λ); 1 + H_{j}(λ)1 + sinh \frac{π}{4} (\frac{i}{4} −λ) ; sinh \frac{π}{4} (\frac{3i}{4} +λ); ind (1 − H_{j}^{2}) = 0]

Fig. 10

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(Received 11.01.2000)

Authors’ addresses:

R. Duduchava A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia E-mail: duduch@rmi.acnet.ge

B. Silbermann Department of Mathematics Technische Universität Chemnitz D–09107 Chemnitz Germany E-mail: B. Silbermann@mathematik.tu–chemnitz.de