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\controldates{30-JAN-2003,30-JAN-2003,30-JAN-2003,30-JAN-2003}
 
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\begin{document}

\title[Levi's parametrix for sub-elliptic
non-divergence operators]{Levi's parametrix for some sub-elliptic
non-divergence form operators}
\author[A. Bonfiglioli]{Andrea Bonfiglioli}
\address{Dipartimento di Matematica,
Universit\`{a} degli Studi di Bologna,
Piazza di Porta S.~Donato 5, 40126 Bologna, Italy}
\email{bonfigli@dm.unibo.it}
\thanks{Investigation supported by the University of Bologna
Funds for selected research topics.}
\author[E. Lanconelli]{Ermanno Lanconelli}
\address{Dipartimento di Matematica,
Universit\`{a} degli Studi di Bologna,
Piazza di Porta S.~Donato 5,  40126 Bologna, Italy}
\email{lanconel@dm.unibo.it}
\author[F. Uguzzoni]{Francesco Uguzzoni}
\address{Dipartimento di Matematica,
Universit\`{a} degli Studi di Bologna,
Piazza di Porta S.~Donato 5, 40126 Bologna, Italy}
\email{uguzzoni@dm.unibo.it}

\subjclass[2000]{Primary 35A08, 35H20, 43A80; Secondary 35A17, 35J70}

\date{November 11, 2002}

\commby{Michael Taylor}


\keywords{Non-divergence sub-elliptic operators,
 stratified groups, fundamental solutions, parametrix method}
\begin{abstract}
 We construct the fundamental solutions for the
 sub-elliptic operators in non-divergence form
  ${\textstyle\sum_{i,j}}
  a_{i,j}(x,t)\,X_iX_j-\partial_t$ and
  ${\textstyle\sum_{i,j}}a_{i,j}(x)\,X_iX_j$,
 where the $X_i$'s form a stratified system of H\"ormander
 vector fields and $a_{i,j}$ are
 H\"older continuous functions belonging to a suitable
 class of ellipticity.
\end{abstract}
\maketitle
\section{Main results}\label{sec:parametrix.note.intro}
 Let $\G$ be a stratified group, i.e., let $\G$ be a connected and simply
 connected Lie group such that its Lie algebra $\g$
 admits the decomposition
 \[\g=\mathfrak{G}_1\oplus\cdots\oplus\mathfrak{G}_r, \text{ where }
  [\mathfrak{G}_1,\mathfrak{G}_i]=\mathfrak{G}_{i+1},
  [\mathfrak{G}_1,\mathfrak{G}_r]=\{0\}\,.\]
 If $\{X_1,\ldots,X_m\}$ is a basis of $\mathfrak{G}_1$,
  we call the second order
 linear operator $\sum_{i=1}^m X_i^2$ a sub-Laplacian on $\G$.
 The study of general second order linear PDE's sum
 of squares of vector fields (started with H\"ormander's paper \cite{Ho})
 has significantly developed after the works by Folland \cite{F}
 and by Rothschild and Stein \cite{RS}; in the latter
 it has been shown that any general H\"ormander
 operator can be locally approximated by a sub-Laplacian on
 a (free) stratified group $\G$. Since the appearance of this result,
 the study of stratified groups (also known as Carnot groups)
 has received great impulse from many authors and
 from different points of view. Second order linear and nonlinear
 PDE's of sub-elliptic type arise in various settings: geometric theory of
 several complex variables, curvature problems for CR-manifolds,
 sub-Riemannian geometry, diffusion processes, control theory,
 and human vision; see, e.g.,
 \cite{CLM,JL,LancPascPol,Montgomery,Petitot,ST,VSC}.
 The underlying algebraic structures of
 all these equations are stratified groups $\G$. 
\enlargethispage*{1000pt}

 In this note, we present a result from a wider project aimed
 to apply analysis on stratified groups
 to the study of \emph{fully non-linear}  PDE's of
 sub-elliptic type arising
 in the geometric theory of several complex variables such as
 the Levi-curvature equation,
 which has achieved rising concern in the last few years (see
 \cite{CLM,HuK,M,ST}).
 The core of this project is to study the
 fundamental solutions for the
 linear parabolic-type operators in \emph{non-divergence}
 form on $\G\times \R\equiv\R^{N+1}$,
\begin{equation}\label{model}
  \mathcal{H}={\textstyle\sum_{i,j}}
  a_{i,j}(x,t)\,X_iX_j-\partial_t\qquad (x\in\G,\,\,t\in\R),
\end{equation}
 where the \pagebreak\ $X_i$'s are as above and $a_{i,j}$ are
 H\"older continuous functions belonging to a suitable
 class of ellipticity. Analogously, we study the fundamental solutions
 for the linear sub-elliptic non-divergence operators
 \[\mathcal{L}={\textstyle\sum_{i,j}} a_{i,j}(x)\,X_iX_j.\]
 Operators such as $\mathcal{H}$ and $\mathcal{L}$ naturally
 intervene in the linearization of non-linear sub-elliptic equations.
 Then, the non-divergence form of
 $\mathcal{H}$ and $\mathcal{L}$ is a crucial
 requirement in view of our applications in non-linear
 PDE's analysis.

 Several results concerning divergence form operators
 $-\sum_{i,j} X_i^*(a_{i,j}X_j)$ are pre\-sent in the literature, both
 for linear and quasi-linear equations.
 Harnack's inequality, regularity results for solutions, existence and size
 estimates of the Green's function can be found, e.g., in the
 papers \cite{Cap1,CDG,CGL,FLW,L2,X}.
 On the contrary, at the authors' knowledge,
 very few papers are devoted to non-divergence form operators.
 We may just quote a work by Bramanti and Brandolini
 \cite{BB1} and the recent paper by Capogna and Han \cite{CapognaHan},
 where a priori estimates in $L^p$ and H\"older spaces, respectively,
 are proved. 

Here we discuss the existence
 and the well-behaved properties of the fundamental solutions
 for $\mathcal{H}$ and $\mathcal{L}$. We construct a fundamental
 solution $\Gamma$ for $\mathcal{H}$ by means of the well-known
 Levi's parametrix method, whereas
 a fundamental solution $\gamma$ for $\mathcal{L}$ is derived by a
 $t$-saturation argument. We explicitly remark that several new difficulties
 arise in the adaptation to our setting of the cited parametrix method.
 The major of these difficulties consists in obtaining suitable
 \emph{uniform} estimates for the relevant frozen operators (see
 \eqref{unif.gaussian.required.param.}). The derivation of these
 estimates is outlined in Section \ref{sec:background}. Here we
 only point out that we make use of a lifting procedure to free
 stratified groups, and of an accurate analysis of the equivalence
 (via well-estimated automorphisms) of all sub-Laplacians on such
 free stratified groups. Moreover, the lack of knowledge of an
 explicit expression for the parametrix clearly
 makes Levi's method
 more involved than in the classical context. 

By integrating $\Gamma$ over the time variable $t$, we
 are then able to construct also a local fundamental solution $\gamma$
 for $\mathcal{L}$. This can be done provided suitable
 \emph{long-time} estimates of $\Gamma$ are established. We stress
 that, whereas optimal small-time estimates of $\Gamma$ and of its
 derivatives can be directly obtained from the construction of
 $\Gamma$, a more delicate matter is to show long-time estimates.
 We are able to obtain the latter up to a suitable modification of the
 coefficients $a_{i,j}$ outside a compact set (see Lemma
 \ref{41.parametrix} below).

We hereafter announce our main results
 (all the notations are explained below).
 For the complete proofs we refer to \cite{BLU2}.
\begin{theorem}\label{main.par.parametrix}
 Suppose the coefficients $a_{i,j}$ of $\mathcal{H}$ belong
 to the ellipticity class $\mathcal{M}_\Lambda$ and to
 the H\"older space $\Gamma^\alpha(\R^{N+1})$.
 Then there exists a fundamental solution $\Gamma$ for
 $\mathcal{H}$, with the properties listed below.
 \begin{itemize}
  \item[{\rm(i)}]
  $\Gamma$ is a continuous function away from the diagonal of
  $\R^{N+1}\times\R^{N+1}$. Moreover, for every fixed $\zeta\in
  \R^{N+1}$, $\Gamma(\cdot;\zeta)\in
  \Gamma^{2+\alpha}_{\text{\emph{loc}}}
  (\R^{N+1}\setminus\{\zeta\})$ and we
  have
  $\mathcal{H}\big(\Gamma(\cdot;\zeta)\big)=0$ in
  $\R^{N+1}\setminus\{\zeta\}$.
  \item[{\rm(ii)}]
  $\Gamma(x,t;\xi,\tau)=0$ for $t\leq \tau$. Moreover, there
  exists a positive constant $M$ and, for every $T>0$, there exists
  a positive constant $c(T)$ such that, for $0T_1$ be such that $(T_2-T_1)\mu$
  is small enough. Then, for every $f\in \Gamma^{\beta}
  (\RN\times[T_1,T_2])$ (where $0<\beta\le \alpha$) and $g\in C(\RN)$ satisfying the growth
  condition
  $|f(x,t)|,|g(x)|\leq c\,\exp(\mu\,d^2(x))$ for some constant
  $c>0$, the function (defined for $x\in \RN$, $t\in(T_1,T_2]$)
  \[\qquad \quad  u(x,t)=
   \int_{\RN}\Gamma(x,t;\xi,T_1)\,g(\xi)\,\myd\xi
   +\int_{\RN\times [T_1,t]}
   \Gamma(x,t;\xi,\tau)\,f(\xi,\tau)\,\myd\xi\,\myd\tau,\]
   belongs to the class
   $\Gamma^{2+\beta}_{\text{\emph{loc}}}
  (\RN\times(T_1,T_2))\cap C(\RN\times[T_1,T_2])$. 
  Moreover, $u$ is a solution to
   the Cauchy problem
   $\mathcal{H}u=-f$ in $\RN\times(T_1,T_2)$, $u(\cdot,T_1)=g$.
 \end{itemize}
\end{theorem}
\begin{theorem}\label{main.ell.parametrix}
 Suppose that the dimension $m$ of $\mathfrak{G}_1$ is greater than two.
 Suppose the coefficients
 $a_{i,j}$ of $\mathcal{L}$ are in the H\"older space $\Gamma^\alpha(\RN)$ and
 in the ellipticity class $\mathcal{M}_\Lambda$.
 For every fixed bounded open set $\Omega\subset \RN$,
 there exists a fundamental solution $\gamma$ for
 $\mathcal{L}$ in $\Omega$, with the properties listed below.
 \begin{enumerate}
  \item[{\rm(i)}]
  $\gamma$ is a continuous function away from the diagonal of
  $\RN\times\RN$. Moreover, for every fixed $\xi\in
  \RN$, $\gamma(\cdot,\xi)\in \Gamma^{2+\alpha}_{\text{\emph{loc}}}
  (\RN\setminus\{\xi\})$ and we
  have $\mathcal{L}\big(\gamma(\cdot,\xi)\big)=0$ in
  $\Omega\setminus\{\xi\}$.
  \item[{\rm(ii)}]
  For every compact set $K\Subset\RN$, there
  exists a positive constant $c$ such that
 \[0\leq \gamma(x,\xi) \leq c\,(1+d(x,\xi)^{2-Q}),\quad
 \xi\in K,\,\,x\in \RN.\]
  \item[{\rm(iii)}]
  For every $\psi\in C_0^\infty(\RN)$, the convolution
  $w(x)=\int_{\RN}\gamma(x;\xi)\,\psi(\xi)\,\myd\xi$
  belongs to the class $\Gamma^{2+\alpha}_{\text{\emph{loc}}}
  (\RN)$ and we have
  $\mathcal{L}w=-\psi$ in $\Omega$.
 \end{enumerate}
\end{theorem}

 In the above theorems, we have denoted
 by $Q=\sum_{j=1}^r j\,\text{dim}(\mathfrak{G}_j)$ the homogeneous dimension of
 $\G$, by $d$ a fixed homogeneous norm on $\G$ and we have set
 $d(x,y)=d(y^{-1}\circ x)$. Moreover, given $\Lambda>1$, we have denoted by
 $\mathcal{M}_\Lambda$ the ellipticity class of the $m\times m$ symmetric
 matrices $A$ such that $\Lambda^{-1}|\xi|^2\leq \langle A\xi,\xi\rangle
 \leq \Lambda |\xi|^2$, for every $\xi\in \R^m$.
 Finally we have denoted by $\Gamma^\beta$, $\Gamma^{2+\beta}$ the
 appropriate sub-elliptic H\"older spaces. 

As is well known, Levi's parametrix method
 requires the knowledge of several good properties of the frozen
 constant coefficient parabolic-type operators
\begin{equation}\label{frozens.parametrix}
 \mathcal{H}_A=\mathcal{L}_A-\partial_t=
 \textstyle\sum_{i,j=1}^m a_{i,j}\,X_iX_j-\partial_t,
\end{equation}
 where $A=(a_{i,j})_{i,j}$ is a fixed matrix in the ellipticity class
 $\mathcal{M}_\Lambda$. For instance, a crucial role
 is played by the following \emph{uniform Gaussian
 estimates} for the fundamental solutions $\Gamma_A$ of
 $\mathcal{H}_A$:
\begin{gather}\label{unif.gaussian.required.param.}
 \begin{split}
  \big|X_{i_1}\cdots X_{i_p}(\partial_t)^q
   \,\Gamma_{A}(x,t)-
   X_{i_1}\cdots X_{i_p}(\partial_t)^q
   \,\Gamma_{B}(x,t)\big|\qquad\qquad\qquad\\
  \qquad\qquad\qquad\qquad
   \leq c_{\Lambda,p,q}\, \|A-B\|^{1/r}\,t^{-(Q+p+2q)/2}\exp\Big(
   -\frac{d^2(x)}{c_\Lambda\,t}\Big),
   \end{split}
\end{gather}
 for every $A,\,B\in \mathcal{M}_\Lambda$ (here, $\|A\|$ denotes the
 matrix norm $\max_{|\xi|=1}|A\xi|$).
  Gaussian estimates, but not uniform,
 for heat kernels on Lie groups were proved
 by Jerison and S\`anchez-Calle \cite{JS},
 by Kusuoka and Stroock \cite{KS2}
 and by Varopoulos, Saloff-Coste and Coulhon \cite{VSC}.
 Uniform
 estimates, but not Gaussian,
 for families of H\"or\-man\-der
 operators generalizing \eqref{model},
 were proved by Rothschild and Stein \cite{RS} and by Bramanti and Brandolini
 \cite{BB1}.

Since the derivation of uniform and Gaussian estimates
 is a non-trivial task and is
 a key point in proving the main results we announce here, in
 Section \ref{sec:background} we briefly recall how
 \eqref{unif.gaussian.required.param.} can be established
 by a direct approach. In Section \ref{sec:parametrixmethod}, we
 briefly outline the proof of Theorem
 \ref{main.par.parametrix}, describing our adaptation of the 
 parametrix method and how uniform estimates naturally intervene.
 Finally, we sketch how long-time estimates are used in order to prove
 Theorem \ref{main.ell.parametrix} employing both a $t$-saturation
 and an approximation argument.
\section{Background material}\label{sec:background}

 The classical Levi's parametrix method
 (i.e., if the underlying stratified group is
 the usual Euclidean space $(\RN, +)$) exploits at various levels the explicit
 knowledge of the fundamental solution for the strictly
 parabolic constant coefficient operator
 $\sum_{i,j=1}^Na_{i,j}\,\partial_i\partial_j-\partial_t$ 
 ($A=(a_{i,j})_{i,j}$ being a positive-definite matrix). Obviously, this
 fundamental solution is given by the composition of the fundamental solution of the
 classical heat operator $\sum_{i=1}^N(\partial_i)^2-\partial_t$
 on $\R^{N+1}$ with a linear change of
 coordinates related to the matrix $A^{-1/2}$. 

A na\"{\i}ve idea in studying the fundamental solutions
 for \eqref{frozens.parametrix} and in
 approaching \eqref{unif.gaussian.required.param.}
 is to ask if something similar may occur
 in the case of general stratified groups $\G$.
 Namely, we ask if all sub-Laplacians on $\G$ can be put
 (via a diffeomorphism) into a fixed canonical form.
 To this end, if $A=(a_{i,j})_{i,j\leq m}$
 is a positive-definite symmetric matrix, we first remark that
 $\mathcal{L}_{A}=\sum_{i,j=1}^m a_{i,j}\,X_iX_j$ may
 be rewritten as
 $\sum_{i=1}^m Y_i^2$, where $Y_i=\sum_{j=1}^m
 (A^{1/2})_{i,j}\,X_j$.
 As a consequence, it is natural to ask if
 there exists a diffeomorphism $T_A:\G\to\G$
 such that (in the new coordinate system defined by $T_A$)
 the vector field $Y_i$ is turned into the (fixed)
 left-invariant vector field $Z_i$ agreeing at the origin with $\partial_i$.
 In this way, if we set $\myL=\sum_{i=1}^m Z_i^2$ (we call
 $\myL$ the canonical sub-Laplacian on $\G$),
 the sub-Laplacian $\sum_{i=1}^m Y_i^2$ is turned into $\myL$, i.e.,
\begin{equation}\label{turning.LA.into.Delta}
 \mathcal{L}_A(u\circ T_A)=(\myL u)\circ T_A,\quad
 \text{for every $u\in C^{\infty}(\G)$}.
\end{equation}
 In the classical case when $X_i=\partial/\partial x_i$,
 this problem always has a solution.
 On the contrary, counterexamples can be given showing that
 $T_A$ may not exist for general stratified groups $\G$
 and, when it exists, it may be non-linear (see \cite{BU2}).
 Broadly speaking, the problem relies on
 the commutativity properties of the $X_i$'s: if the linear
 dependence relations among
 commutators up to order $r$ of $X_1,\ldots,X_m$
 are the least possible, then the above problem
 does have a solution.
 More explicitly, if
 $\G$ is a \emph{free stratified group} (i.e., its Lie algebra is isomorphic
 to a free nilpotent Lie algebra) then there exists an
 automorphism $T_A$ of the group $\G$ satisfying
 \eqref{turning.LA.into.Delta}. 

As a straightforward consequence, when $\G$ is free,
 we are able to obtain
 the fundamental solution
 $\Gamma_A$ for $\mathcal{H}_A=\mathcal{L}_A-\partial_t$ simply
 as the composition of $T_A$ with the
 fundamental solution $\Gamma_\G$ for the \emph{fixed} canonical
 heat operator $\myL-\partial_t$. Indeed, if $\G$ is free,
 it turns out that
 \begin{equation}\label{gammaAgammaG}
 \Gamma_A(x,t;\xi,\tau)=
 |\det \mathcal{J}_{T_A}(x)|\,\,
 \Gamma_{\G}(T_A(x),t;T_A(\xi),\tau),
 \quad
 x,\,\xi\in \RN,\,\,t,\,\tau\in \R
 \end{equation}
 ($\mathcal{J}_{T_A}$ denotes the Jacobian matrix of $T_A$).
 Thanks to this somewhat explicit representation of the fundamental solution
 for $\mathcal{H}_A$, the next step in order to obtain the uniform estimates in
 \eqref{unif.gaussian.required.param.} is to
 establish \emph{ad hoc} uniform estimates for $T_A$.
 To this end it can be proved that
 $|\det\mathcal{J}_{T_A}(x)|$ turns out to be a uniformly bounded
 constant and that
 $d(T_A(x),T_{B}(x))\leq c_{\Lambda}\,
  \|A-B\|^{1/r}\,d(x)$ for every $A,\,B\in\mathcal{M}_\Lambda$ and
  $x\in \G$. Consequently (see \cite{BLU}), when $\G$ is free,
  the uniform Gaussian estimates
  \eqref{unif.gaussian.required.param.} follow from
  \eqref{gammaAgammaG} and from the following
 Gaussian estimates of the (fixed) fundamental solution $\Gamma_\G$:
   \[\big| X_{i_1}\cdots X_{i_p}(\partial_t)^q
   \,\Gamma_\G(x,t)\big|\leq
   c_{\G}\, t^{-(Q+p+2q)/2}\,\exp\Big(
   -\frac{d^2(x)}{c_{\G}\,t}\Big).\]

In order to handle the case of an arbitrary stratified
 group $\G$, our main tool is to \emph{lift} $\G$ to a free
 stratified group $\widetilde{\G}$ in such a way that $\myL$ is lifted
 to $\Delta_{\widetilde{\G}}$. The lifting technique introduced
 by Rothschild and Stein in \cite{RS} together with some further remarks on
 the homogeneity properties of stratified groups,
 allows us to prove the following lifting result (see also
 \cite{BU1}): If $\G$ is an $N$-dimensional stratified group then there
 exists an $H$-dimensional free stratified group $\widetilde{\G}$
 (with $H\geq N$) such that, denoting by $\pi:\R^{H}\to\RN$
 the projection on the first $N$ coordinates,
 for every $u\in C^{\infty}(\G)$, we have
 $\widetilde{Z}_i(u\circ \pi)=
  (Z_i u)\circ \pi$, 
  where $\sum_{i=1}^m Z_i^2$ and $\sum_{i=1}^m \widetilde{Z}_i^2$
  are the canonical sub-Laplacians $\myL$ and
  $\Delta_{\widetilde{\G}}$, respectively.
  Since $\mathfrak{G}_1=\text{span}\{Z_1,\ldots,Z_m\}$,
  the lifting result leads to a correspondence
  between the operators $\mathcal{H}_A=
  \sum_{i,j=1}^m a_{i,j}\,X_iX_j-\partial_t$ on $\G\times\R$ and
  $\widetilde{\mathcal{H}}_A=
  \sum_{i,j=1}^m a_{i,j}\,\widetilde{X}_i\widetilde{X}_j-\partial_t$
  on $\widetilde{\G}\times\R$. 

The lifting result also allows us to establish a natural
  relation between the fundamental solution $\Gamma_A$ for $\mathcal{H}_A$
  and $\widetilde{\Gamma}_A$ for $\widetilde{\mathcal{H}}_A$.
  Indeed, we have
\begin{equation}\label{fundsol.liftate}
 \textstyle
  \Gamma_A(x,t)=\int_{\R^{H-N}}\widetilde{\Gamma}_A((x,\widehat{x}),t)
   \,\myd \widehat{x},\qquad \text{for every $x\in \G,\,t\in \R$},
\end{equation}
 where $(x,\widehat{x})$ denotes the point of $\RN\times
 \R^{H-N}$. This fact, together with the established uniform
 estimates for free groups, allows us to prove
 \eqref{unif.gaussian.required.param.} in the general case.
 Indeed, by means of the integral representation in
 \eqref{fundsol.liftate}, it is possible to transfer the uniform
 estimates for $\{\widetilde{\Gamma}_A\}$ to the
 uniform estimates for $\{\Gamma_A\}$.
\section{Levi's parametrix method}\label{sec:parametrixmethod}
 We first fix some notation. The point of $\R^{N+1}$ will be denoted by
 $z=(x,t)$ ($x\in \RN$, $t\in \R$) and analogously
 $\zeta=(\xi,\tau)$. The coefficients
 $a_{i,j}$ of the operator $\mathcal{H}$ in \eqref{model}
 will be assumed to satisfy
\begin{equation*}
  \big(a_{i,j}(x,t)\big)_{i,j}\in\mathcal{M}_\Lambda,\quad
  |a_{i,j}(x,t)-a_{i,j}(x',t')|\leq
  L\,(d(x,x')^\alpha+|t-t'|^{\alpha/2}).
\end{equation*}
 All constants will be meant to depend
 on $\Lambda, \,L,\,\alpha$. Following
 \eqref{frozens.parametrix}, we set for brevity
 $\mathcal{H}_{\zeta_0}:=
 \mathcal{H}_{A(\zeta_0)}$ (and analogously, $\Gamma_{\zeta_0}:=
 \Gamma_{A(\zeta_0)}$). The uniform Gaussian estimates discussed
 in Section \ref{sec:background} (see, e.g., \eqref{unif.gaussian.required.param.})
 allow us to prove the
 following estimates of the fundamental solutions $\Gamma_{\zeta_0}$ for the frozen
 operators $\mathcal{H}_{\zeta_0}$ (uniform in $\zeta_0$):
\begin{gather}\label{unif.bis.totali}
 \begin{split}
  & c^{-1}\,\mathbf{E}(x, c^{-1}\,t)\leq \Gamma_{\zeta_0}(x,t)\leq
  c\,\mathbf{E}(x, c\,t);\\
 & \big|X_{i_1}\cdots X_{i_p}(\partial_t)^q
   \,\Gamma_{\zeta_0}(x,t)\big|\leq
    c_{p,q}\,t^{-(p+2q)/2}\,\mathbf{E}(x, c\,t);\\
 & \big|X_{i_1}\cdots X_{i_p}(\partial_t)^q
   \,\Gamma_{\zeta_0}(x,t)-
   X_{i_1}\cdots X_{i_p}(\partial_t)^q
   \,\Gamma_{\zeta_1}(x,t)\big|\\
  &\qquad \qquad\qquad \qquad \qquad\leq \myc_{p,q}\,\big(d(\xi_0,\xi_1)^{\frac{\alpha}{r}}+
   |\tau_0-\tau_1|^{\frac{\alpha}{2r}}
   \big)\,t^{-(p+2q)/2}\,\mathbf{E}(x, c\,t),
\end{split}
\end{gather}
 where $\mathbf{E}(x,t)=
 t^{-Q/2}\exp(-d(x)^2/t)$.
 We remark that the parametrix method
 outlined below is classical, but several technical
 complications arise in our setting.
 We set (for $z\neq \zeta$)
\begin{equation*}
  Z_1(z;\zeta)=\mathcal{H}\big(
  z\mapsto\Gamma_\zeta(\xi^{-1}\circ x,t-\tau)
  \big)(z).
\end{equation*}
 Since $\mathcal{H}=
 \mathcal{L}-\mathcal{L}_\zeta+\mathcal{H}_\zeta$, we have
 \[Z_1(z;\zeta)=
   \sum_{i,j=1}^m(a_{i,j}(z)-a_{i,j}(\zeta))\,X_iX_j\Gamma_\zeta
  (\xi^{-1}\circ x,t-\tau),\] 
whence \eqref{unif.bis.totali} gives
  $|Z_1(z;\zeta)|\leq c\,(t-\tau)^{\frac{\alpha}{2}-1}\,
   \Gamma_{\zeta_0}(\xi^{-1}\circ x, c\,(t-\tau))$.
   If we inductively define
 \[\textstyle Z_{j+1}(z;\zeta)=
 \int_{\RN\times[\tau,t]}Z_{1}(z;\eta)\,Z_{j}(\eta;\zeta)\,\myd\eta
 \quad (t>\tau),\]
 the following estimate holds (for suitable constants $c_1,\,c_2,\,b_j$)
\begin{equation*}
  |Z_j(z;\zeta)|\leq  c_1^j\,b_j(\alpha)
  \,(t-\tau)^{-1+j\alpha/2}\,
   \Gamma_{\zeta_0}(\xi^{-1}\circ x, c_2\,(t-\tau)).
\end{equation*}
 By means of \eqref{unif.bis.totali}, it can then be proved that the series
 $\Phi(z;\zeta)=\textstyle\sum_{j=1}^\infty Z_j(z;\zeta)$
 totally converges on a suitable domain and
 satisfies the estimate (here $T>0$ and $c(T)>0$ is a constant)
\begin{equation}\label{9b.parametrix}
     |\Phi(z;\zeta)|\leq  c(T)\,(t-\tau)^{\frac{\alpha}{2}-1}\,
   \mathbf{E}(\xi^{-1}\circ x, c\,(t-\tau)),\quad 0\tau$)
 $\Phi(z;\zeta)=Z_1(z;\zeta)+
 \int_{\RN\times[\tau,t]}Z_{1}(z;\eta)\,\Phi(\eta;\zeta)\,\myd\eta$.
 A crucial tool in the adaptation of the parametrix method is
 played by the following non-trivial regularity properties of $\Phi$:
 $\Phi(\cdot;\zeta)$
 and $\Phi(z;\cdot)$ are continuous functions and (for $0\tau$,
\begin{equation}\label{11.parametrix}
  J(z;\zeta)=
  \int_{\RN\times[\tau,t]}\Gamma_\eta(z;z')\,\Phi(z';\zeta)\,\myd z',
  \qquad \Gamma(z;\zeta)=J(z;\zeta)+\Gamma_\zeta(z;\zeta),
\end{equation}
 and extend $\Gamma(z;\zeta)$ to be zero for $t\leq
 \tau$. The good property \eqref{9b.parametrix} of $\Phi$ and the Gaussian estimates
 \eqref{unif.bis.totali} ensure that $\Gamma$ is well posed.
 Exploiting again the estimates \eqref{unif.bis.totali}
 and \eqref{9b.parametrix}, it is not difficult to show that
 $\Gamma$ is continuous away from the diagonal of
 $\R^{N+1}\times\R^{N+1}$ and satisfies
\begin{equation*}
 \begin{split}
  |\Gamma(z;\zeta)|\leq
   c(T)\,\mathbf{E}(\xi^{-1}\circ x, c(t-\tau))\qquad
   (0\tau+1$ and $\xi^{(1)}\in K$). We have denoted by $x^{(1)}$
 the vector of the first $m$ coordinates of $x$.
\end{lemma}
 The proof of Lemma \ref{41.parametrix}
 relies on the weak
 maximum principle for $\mathcal{H}$
 in the class $\mathfrak{C}^2$ (see Proposition
 \ref{prop.max.pr.})
 and on a direct comparison argument. The above long-time estimate of $\Gamma$
 is certainly not optimal, but it is indeed sufficient to ensure
 the convergence of the integral $\int_\R\Gamma(x,t;\xi,0)\,\myd t$ (for $x\neq \xi$)
 if $m>2$. This allows us to obtain the result in Theorem \ref{main.ell.parametrix}
 by arguing as sketched below. 

 We fix a cut-off function $\varphi\in C_0^\infty(\RN)$
 such that $0\leq \varphi\leq 1$ and $\varphi\equiv 1$ in $\Omega$.
 We next define a suitable ``$\G$-regularization" $A^\e=(a^\e_{i,j})_{i,j}$
 of the coefficients $a_{i,j}$ of $\mathcal{L}$,
 and we set
 \[\widetilde{A}^\e(x)=\varphi(x)\,A^\e(x)+
 \big(1-\varphi(x)\big)\,(4\,\Lambda)^{-1}\,\mathbb{I}_m\]
 (where $\mathbb{I}_m$ denotes the $m\times m$ identity matrix)
 and \[\widetilde{\mathcal{H}}^\e=
 \widetilde{\mathcal{L}}^\e-\partial_t=
 \textstyle\sum_{i,j=1}^m
 \widetilde{a}_{i,j}^\e(x)\,X_iX_j-\partial_t.\]
 The coefficients have been suitably modified outside $\Omega$ so
 that we can apply Lemma \ref{41.parametrix} to ensure that the
 integral
\begin{equation*}
  \textstyle\widetilde{\gamma}^\e(x,\xi)=\int_0^\infty\widetilde{\Gamma}^\e
  (x,t;\xi,0)\,\myd t,\quad x\neq \xi\in \RN,
\end{equation*}
 is convergent. Moreover, since the approximating operator
 $\widetilde{\mathcal{L}}^\e$ has smooth coefficients,
 one can prove that $\widetilde{\gamma}^\e$ is a fundamental
 solution for $\widetilde{\mathcal{L}}^\e$ by making use of the
 well-posedness of the adjoint operators
 $(\widetilde{\mathcal{H}}^\e)^*$ and
 $(\widetilde{\mathcal{L}}^\e)^*$. Indeed, for every test function
 $\psi\in C_0^\infty(\RN)$, setting $\psi_n(x,t)=\psi(x)\,
  \theta(|t|/n)$ (where $\theta$ is a smooth function
  defined on $\R$, such that $\theta(s)=1$ for $s\leq 1$, and
  $\theta(s)=0$ for $s\geq 1$) we obtain
\begin{equation*}
 \begin{split}
  \textstyle
  -\psi(\xi)&=-\psi_n(\xi,0)=\int_{\R^{N+1}}
  \widetilde{\Gamma}^\e(x,t;\xi,0)\,
  (\widetilde{\mathcal{H}}^\e)^*\psi_n(x,t)\,\myd\xi\,\myd t\\
  &\xrightarrow{n\to\infty}
  \textstyle
  \int_{\R^{N+1}}
  \widetilde{\Gamma}^\e(x,t;\xi,0)\,
  (\widetilde{\mathcal{L}}^\e)^*\psi(x)\,\myd x\,\myd t=
  \int_{\RN}
  \widetilde{\gamma}^\e(x,\xi)\,
  (\widetilde{\mathcal{L}}^\e)^*\psi(x)\,\myd x.
 \end{split}
\end{equation*}
 We finally find a fundamental solution $\widetilde{\gamma}$ of
 \[\textstyle\widetilde{\mathcal{L}}=\sum_{i,j=1}^m
 \widetilde{a}_{i,j}(x)X_iX_j\]
 (where $\widetilde{A}=\varphi A+
 (1-\varphi)(4\Lambda)^{-1}\mathbb{I}_m$), by an approximation argument, by
 letting $\e$ tend to zero in $\widetilde{\gamma}^\e$.
 This approximation argument is not trivial and it
 requires
 suitable Schauder-type a priori estimates and a careful study of the
 $\e$-dependence in the construction of $\widetilde{\Gamma}^\e$.
 We omit details here.
 Finally, observing
 that $\mathcal{L}=
 \widetilde{\mathcal{L}}$ in $\Omega$, we are then able to prove
 Theorem \ref{main.ell.parametrix}.
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