Group invariance and L^{p}-bounded operators

Toshiyuki Kobayashi^{∗} RIMS, Kyoto University

         E-mail address: toshi@kurims.kyoto-u.ac.jp

and

Andreas Nilsson^{†} University of Kalmar

           E-mail address: andreas.nilsson@hik.se

Abstract

In this paper we consider translation invariant operators with additional symmetry coming from group actions. As the classic Hilbert and Riesz transforms can be characterized up to scalar by means of relative invariance of conformal transformation groups, certain multiplier operators are characterized by relative invariance of some other affine subgroups. In this article, we formalize a geometric condition that characterizes specific multiplier operators uniquely up to scalar, and provide several examples of multiplier operators having ‘large symmetry’. Finally, we classify which of these examples are L^{p}-bounded.

∗Partially supported by Grant-in-Aid for Scientific Research 18340037, Japan Society for the Promotion of Science.

†Partially supported by Japan Society for the Promotion of Science.

2000 Mathematics Subject Classification. 42B15; Secondary 22E46, 42B20.

Key words and phrases. Multiplier, translation invariant operator, group invariance, relative invariants, prehomogeneous vector space

Contents

1 Introduction 2 1.1 Hilbert and Riesz transforms . . . . . . . . . . . . . . . . . . . 2 1.2 Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Formulation of relative invariance 6 2.1 Affine actions and translation invariant operators . . . . . . . 6 2.2 Classic examples . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 L^{2}-boundedness and unitarizability . . . . . . . . . . . . . . . 9 2.4 Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.5 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Examples of invariant multipliers (dim V =1) 13 3.1 Invariant multipliers for (GL(2, R), R^{3}) . . . . . . . . . . . . . 14 3.2 Invariant multipliers for (GL(2) × GL(2), R^{4}) . . . . . . . . . 16 3.3 Invariant multipliers for (SO_{0}(p, q) × R_{+}, R^{p+q}) . . . . . . . . 18

4 Invariant multipliers for (O(m) × GL_{+}(k, R), R^{mk}) 20 4.1 Exterior Riesz transforms . . . . . . . . . . . . . . . . . . . . 20 4.2 Proof of Theorem 5 . . . . . . . . . . . . . . . . . . . . . . . . 22

5 Classification of invariant L^{p}-bounded operators 23 5.1 Algebra of L^{p}-bounded operators—quick review . . . . . . . . 24 5.2 Classification of L^{p}-bounded operators from Sections 3 and 4 . 25

1 Introduction

Our object of study is translation invariant operators bounded on L^{p}(R^{n}) from the viewpoint of group invariance, with emphasis on ‘maximal symmetry’ that is satisfied by specific operators.

1.1 Hilbert and Riesz transforms

Classic examples of translation invariant singular integrals are the Hilbert transform H, which is defined on (a dense subspace of) L^{2}(R^{n}) by

Hf(x) := lim_{ϵ→0} \frac{1}{π} ∫_{|y|≥ϵ} \frac{f(x − y)}{y} dy,

and the Riesz transforms R_{j} as its higher dimensional generalization:

R_{j}f(x) := lim_{ϵ→0} \frac{Γ(\frac{n+2}{2})}{π^{\frac{n+2}{2}}} ∫_{|y|≥ϵ} \frac{y_{j}}{|y|^{n+1}} f(x − y) dy (1 ≤ j ≤ n).  (1.1.1)

Hilbert and Riesz transforms have been used in various aspects of analysis such as

1) (harmonic analysis) L^{p}-convergence of Fourier series,

2) (differential equations) regularity properties of solutions to the Laplace equation.

For more details of applications and perspectives of these operators in analysis, we refer the reader to the survey papers [F2] and [S2].

On the other hand, these translation operators enjoy further group symmetry. We begin with an observation that the Hilbert transform satisfies the following two properties:

τ_{a} ∘ T = T ∘ τ_{a}  for all a ∈ R,  (1.1.2)
D_{η} ∘ T = sgn(η)T ∘ D_{η} for all η ∈ R^{∗}.  (1.1.3)

Here we have used the notation:

(τ_{a} ∘ f)(x) := f(x − a) for a ∈ R,
(D_{η}f)(x) := f(ηx)  for η ∈ R^{∗},

for translation and dilation, respectively. By translation invariant operators we mean a bounded operator satisfying the condition (1.1.2). The condition (1.1.3) is regarded as an additional group invariance, on which we shall focus in this article. The viewpoint here is that the group invariance (1.1.3) is strong enough to characterize the Hilbert transform in the sense that any translation invariant operator acting on L^{2}(R) and satisfying (1.1.3) must be the Hilbert transform up to scalar multiple (see [S, Section 3.1] or [EG, Section 6.8]).

The Riesz transforms can be also characterized in a similar manner. For f ∈ L^{2}(R^{n}) and g ∈ O(n), we set l_{g}(f)(x) := f(g^{−1}x). Let π(g) be the standard representation of O(n) on R^{n}. We then have:

Fact 1.1 ( [S, Section 3.1, Proposition 2]). ^{2} A family of translation invariant operators T̄ = ^{t}(T_{1}, …, T_{n}) bounded on L^{2}(R^{n}) and commuting with positive dilations, satisfies the identity l_{g^{−1}} ∘ T̄ ∘ l_{g} = π(g) ∘ T̄ for g ∈ O(n), if and only if, up to a constant multiple, it is the family of Riesz transforms.

We shall come back to these examples in Subsection 2.3 after we formalize a general framework to work in.

1.2 Strategies

In light of the aforementioned invariance properties of the Hilbert and Riesz transforms, we may expect that ‘nice translation invariant operators’ ought to enjoy additional symmetry. To reveal such invariance conditions arising from the affine transformation group, we propose the following strategies:

Strategy 1 (Characterization of translation invariant operators). Suppose we are given a translation invariant operator bounded on L^{2}(R^{n})(or a family of such operators):

Step 1. Find a (maximal) group of relative invariance of this operator.

Step 2. Conversely, find all bounded translation invariant operators that satisfy the same condition of relative invariance.

We are particularly interested in the case where Step 2 yields a finite dimensional (or even preferably, one dimensional) space of operators. Then, we might say that Strategy 1 gives a characterization of the original operator.

This idea could be used in reverse to find new operators by starting from group invariance:

Strategy 2 (Finding nice operators). Suppose we are given an invariance condition by means of a subgroup of the affine transformation group Aff(R^{n}):

Step 1. Find explicitly all solutions that satisfy the invariance conditions.

Step 2. Choose the solutions that yield L^{2}-bounded (or L^{p}-bounded) operators.

2It was stated as l_{g} ∘ T̄ ∘ l_{g^{−1}} = π(g) ∘ T̄ in our notation, but g should read as g^{−1} in the left-hand side.

The point of Strategy 2 is to find a nice invariance condition such that the resulting space of operators in Step 2 is one dimensional, or at least non-zero and finite dimensional.

We will give a rigorous formulation in Theorem 1 in Subsection 2.1 to pursue Strategies 1 and 2. The aforementioned characterization of the Hilbert and Riesz transforms (Fact 1.1) is reexamined in Subsection 2.2. Furthermore, Stein’s higher Riesz transforms (see Example 2.2.1 (2)) are obtained in this framework. Relative invariants of Sato’s prehomogeneous vector spaces (see [Sa]) are also examples of the solutions in Strategy 2. The formalization of Theorem 1 is built on ‘vector valued relative invariants’ of prehomogeneous vector spaces.

In Sections 3 and 4, we shall illustrate our general framework (Theorem 1) by the examples of translation invariant operators with additional group invariance defined by the following affine subgroups

(R_{+}^{∗} × SO(p, q)) ⋉ R^{p+q}  ⊂ Aff(R^{p+q}) (see Theorems 2, 3 and 4),
(O(m) × GL_{+}(k, R)) ⋉ R^{km} ⊂ Aff(R^{km}) (see Theorem 5).

The latter example reproduces the Riesz transforms when k = 1.

In Subsection 5.2, we shall determine which of the L^{2}-bounded invariant operators obtained in Theorems 2, 3, 4 and 5 give L^{p}-bounded operators (1 < p < ∞). The classification is given in Theorems 6, 7, 8 and 9, respectively.

Generalizations:

Theorem 1 deals with invariance conditions of operators defined by finite dimensional representations of (almost) connected subgroups of the affine transformation groups. In subsequent papers we shall consider two directions of generalization of our strategies:

1) (A generalization from continuous to discrete)

In [KN], we shall consider the relative invariance for semigroup actions in place of the relative invariance for group actions. This generalization allows us, for example, to give a characterization of discrete Riesz transforms on T^{n} and Z^{n}, extending previous work by Edwards and Gaudry [EG].

2) (A generalization from finite dimensional to infinite dimensional representations)

In [KN2], we shall use unitary representations in place of finite dimensional representations for the ‘symmetry’ of operators. This generalization yields much more bounded invariant multiplier operators. Besides, we also generalize the formulation of our strategies by means of differential equations rather than the group action itself. A typical example is when the group R_{+} × O(p, q) acts on R^{n} by the natural action and the discrete series representations for hyperboloids play an important role in constructing invariant L^{p}-bounded operators.

2 Formulation of relative invariance

2.1 Affine actions and translation invariant operators

In this section we will generalize the setting of the Introduction, and introduce the notion of translation invariant operators with additional symmetry by using group representations.

For f ∈ L^{2}(R^{n}) we define (l_{g}f)(t) = f(g^{−1}t), for g ∈ GL(n, R). Let H be a subgroup of GL(n, R) and take a finite dimensional representation (π, V) of H. We write (π^{∗}, V^{∗}) for the contragredient representation of (π, V). As H acts on R^{n}, so does it on the character group (R^{n})^{∗} by the contragredient action: λ ↦ ^{t}h^{−1}λ. We will assume that H acts on (R^{n})^{∗} with finitely many open orbits, O_{1}, …, O_{N} such that their union is conull in (R^{n})^{∗}. The orbits O_{j} are expressed as homogeneous spaces H/H_{j}. Let C_{bdd}(O_{j}) denote the complex vector space consisting of bounded continuous functions on O_{j}, on which the group H acts by pullback of functions. By B_{H}(L^{2}(R^{n}), V ⊗L^{2}(R^{n})) we denote the vector space consisting of bounded, translation invariant operators T : L^{2}(R^{n}) → V ⊗ L^{2}(R^{n}) satisfying

[figure: L^{2}(R^{n}); V ⊗ L^{2}(R^{n}); L^{2}(R^{n}); V ⊗ L^{2}(R^{n}); T; l_{g}; π(g)⊗l_{g}; T]  (2.1.1)

for all g ∈ H.

Theorem 1 (Description of invariant bounded operators).

1) There is a natural isomorphism of vector spaces:

B_{H}(L^{2}(R^{n}), V ⊗ L^{2}(R^{n})) ≅ ⨁_{j=1}^{N} Hom_{H}(V^{∗}, C_{bdd}(O_{j})).  (2.1.2)

2) The left-hand side of (2.1.2) is one dimensional if H acts on (R^{n})^{∗} with an open dense orbit O_{1} and if

dim Hom_{H}(V^{∗}, C_{bdd}(O_{1})) = 1.

3) (Upper bound) Let V^{H_{j}} := {v ∈ V : π(h)v = v for any h ∈ H_{j}}.

dim B_{H}(L^{2}(R^{n}), V ⊗ L^{2}(R^{n})) ≤ ∑_{j=1}^{N} dim V^{H_{j}}.  (2.1.3)

Corollary 2.1.1. If dim V = 1 then dim B_{H}(L^{2}(R^{n}), V ⊗ L^{2}(R^{n})) ≤ N. In particular, the operator is unique, up to a scalar, on each orbit if it exists.

Proof. On each open orbit O_{j}, we have

dim Hom_{H}(V^{∗}, C_{bdd}(O_{j})) ≤ dim Hom_{H}(V^{∗}, C(O_{j})) ≤ 1.  (2.1.4)

It is natural to seek for geometric conditions to ensure that dim Hom_{H}(V^{∗}, C(O_{j})) ≤ 1, even if dim V > 1. Here is a sufficient condition:

Corollary 2.1.2. Suppose H is a reductive Lie group. If O_{j} is a symmetric space of H, then

dim Hom_{H}(V^{∗}, C(O_{j})) ≤ 1.

If all the orbits O_{j} are symmetric spaces, then dim B_{H}(L^{2}(R^{n}), V ⊗L^{2}(R^{n})) ≤ N for any irreducible finite dimensional representation (π, V) of H.

Proof. Suppose O_{j} ≃ H/H_{j} is a reductive symmetric space. Then, for any irreducible finite dimensional representation (π, V), we have dim V^{H_{j}} ≤ 1 by a theorem of É. Cartan ([C, Sect. 17], see also [K, Fact 29] and references therein for related results). The result then follows from (2.1.3).

2.2 Classic examples

Example 2.2.1.

1) (Riesz transforms) Stein’s theorem (Fact 1.1) can be explained in the framework of Theorem 1 where H = R_{+} × O(n), π is the trivial extension of the standard representation of O(n) to H, and V = R^{n}. Then, H has an open dense orbit O_{1} = (R^{n})^{∗} \ {0} in (R^{n})^{∗}, and therefore N = 1. O_{1} ≃ H/ O(n − 1) is a symmetric space.

2) (Higher Riesz transforms) Observe that the standard representation of O(n) on R^{n} is equivalent to the spherical harmonics representation of degree one. This observation leads us to a family of invariant operators that enjoy the same symmetry as spherical harmonics representations. Stein called these operators higher Riesz transforms. Higher Riesz transforms appear in the algebra generated by the Riesz transforms. See [S, Section 3.3 and 3.4.8] for further properties on these operators.

Remark 2.2.2. Fact 1.1 still holds if we replace O(n) by SO(n) for n ≥ 3.

Remark 2.2.3. Suppose we are in the setting of Example 2.2.1 (1), but we let π, instead of being the trivial extension, be the extension taking elements of (r, ϕ) ∈ R_{+} × O(n) to r^{a}π(ϕ). If 0 < a ≤ n/2 then we have

Hom_{H}(L^{2}(R^{n}), R^{n} ⊗ L^{2}(R^{n})) = 0,
dim Hom_{H}(L^{2}(R^{n}), R^{n} ⊗ L^{p}(R^{n})) = 1 (1/p = 1/2 − a/n).

This follows essentially from the proof of Fact 1.1 (or Theorem 1) and the Hardy–Littlewood–Sobolev theorem, see [S, Theorem 1.2.1].

Example 2.2.4. Assume dim V = 1. In the theory of prehomogeneous vector spaces, a non-trivial function on O_{j} contained in the image of Hom_{H}(V^{∗}, C(O_{j})), is called a relative invariant. The corresponding one dimensional representation (π^{∗}, V^{∗}) defines a function on H by h ↦ π^{∗}(h), which is called Sato–Bernstein’s b-function, see [Sa] for more details. We shall give some examples in Subsections 3.1 and 3.2.

The above three examples treat the cases where either dim V = 1 or the orbits O_{j} are symmetric spaces. Corollaries 2.1.1 and 2.1.2 ensured that the dimension of B_{H}(L^{2}(R^{n}), V ⊗ L^{2}(R^{n})) does not exceed the number of open H-orbits on R^{n}, giving a characterization of such operators by group invariance. However, there are also interesting examples where dim B_{H}(L^{2}(R^{n}), V ⊗ L^{2}(R^{n})) ≤ 1, even though dim V can be greater than one and the orbit is not a symmetric space. In Subsection 4.2 we will provide such an example where O(k) × GL(m, R) is acting on R^{mk}.

Example 2.2.5. Consider the action on R^{n} by the group H = R_{+} ×O(p, q), pq > 0. Let π be the standard representation of O(p, q) on V := C^{p+q} extended trivially to R_{+}. In this case H has two open orbits, namely, O_{1} = R_{+} × O(p, q)/ O(p − 1, q) and O_{2} = R_{+} × O(p, q)/ O(p, q − 1). Both quotients are reductive symmetric spaces and the representation π appears in C(O_{1}) as well as C(O_{2}). Hence, Example 2.1.2 tells us that dim Hom_{H}(V, C(O_{1})⊕C(O_{2})) = 2. However, in this case, the space B_{H}(L^{2}(R^{n}), V ⊗L^{2}(R^{n})) is in fact trivial.

Example 2.2.5 shows a typical feature of the action of a non-compact group. See Proposition 2.3.1.

2.3 L^{2}-boundedness and unitarizability

Suppose π : H → GL_{C}(V) is a finite dimensional representation of a group H. We say (π, V) is unitarizable if there exists an H-invariant Hermitian inner product on V, and is non-unitarizable if not.

For compact H, any finite dimensional representation is unitarizable. However, this is not the case for noncompact H. For example, if H = SL(n, R) and π is the natural representation of H on V = R^{n}, then (π, V) is non-unitarizable for n > 1.

Proposition 2.3.1. Retain the notation of Theorem 1.

1) If (π, V) is a unitarizable representation, then we have

dim B_{H}(L^{2}(R^{n}), V ⊗ L^{2}(R^{n})) = ∑_{j=1}^{N} dim V^{H_{j}}.

2) If (π, V) is a non-unitarizable representation of a reductive Lie group H, then B_{H}(L^{2}(R^{n}), V ⊗ L^{2}(R^{n})) = {0}.

Proof. Theorem 1 shows that

B_{H}(L^{2}(R^{n}), V ⊗ L^{2}(R^{n})) ≅ ⨁_{j=1}^{N} Hom_{H}(V^{∗}, C_{bdd}(O_{j})).

In general we have the following:

Lemma 2.3.2. Suppose O ≃ H/H_{0} is a homogeneous space of H. Then, there is a natural isomorphism between the two vector spaces Hom_{H}(V^{∗}, C(O)) and V^{H_{0}}.

Proof of Lemma. Let ϕ be an element in Hom_{H}(V^{∗}, C(O)). We define a V - valued function F : O → V by the relation ϕ(v^{∗})(x) = ⟨F (x), v^{∗}⟩, for x ∈ O and v^{∗} ∈ V^{∗}. Then, F is a V -valued continuous function satisfying the relation

F (h^{−1}x) = π(h)F (x) for h ∈ H and x ∈ O.

We denote by C(O, V)^{H} the vector space of such V -valued continuous functions on O. Next, let o := eH_{0} ∈ O ≃ H/H_{0}. Then, u := F (o) satisfies

ϕ(v^{∗})(h^{−1}o) = ⟨F (h^{−1}o), v^{∗}⟩ = ⟨π(h)u, v^{∗}⟩ for any v^{∗} ∈ V^{∗}. (2.3.1)

In particular, π(h)u = u if h ∈ H_{0}. Then, ϕ is recovered from u ∈ V^{H_{0}} by the relation ϕ(v^{∗})(x) = ⟨π(h^{−1})u, v^{∗}⟩ if x = h · o (the right-hand side does not depend on the choice of h ∈ H such that x = h · o). It is now readily seen that the correspondence ϕ ↦ F ↦ F (o) gives the following bijections:

Hom_{H}(V^{∗}, C(O)) →^{∼} C(O, V)^{H} →^{∼} V^{H_{0}}.

1) Suppose (π, V) is a unitary representation. Then, any matrix coefficient is bounded because the operator norm ∥π(g)∥ = 1. Thus from (2.3.1) and Lemma 2.3.2 it follows that

Hom_{H}(V^{∗}, C_{bdd}(O)) ≅ Hom_{H}(V^{∗}, C(O)) ≅ V^{H_{0}}.

2) Suppose (π, V) is non-unitarizable. Then, we have

Hom_{H}(V^{∗}, C_{bdd}(O)) = {0}.

This is a consequence of the lemma below which shows that matrix coefficients of a non-unitary representation of a reductive Lie group are unbounded.

Lemma 2.3.3. Let G be a simple, connected and non-compact Lie group, and (π, V) a finite dimensional representation of G. Assume further that all the matrix-coefficients of π are bounded functions on G. Then π is trivial.

Proof of Lemma. By replacing G with π(G)(⊂ GL_{C}(V)) if necessary, we may and do assume that G is a linear group contained in its complexification G_{C}. We extend the representation holomorphically to G_{C}. Let K be a maximal compact subgroup of G, 𝔤 = 𝔨 + 𝔭 the corresponding Cartan decomposition of the Lie algebra 𝔤 of G, and G_{U} a maximal compact subgroup of G_{C} containing K. As G_{U} is compact, there exists a G_{U}-invariant inner product on V. (In fact, for an arbitrary inner product on V, the new inner product defined by taking the average over G_{U} becomes G_{U}-invariant.)

Let A be a maximally split subgroup of G, and 𝔞 its Lie algebra, and T := exp(i𝔞) ⊂ G_{U}. As π|_{T} is unitarizable the differential dπ|_{𝔞} has only real eigenvalues. The assumption that the matrix coefficients are bounded implies that π|_{A} is trivial. This means also that π is trivial on exp(𝔭) because all elements in exp(𝔭) are conjugate to an element in A. As exp(𝔭) builds up G as a group, we obtain that π|_{G} is trivial. This is what we wanted.

2.4 Multipliers

We write F : L^{2}(R^{n}) → L^{2}(R^{n}) for the Fourier transform

F(f)(λ) = ∫_{R^{n}} e^{−2πi⟨x,λ⟩}f(x) dx,

and F^{−1} for its inverse. For a bounded measurable function m(λ), we set

T_{m}(f) = F^{−1}(m(·)F(f)(·)).

Then, T_{m} : L^{2}(R^{n}) → L^{2}(R^{n}) is a bounded translation invariant operator. Such an operator T_{m} is called a multiplier operator associated to the multiplier m. Conversely, any bounded translation invariant operator

T : L^{p}(R^{n}) → L^{p}(R^{n}), 1 < p < ∞

is bounded on L^{2}(R^{n}) as well, and has the form T_{m} with a bounded function m(λ). In other words, f ↦ F ∘ T ∘ F^{−1}(f) is given by a multiplication of a bounded measurable function m(λ) if T is a bounded translation invariant operator.

From now on, we shall identify bounded translation invariant operators with multiplier operators.

2.5 Proof of Theorem 1

Proof. Suppose T : L^{2}(R^{n}) → V ⊗L^{2}(R^{n}) is a bounded translation invariant operator. Then, there exists a bounded measurable V -valued function m on R^{n} such that

m(λ)(Ff)(λ) = (id ⊗F) ∘ (T f)(λ) for f ∈ L^{2}(R^{n}),  (2.5.1)

that is, the multiplication by m gives the multiplier operator (we use the same letter m):

m = (id ⊗F) ∘ T ∘ F^{−1}.

We recall that the Fourier transform F satisfies

(Ff(g^{−1}·))(λ) = ∫_{R^{n}} f(g^{−1}x)e^{−2πi⟨x,λ⟩}dx = | det g| (Ff)(^{t}gλ),

that is,

F ∘ l_{g} = | det g| l_{^{t}g^{−1}} ∘ F

for g ∈ GL(n, R). Suppose now T ∈ B_{H}(L^{2}(R^{n}), V ⊗ L^{2}(R^{n})). Then, we have

m ∘ | det g| l_{^{t}g^{−1}} ∘ F = m ∘ F ∘ l_{g}
= (id ⊗F) ∘ T ∘ l_{g}
= (id ⊗F) ∘ (π(g) ⊗ l_{g}) ∘ T
= (π(g) ⊗ (F ∘ l_{g})) ∘ T
= (π(g) ⊗ | det g| l_{^{t}g^{−1}} ∘ F) ∘ T
= | det g| (π(g) ⊗ l_{^{t}g^{−1}}) ∘ (id ⊗F) ∘ T
= | det g| (π(g) ⊗ l_{^{t}g^{−1}}) ∘ m ∘ F

for g ∈ H. Cancelating the determinant factor on both sides, we obtain

m(λ) = π(g)m(^{t}gλ) for g ∈ H,  (2.5.2)

where we have kept the same notation, m, for the bounded vector valued function corresponding to the multiplier operator. Also, any bounded vector valued function satisfying (2.5.2) gives rise to a translation invariant operator, T, satisfying (2.1.1). Thus, we have proved the following isomorphism of vector spaces:

B_{H}(L^{2}(R^{n}), V ⊗ L^{2}(R^{n}))
≃{m : R^{n} → V, bounded and satisfying (2.5.2)}.

Since ⋃_{j=1}^{N} O_{j} is conull in V, the right-hand side is isomorphic to

⨁_{j=1}^{N}{m : O_{j} → V, bounded and satisfying (2.5.2)}.

A function satisfying (2.5.2) is continuous on each orbit O_{j}(to be more precise, we identify functions which coincide almost everywhere). Hence, we obtain

≃ ⨁_{j=1}^{N}(C_{bdd}(O_{j}) ⊗ V)^{H},

the subspace of H-fixed vectors in the tensor product representation. By duality this can be rewritten as

≃ ⨁_{j=1}^{N} Hom_{H}(V^{∗}, C_{bdd}(O_{j})).

This proves the first statement of the theorem. The second statement is clear from (2.1.2). The upper estimate (2.1.3) follows from Lemma 2.3.2. Thus the theorem follows.

3 Examples of invariant multipliers (dim V =1)

Our main results in this section, Theorems 2, 3 and 4, all exemplify Strategy 2 of Introduction. In Subsection 3.1 we consider the case when the group GL(2, R) is acting on R^{3}. In Subsection 3.2, the group is GL(2) × GL(2) and the space is R^{4}. In Subsection 3.3 these two examples are generalized by considering the group SO_{0}(p, q)×R_{+} acting on R^{p+q}. In these three examples the dimension of the representation space, V, is 1. By Corollary 2.1.1 we know that the dimension of the space of invariant operators will be at most 1 for each orbit. Hence, in order to determine all the invariant operators, it will be enough to find a single operator which satisfies the given invariance condition for each orbit. This will be carried out in Theorems 2, 3 and 4.

We will consider L^{p}-boundedness of the operators characterized in these examples later in Subsection 5.2. In contrast to this section, Section 4 provides an example where the dimension of the representation space is greater than 1.

3.1 Invariant multipliers for (GL(2, R), R^{3})

We will identify the set of real symmetric matrices S = Symm(2) with R^{3} by the map

(\begin{matrix}x & z \\ z & y \end{matrix}) ↦ (x, y, z).

We define three open subsets in the dual space S^{∗} ≅ R^{3} by

O_{++} = {λ = (λ_{1}, λ_{2}, λ_{3}) : λ_{1} + λ_{2} > 0, λ_{1}λ_{2} − λ_{3}^{2} > 0},
O_{+−} = {λ = (λ_{1}, λ_{2}, λ_{3}) : λ_{1}λ_{2} − λ_{3}^{2} < 0},
O_{−−} = {λ = (λ_{1}, λ_{2}, λ_{3}) : λ_{1} + λ_{2} < 0, λ_{1}λ_{2} − λ_{3}^{2} > 0}.

Their union O_{++} ∪ O_{+−} ∪ O_{−−} is open dense.

We let GL(2, R) act on S by l_{g} : X ↦ gX^{t}g. For simplicity, we shall write GL(2) instead of GL(2, R). Consider the contragredient representation of GL(2) on S^{∗} ≃ R^{3}.

For β ∈ R and δ ∈ {++, +−, −−} we define a function supported on the orbit O_{δ} by

m_{δ}^{β}(λ) = \begin{cases}|λ_{1}λ_{2} − λ_{3}^{2}|^{− \frac{iβ}{2}} & (λ ∈ O_{δ}), \\ 0 & (λ ∉ O_{δ}). \end{cases}

The group GL(2) has two natural families of one dimensional unitary representations:

π_{ϵ,α} : g ↦ sgn(det g)^{ϵ}| det g|^{iα},  (3.1.1)

where ϵ ∈ {0, 1} and α ∈ R.

Theorem 2. Fix a one dimensional unitary representation π_{ϵ,α} : GL(2) → C^{∗}. Let T : L^{2}(R^{3}) → L^{2}(R^{3}) be a bounded, translation invariant operator, which satisfies

T ∘ l_{g} = π_{ϵ,α}(g) l_{g} ∘ T  (3.1.2)

for all g ∈ GL(2).

1) If ϵ = 0, then T is a multiplier operator associated to m(λ) of the form

m(λ) = C_{1}m_{++}^{α}(λ) + C_{2}m_{+−}^{α}(λ) + C_{3}m_{−−}^{α}(λ),

for some C_{1}, C_{2}, C_{3} ∈ C.

2) If ϵ = 1, then T = 0.

Proof. By using the bilinear map

⟨ , ⟩ : Symm(2) × Symm(2) ↦ R, (u, v) ↦ Trace(uv),

we shall identify S^{∗} with Symm(2), and hence also with R^{3}. The contragredient representation of GL(2) on S^{∗} is given by

l_{g}^{∗}λ = ^{t}g^{−1}λg^{−1},

for λ ∈ Symm(2). Via the isomorphism S ≃ S^{∗}, O_{++} corresponds to symmetric matrices with both eigenvalues positive, O_{−−} to those with both eigenvalues negative and O_{+−} to those with eigenvalues of different signature. Then, each of O_{++}, O_{+−}, and O_{−−} is a single orbit of GL(2), since matrices with the same signature are conjugate. We note that

⟨l_{g}u, l_{g}^{∗}λ⟩ = ⟨u, λ⟩.

For δ ∈ {++, +−, −−} and α ∈ R, we claim:

Hom_{GL(2)}(π_{0,α}^{∗}, C_{bdd}(O_{δ})) ≃ Cm_{δ}^{α},  (3.1.3)
Hom_{GL(2)}(π_{1,α}^{∗}, C_{bdd}(O_{δ})) = {0}.  (3.1.4)

First we note that the dimension of the left-hand side is at most one dimensional by (2.1.4) in the proof of Corollary 2.1.1.

To see (3.1.3), it is now sufficient to show m_{δ}^{α} belongs to the left-hand side of (3.1.3). For λ ∈ O_{δ} and g ∈ GL(2), we have from the definition of m_{δ}^{α}:

π_{0,α}^{∗}(g)m_{δ}^{α}(λ) = | det g|^{−iα}m_{δ}^{α}(λ)
= | det(^{t}gλg)|^{− \frac{iα}{2}}
= m_{δ}^{α}(^{t}gλg)
= m_{δ}^{α}(l_{g^{−1}}^{∗}λ),

whence (3.1.3). Hence the result for ϵ = 0 follows from Theorem 1. To see (3.1.4) for ϵ = 1, we just note that

g_{0} := (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix})

satisfies det(g_{0}) = −1 and that g_{0} leaves some element λ_{0} of each orbit invariant. Since m(^{t}gλg) = m(l_{g−1}^{∗}λ) = π_{1,α}^{∗}(g)m(λ) = sgn(det g)| det g|^{−iα}m(λ), we then have

m(^{t}g_{0}λ_{0}g_{0}) = −m(λ_{0}).

For O_{++} we observe

tg_{0} (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) g_{0} = (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) ,

which implies that −m(1, 1, 0) = m(1, 1, 0), i.e. m has to be equal to zero on O_{++}. For O_{+−} we observe

tg_{0} (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) g_{0} = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) ,

which implies that −m(0, 0, 1) = m(0, 0, 1), i.e. m has to be equal to zero on O_{+−}. Finally, the case O_{−−} is similar to O_{++}, and −m(−1, −1, 0) = m(−1, −1, 0) shows m = 0.

3.2 Invariant multipliers for (GL(2) × GL(2), R^{4})

Next, we consider the action of the direct product group GL(2) × GL(2) on the set M(2) of 2 × 2 matrices by

X ↦ g_{1}X^{t}g_{2}

for g = (g_{1}, g_{2}) ∈ GL(2) × GL(2). Via the isomorphism

M(2) ≃ R^{4} (\begin{matrix}λ_{1} & λ_{2} \\ λ_{3} & λ_{4} \end{matrix}) ↦ ^{t}(λ_{1}, λ_{2}, λ_{3}, λ_{4}),  (3.2.1)

we then have a group homomorphism

GL(2) × GL(2) → GL(4),

whose kernel is K = {(sI_{2}, s^{−1}I_{2}) : s ∈ R^{∗}}. Here, I_{2} is the 2 × 2 identity matrix.

By using the non-degenerate bilinear symmetric form

⟨ , ⟩ : M(2) × M(2) → R, (X, Y) ↦ Trace X^{t}Y,

we identify the dual space M(2)^{∗} with M(2). Then the contragredient representation is given by

Y ↦ ^{t}g_{1}^{−1}Y g_{2}^{−1}

because ⟨g_{1}X^{t}g_{2},^{t}g_{1}^{−1}Y g_{2}^{−1}⟩ = ⟨X, Y ⟩ for any g_{1}, g_{2} ∈ GL(2). We note that any one dimensional unitary representation of GL(2) × GL(2) is of the form π_{ϵ_{1},α_{1}}⊗π_{ϵ_{2},α_{2}} for some ϵ_{1}, ϵ_{2} ∈ {0, 1} and α_{1}, α_{2} ∈ R, where the representation π_{ϵ,α} is given in (3.1.1).

Theorem 3. Fix ϵ_{1}, ϵ_{2} ∈ {0, 1} and α_{1}, α_{2} ∈ R. Let T : L^{2}(R^{4}) → L^{2}(R^{4}) be a bounded, translation invariant operator, which satisfies the relation

T ∘ l_{(g_{1},g_{2})} = π_{ϵ_{1},α_{1}}(g_{1})π_{ϵ_{2},α_{2}}(g_{2}) l_{(g_{1},g_{2})} ∘ T  (3.2.2)

for all g_{1}, g_{2} ∈ GL(2). Then T is non-zero if and only if ϵ_{1} = ϵ_{2} and α_{1} = α_{2}. In this case, we set ϵ := ϵ_{1} = ϵ_{2} and α := α_{1} = α_{2}. Then, T is a multiplier operator corresponding to a multiplier function of the form

m(λ_{1}, λ_{2}, λ_{3}, λ_{4}) = C sgn(λ_{1}λ_{4} − λ_{2}λ_{3})^{ϵ}|λ_{1}λ_{4} − λ_{2}λ_{3}|^{iα},  (3.2.3)

where C is a constant.

Proof. We claim that any bounded, translation invariant operator T satisfying (3.2.2) must be zero if ϵ_{1} ≠ ϵ_{2} or α_{1} ≠ α_{2}. Transferring the relation (3.2.2) to the Fourier transform side, we see that the corresponding multiplier function m must satisfy

m(^{t}g_{1}^{−1}, λg_{2}^{−1}) = (sgn(det g_{1}))^{ϵ_{1}}(sgn(det g_{2}))^{ϵ_{2}}| det g_{1}|^{iα_{1}}| det g_{2}|^{iα_{2}}m(λ),

(3.2.4)

for almost everywhere λ ∈ M(2) for each g_{1}, g_{2} ∈ GL(2).

Since GL(2) × GL(2) has an open dense orbit GL(2) on M(2), we may and do assume that m is continuous on GL(2). Then, the condition (3.2.4) applied to λ := I_{2} and (g_{1}, g_{2}) := (^{t}g^{−1}, I_{2}) or (I_{2}, g^{−1}) amounts to

m(g) = (sgn(det g))^{ϵ_{1}}| det g|^{−iα_{1}}m(I_{2}),
m(g) = (sgn(det g))^{ϵ_{2}}| det g|^{−iα_{2}}m(I_{2}),

respectively. Hence, if m is not identically zero, we must have

ϵ_{1} = ϵ_{2} and α_{1} = α_{2}.

Thus, from now on we consider the case where ϵ := ϵ_{1} = ϵ_{2} and α := α_{1} = α_{2}. The identity (3.2.4) then becomes

m(^{t}g_{1}^{−1}λg_{2}^{−1}) = (sgn(det g_{1}g_{2}))^{ϵ}| det g_{1}g_{2}|^{iα}m(λ).  (3.2.5)

On the other hand, it is obvious that the function in (3.2.3) satisfies (3.2.5). By Corollary 2.1.1, the proof of Theorem 3 is completed. This completes the proof.

3.3 Invariant multipliers for (SO_{0}(p, q) × R_{+}, R^{p+q})

In light of local isomorphisms of Lie groups

SL(2, R) ≈ SO_{0}(2, 1),
SL(2, R) × SL(2, R) ≈ SO_{0}(2, 2),

the previous two examples can be extended to a more general setting by using the indefinite orthogonal group O(p, q) as follows.

For p, q ≥ 1, we let G_{1} := SO_{0}(p, q), the identity component of the indefinite orthogonal group

O(p, q) = {g ∈ GL(p + q, R) : Q(gx) = Q(x) for any x ∈ R^{p+q}},

where Q is the quadratic form given by

Q(x) := x_{1}^{2} + ⋯ + x_{p}^{2} − x_{p+1}^{2} − ⋯ − x_{p+q}^{2} .

We shall consider a direct product group

G := G_{1} × R_{+},

the group acting conformally on the standard flat pseudo-Riemannian manifold R^{p,q} equipped with the indefinite metric ds^{2} = dx_{1}^{2} + ⋯ + dx_{p}^{2} − dx_{p+1}^{2} − ⋯ − dx_{p+q}^{2}.

We define a family of one dimensional unitary representations of G by

π_{α} : G → C^{×}, (h, a) ↦ a^{iα}  (p + q ≥ 3) (3.3.1)

π_{α,β} : G → C^{×}, ((\begin{matrix}cosh t & sinh t \\ sinh t & cosh t \end{matrix}) , a) ↦ a^{iα}e^{itβ}  (p + q = 2) (3.3.2)

for α, β ∈ R.

We also define bounded functions on R^{p+q} by

Q_{+}(λ)^{iα} := \begin{cases}Q(λ)^{iα} & if Q(λ) > 0 \\ 0 & otherwise \end{cases}

Q_{−}(λ)^{iα} := \begin{cases}|Q(λ)|^{iα} & if Q(λ) < 0 \\ 0 & otherwise \end{cases}

Q_{+}^{(±)}(λ)^{iα} := \begin{cases}Q(λ)^{iα} & if Q(λ) > 0 and ± λ_{1} > 0 \\ 0 & otherwise. \end{cases}

We also use the following notation for ϵ = + or −.

a_{ϵ}^{iα} = \begin{cases}|a|^{iα} & if ϵa > 0, \\ 0 & otherwise. \end{cases}

Theorem 4. Let p, q ≥ 1. Let T : L^{2}(R^{p+q}) → L^{2}(R^{p+q}) be a bounded translation invariant operator, which satisfies the following relation

T ∘ l_{g} = \begin{cases}π_{α}(g)l_{g} ∘ T & (p + q ≥ 2) \\ π_{α,β}(g)l_{g} ∘ T & (p + q = 2) \end{cases}

for all g ∈ SO_{0}(p, q) × R_{+}, where the representations π_{α} and π_{α,β} are defined by (3.3.1) and (3.3.2) respectively. Then T is a multiplier operator associated to the multiplier of the form:

m(λ)  (3.3.3)

= \begin{cases}c_{1}Q_{+}(λ)^{− \frac{1}{2} iα} + c_{2}Q_{−}(λ)^{− \frac{1}{2} iα} & (p, q ≥ 2) \\ c_{1}Q_{+}^{(+)}(λ)^{− \frac{1}{2} iα} + c_{2}Q_{+}^{(−)}(λ)^{− \frac{1}{2} iα} + c_{3}Q_{−}(λ)^{− \frac{1}{2} iα} & (p = 1, q ≥ 2) \\ ∑_{ε_{1}=±,ε_{2}=±} c_{ε_{1},ε_{2}}(λ_{1} + λ_{2})_{ε_{1}}^{− \frac{1}{2} i(α+β)}(λ_{1} − λ_{2})_{ε_{2}}^{− \frac{1}{2} i(α−β)} & (p = q = 1) \end{cases}

for some constants c_{1}, c_{2}, c_{3}, c_{ε_{1},ε_{2}} ∈ C. The case p ≥ 2 and q = 1 is similar to the second case.

Remark 3.3.1. Here we have treated the connected group SO_{0}(p, q). The cases SO(p, q) and O(p, q) can be reduced to this one. However, the number of orbits are different for p = 1 or q = 1 Proof. Consider the natural action of G = SO_{0}(p, q) × R_{+} on R^{p+q}. Then, the following unions of open G-orbits

O_{+} ∪ O_{−}  (p, q ≥ 3),
O_{+}^{(+)} ∪ O_{+}^{(−)} ∪ O_{−}  (p = 1, q ≥ 2),
O_{+}^{(+)} ∪ O_{+}^{(−)} ∪ O_{−}^{(+)} ∪ O_{−}^{(−)} (p = q = 1)

are dense in R^{p+q}, respectively, where we set

O_{±} := {λ ∈ R^{p+q} : ±Q(λ) > 0},
O_{+}^{(±)} := {λ ∈ O_{+} : ±λ_{1} > 0}  (p = 1),
O_{−}^{(±)} := {λ ∈ O_{−} : ±λ_{p+1} > 0} (q = 1).

Owing to Corollary 2.1.1, Theorem 4 follows if we show that the function m in equation (3.3.3) satisfies the relation

m(^{t}gλ) = \begin{cases}π_{−α}(g)m(λ) & (p + q ≥ 2) \\ π_{−α,−β}(g)m(λ) & (p + q = 2) \end{cases}

for any g ∈ G on each orbit. A simple computation shows that this is indeed the case.

4 Invariant multipliers for (O(m)×GL_{+}(k, R), R^{mk})

This section provides an example of Theorem 1 where the invariance conditions determine multiplier operators up to scalar, even in the setting that (π, V) is not one dimensional. The main result of this section is Theorem 5. It shall be noted that the open H-orbits are not symmetric in this example.

4.1 Exterior Riesz transforms

Let n = mk (m ≥ k), and

H := G_{1} × G_{2} = O(m) × GL_{+}(k, R).

Then H acts on R^{n} ≃ M(m, k; R) in the following manner: for (a, b) ∈ H,

X ↦ aXb^{−1}.

We define a subset of M(m, k; R) by

O = {X ∈ M(m, k; R) : rank X = k}.

Then O is open dense in M(m, k; R). Furthermore, if X ∈ O, then the k × k matrix ^{t}XX is positive definite, and in particular det(^{t}XX) > 0.

Associated to a subset I ⊂ {1, 2, …, m} with |I| = k, we define a function

m_{I} : O → R, X ↦ \frac{det(X_{ij})_{i∈I,1≤j≤k}}{det(^{t}XX)^{\frac{1}{2}}} ,  (4.1.1)

where X_{ij} is the (i, j) component of the matrix X.

Let {e_{1}, …, e_{m}} be the standard basis of R^{m}. For I = {i_{1}, …, i_{k}} (1 ≤ i_{1} < ⋯ < i_{k} ≤ m), we set e_{I} := e_{i_{1}} ∧ ⋯ ∧ e_{i_{k}}. Then, {e_{I} : |I| = k} forms a basis of the kth exterior tensor space ∧^{k}(R^{m}). Thus, we regard a family of functions m = {m_{I}} as a ∧^{k}(R^{m})-valued function on O. Let π be the standard representation of O(n) on R^{m}. We use the same letter π to denote the kth exterior tensor representation of O(m) on ∧^{k}(R^{m}). Then, the function m : O → ∧^{k}(R^{m}) satisfies

m(aXb^{−1}) = π(a)m(X) for a ∈ O(m) and b ∈ GL_{+}(k, R).

We extend the representation (π, ∧^{k}(R^{m})) of O(m) to H by letting GL_{+}(k, R) act trivially on ∧^{k}(R^{m}). With this notation, we have

m(gX) = π(g)m(X) for g ∈ H.

We now recall a minor summation formula (see [B, exercise III.8.6] for instance):

∑_{I}(det(X_{ij})_{i∈I,1≤j≤k})^{2} = det(^{t}XX).

Hence, |m_{I}(X)| ≤ 1 for any X ∈ O and any I. As O is open dense in M(m, k; R) ≃ R^{n}, we shall regard m_{I} as a bounded function on R^{n} and m as a ∧^{k}(R)-valued bounded function on M(m, k; R).

Theorem 5. Let H = O(m) × GL_{+}(k, R) (m ≥ k), π the representation of H on ∧^{k}(R^{m}) as above, and n = mk. Then the set of multipliers {m_{I}} defines a bounded translation invariant operator

T : L^{2}(R^{n}) → ∧^{k}(R^{m}) ⊗ L^{2}(R^{n}).

This operator satisfies the invariance condition

(π(g) ⊗ l_{g})T = T ∘ l_{g} for g ∈ H.  (4.1.2)

Conversely, any bounded translation invariant operator L^{2}(R^{n}) → ∧^{k}(R^{m})⊗ L^{2}(R^{n}) satisfying (4.1.2) is a scalar multiple of T.

We will say the operator characterized by this theorem is the exterior Riesz transform.

Remark 4.1.1. If k = 1 then det(^{t}XX)^{\frac{1}{2}} is nothing but the norm |X| of a vector X ∈ R^{n} and m_{I}(X) = \frac{X_{i}}{|X|} for I = \{i\} (1 ≤ i ≤ n). Thus, Theorem 5 in the case k = 1 corresponds to Stein’s Theorem characterizing the usual Riesz transforms (see Fact 1.1).

Remark 4.1.2. Theorem 5 has the following two distinguishing features: 1) the dimension of the representation space ∧^{k}(R^{m}) is no longer one dimensional, thus Corollary 2.1.1 does not apply; 2) the orbit is not a reductive symmetric space, thus it does not fit with Corollary 2.1.2 either. Nevertheless, Theorem 5 asserts that one can characterize invariant multipliers up to scalar by the invariance condition. The idea of the following proof is to show that there is a reductive symmetric space for which the dimension of the space of homomorphisms dominate the dimension of the space of homomorphisms for our space.

4.2 Proof of Theorem 5

Proof of Theorem 5. We apply Theorem 1. Since O is open dense in R^{n}, Theorem 5 is a consequence of the following multiplicity-free result:

Lemma 4.2.1. For a representation π of O(m), we shall denote by π̃ its extention to H = O(m) × GL_{+}(k, R) by letting GL_{+}(k, R) act trivially.

1) For any irreducible (finite dimensional) representation π of O(m), we have

Hom_{H}(π̃, C_{bdd}(O)) ≤ 1.

2) Furthermore, if π is the natural representation of O(m) on the exterior algebra ∧^{k}(R^{m}), then

Hom_{H}(π̃, C_{bdd}(O)) = 1,

and the image of π̃ in C_{bdd}(O) coincides with the complex vector space spanned by the basis {m_{I} : |I| = k}.

Proof of Lemma 4.2.1. We recall G_{1} = O(m) and G_{2} = GL_{+}(k, R), and H = G_{1} × G_{2}. We write C(O)^{G_{2}} for the set of G_{2}-invariant continuous functions of O. Then, C(O)^{G_{2}} is a G_{1}-submodule of C(O), and we have a natural bijection:

Hom_{H}(π̃, C(O)) ≃ Hom_{G_{1}}(π, C(O)^{G_{2}}).

Let us consider the right-hand side. We begin with the H-action on M(m, k; R). It follows from the Gram–Schmidt orthogonalization procedure that H acts

transitively on O. Let L be the isotropy subgroup at (\begin{matrix}I_{k} \\ O \end{matrix}) ∈ O. Then, L is

given by

L = \{((\begin{matrix}b & 0 \\ 0 & c \end{matrix}) , b) : b ∈ SO(k), c ∈ O(m − k)\}
≃ SO(k) × O(m − k).

Thus, we can identify O with the homogeneous space H/L.

Let ι : G_{1} → H, a ↦ (a, I_{k}) be the natural injection. Then, it is not difficult to see that the pull-back ι^{∗} induces isomorphisms of G_{1}-modules:

C(O)^{G_{2}} ≃ C(H/L)^{G_{2}} ≃ C(G_{1}/ι^{−1}(L(I_{n} ×G_{2}))).

In our setting, L(I_{k} ×G_{2}) = (SO(k) × O(m − k)) × GL_{+}(k, R), and therefore

C(O)^{G_{2}} ≃ C(O(m)/(SO(k) × O(m − k))).

Thus we have shown

Hom_{H}(π̃, C(O)) ≃ Hom_{O(m)}(π, C(O(m)/(SO(k) × O(m − k)))).

Since O(m)/(SO(k)×O(m−k)) is a reductive symmetric space, the dimension of the right-hand side is not greater than one by a theorem of É. Cartan. Hence,

dim Hom_{H}(π̃, C_{bdd}(O)) ≤ dim Hom_{H}(π̃, C(O)) ≤ 1.

This shows the first statement. We have already seen that the representation of H on C-span {m_{I} : |I| = k} is isomorphic to the kth exterior representation tensor ∧^{k}(R^{m}). Hence, the second statement follows.

In Theorem 9, we shall discuss L^{p}-boundedness of the exterior Riesz multipliers.

5 Classification of invariant L^{p}-bounded operators

We have found some explicit examples of invariant multipliers (Theorems 2, 3, 4, and 5) in the framework of Strategy 2. We shall determine for which p they define L^{p}-multipliers. The main results of this section are Theorems 6, 7, 8, and 9. We find a feature that L^{p}-bounded invariant operators arising from Strategy 2 are ‘rare’ if p ≠ 2, in the sense that they are built from known examples such as Riesz transforms.

5.1 Algebra of L^{p}-bounded operators—quick review

Standard multiplier theory tells us that a multiplier operator bounded on L^{p}(R^{n}) must also be bounded on L^{2}(R^{n}), see for example [Ho, Corollary 1.3]. This also holds in the vector valued case, namely, for a finite dimensional vector space V, a multiplier operator bounded from L^{p}(R^{n}) → V ⊗ L^{p}(R^{n}) must be also bounded from L^{2}(R^{n}) → V ⊗L^{2}(R^{n}). There are some sufficient conditions for a bounded function to be an L^{p}-multiplier, but there are no general criteria. Hence, we are tempted to ask for which set of p the multiplier operators we have seen remain bounded.

We begin with a brief summary of some known results. For 1 ≤ p ≤ ∞ we denote by M_{p}(R^{n}) the set of bounded functions m on R^{n} such that the corresponding translation invariant operators T_{m} are bounded on L^{p}(R^{n}).

Fact 5.1. Suppose 1 < p, q < ∞.

1) M_{p}(R^{n}) = M_{q}(R^{n}) if \frac{1}{p} + \frac{1}{q} = 1.

2) M_{q}(R^{n}) ⊂ M_{p}(R^{n}) if |\frac{1}{p} − \frac{1}{2} | < |\frac{1}{q} − \frac{1}{2} | .

3) (deLeeuw) If m ∈ M_{p}(R^{l+n}) then m(a, ·) ∈ M_{p}(R^{n}) for a.e. a ∈ R^{l}.

4) (Fefferman’s ball multiplier theorem) χ_{B} ∉ M_{p}(R^{n}), if p ≠ 2 and n ≥ 2. Here χ_{B} is the characteristic function of the unit ball, B, in R^{n}.

5) If m ∈ M_{p}(R^{n}) and A ∈ Aff(R^{n}) then m ∘ A ∈ M_{p}(R^{n}) and A ∘ m ∈ M_{p}(R^{n}).

6) If m_{1} and m_{2} are elements in M_{p}(R^{n}) then m_{1} · m_{2} ∈ M_{p}(R^{n}) and m_{1} + m_{2} ∈ M_{p}(R^{n}).

7) For α ∈ R we have λ_{+}^{iα} ∈ M_{p}(R) (1 < p < ∞).

8) If m ∈ M_{p}(R^{n}) then the function M, defined by M(a, b) = m(a), is in M_{p}(R^{n+l}).

Proof. 1) and 2) See [Ho, Theorem 1.3]. 3) See [T, Theorem 2.4]. 4) See [F1]. The proofs of 5), 6) and 8) are straightforward. 7) See [S, page 96].

5.2 Classification of L^{p}-bounded operators from Sections 3 and 4

Suppose we are in the setting of Subsection 3.1.

Theorem 6 ((GL(2), R^{3}) case). The operator characterized by Theorem 2 does not extend to a bounded operator on L^{p}(R^{3}) (1 < p < ∞) except for p = 2.

Proof. Let m_{δ}^{β} be the multiplier operator in Theorem 2. Assume m_{δ}^{β} ∈ M_{p}(R^{3}) for some δ = ++, +−, −− and β ∈ R. Then, also m_{δ}^{−β} ∈ M_{p}(R^{n}), because m_{δ}^{−β} is the complex conjugate of m_{δ}^{β}. It then follows from Fact 5.1 (6) that their product m_{δ}^{β} · m_{δ}^{−β} is also in M_{p}(R^{n}). Now we observe that the product m_{δ}^{β} · m_{δ}^{−β} is the characteristic function χ_{O_{δ}} of the orbit O_{δ}. Thus, we have proved the implication:

m_{δ}^{β} ∈ M_{p}(R^{3}) for some β ∈ R ⇒ O_{δ} ∈ M_{p}(R^{3}).  (5.2.1)

Let us show that χ_{O_{δ}} ∈ M_{p}(R^{3}) only if p = 2. The case δ = +− can be reduced to the others because T_{χ_{O_{+−}}} = id −T_{χ_{O_{++}}} − T_{χ_{O_{−−}}}. Fix a > 0. For δ = ++ or = −−, the intersection of O_{δ} with the hyperplane λ_{1} + λ_{2} = a (δ = ++), = −a (δ = −−) is the ellipse {(x, y) ∈ R^{2} : a^{2} − x^{2} − 4y^{2} > 0} where x = λ_{1} − λ_{2} and y = λ_{3}. Hence, p has to be equal to 2 by Facts 5.1 (3), (4) and (5).

Next, we consider the setting of Subsection 3.2. In the same way we have

Theorem 7 ((GL(2) × GL(2), R^{4}) case). The operator characterized by Theorem 3 does not extend to a bounded operator on L^{p}(R^{4}) (1 < p < ∞) except for p = 2.

In this case the relevant operator, after a suitable change of variables, is the one corresponding to the characteristic function of the set {λ : λ_{1}^{2} + λ_{2}^{2} ≥ λ_{3}^{2} + λ_{4}^{2}}. Taking the intersection with the two plane λ_{1} = λ_{2} = 1 or alike, we see that the operators are bounded on L^{p}(R^{4}) only if p = 2 by Fact 5.1 (3) and (4).

It also follows in a similar manner in the setting of Subsection 3.3:

Theorem 8 ((SO(p, q) × R_{+}, R^{p+q}) case).

1) The operator characterized by Theorem 4 does not extend to a bounded operator on L^{r}(R^{p+q}) (1 < r < ∞) except for r = 2 if p + q ≥ 3.

2) If p + q = 2, the operator is bounded on L^{r}(R^{2}), for all 1 < r < ∞.

Proof. 1) For p + q ≥ 3 the guiding operator is the one given by the characteristic function of the set {λ : λ_{1}^{2} + ⋯ + λ_{p}^{2} ≥ λ_{p+1}^{2} + ⋯ + λ_{p+q}^{2}}, where we might assume that p ≥ q. The first statement then follows as before.

2) If p = q = 1 we are considering the multiplier

∑_{ε_{1}=±,ε_{2}=±} c_{ε_{1},ε_{2}}(λ_{1} + λ_{2})_{ε_{1}}^{− \frac{1}{2} i(α+β)}(λ_{1} − λ_{2})_{ε_{2}}^{− \frac{1}{2} i(α−β)}.

We want to show that the corresponding multiplier operator is bounded on L^{r}(R^{2}) for all 1 < r < ∞. To do this it is enough to consider the factors separately

m_{1,ϵ}^{α}(λ) = (λ_{1} + λ_{2})_{ϵ}^{iα},
m_{2,ϵ}^{α}(λ) = (λ_{1} − λ_{2})_{ϵ}^{iα},

because of Fact 5.1 (6). Clearly, they are all simple rotations of the multiplier

m(λ) = \begin{cases}|λ_{1}|^{iα} & if λ_{1} > 0, \\ 0 & otherwise, \end{cases}

which, by Facts 5.1 (7) and (8), is in M_{r}(R^{n}) for 1 < r < ∞.

Theorem 9 (exterior Riesz multipliers). Suppose n = mk (m ≥ k), and we are in the setting of Subsection 4.2. Then there exists a non-trivial bounded translation invariant operator

T : L^{p}(R^{n}) → ∧^{k}(R^{m}) ⊗ L^{p}(R^{n})

satisfying the O(m) × GL_{+}(k, R)-invariance condition (4.1.2), if and only if one of the following conditions are satisfied:

• 1 < p < ∞ and k = 1 (Riesz transforms, see [S, page 57 and Theorem 3] or [T, page 269]).

• p = 2 and k arbitrary (Theorem 5).

Proof. Since a bounded translation invariant operator on L^{p} is automatically bounded on L^{2}, it follows from Theorem 5 that T must be a scalar multiple of the exterior Riesz transform defined by the set of multipliers

m_{I} : O → R, X ↦ \frac{det(X_{ij})_{i∈I,1≤j≤k}}{det(^{t}XX)^{\frac{1}{2}}} ,

(see (4.1.1)) for I ⊂ {1, 2, …, m} with p ≠ 2 and |I| = k. All we have to do is to prove that this operator is not L^{p}-bounded if k ≥ 2. To see this, we restrict m_{I} on the two dimensional subspace of M(m, k; R)(≃ R^{n}) defined by the system of linear equations:

X_{11} − X_{22} = 2 + ϵ_{1}
X_{12} − X_{21} = ϵ_{2}
X_{ii} = 1 + ϵ_{i} (3 ≤ i ≤ k)
X_{ij} = ϵ_{ij} (if max(i, j) ≥ 3 and i ≠ j),

where ϵ_{i} and ϵ_{ij} are parameters. If all of ϵ_{i} and ϵ_{ij} are zero this subspace admits the following coordinates

(\begin{matrix}A & O_{2,k−2} \\ O_{k−2,2} & I_{k−2} \\ O_{m,2} & O_{m,k−2} \end{matrix}) ,

where

A = (\begin{matrix}x + 1 & y \\ −y & x − 1 \end{matrix})

and O_{m,k} is the zero m × k matrix. Then m_{{1,...,k}}(X) = sgn(x^{2} + y^{2} − 1) (= 2χ_{B} − 1) which does not define an L^{p}-bounded operator by Fefferman’s ball multiplier theorem, see Fact 5.1 (4). For sufficiently small ϵ_{i} and ϵ_{ij}, the restriction of m_{I} to the corresponding two dimensional vector space is of the form χ_{B′} − 1, where B′ is the interior of a certain ellipse depending on the parameters ϵ_{i} and ϵ_{ij}. Again, it is not L^{p}-bounded by Fact 5.1 (4) and (5). By deLeeuw’s theorem, Fact 5.1 (3), m_{{1,...,k}} ∉ M_{p}(R^{n}). Similarly for m_{I} for any I with |I| = k (k ≥ 2). This completes the proof of Theorem 9.

Hence, we have determined for which the invariant multipliers in Sections 3 and 4 define L^{p}-multipliers (1 < p < ∞).

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