\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2001(2001), No. 51, pp. 1--25.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu	(login: ftp)}
\thanks{\copyright 2001 Southwest Texas State University.}
\vspace{1cm}}

\begin{document}

\title[\hfilneg EJDE--2001/51\hfil Parabolic equations with VMO coefficients]
{Parabolic equations with VMO coefficients in Morrey spaces}

\author[Lubomira G. Softova\hfil EJDE--2001/51\hfilneg]
{Lubomira G. Softova}

\address{Lubomira G. Softova \hfill\break
Bulgarian Academy of Sciences,
Institute of Mathematics and Informatics, \hfill\break
 ``Acad.~G.~Bonchev'' Str., bl. 8, 1113 Sofia, Bulgaria}
\email{luba@pascal.dm.uniba.it}

\date{}
\thanks{Submitted March 10, 2001. Published July 16, 2001.}

\subjclass[2000]{35K20, 35B65, 35R05}
\keywords{Uniformly parabolic operator, regular oblique derivative,
VMO, \hfil\break\indent
Morrey spaces, singular integrals and commutators, parabolic
Calderon-Zygmund kernel}


\begin{abstract}
 Global regularity in Morrey spaces is derived for the regular 
 oblique derivative for linear uniformly parabolic operators. 
 The principal coefficients of the operator are supposed to be 
 discontinuous, belonging to Sarason's class of functions with 
 vanishing mean oscillation (VMO).
\end{abstract}

\maketitle

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\numberwithin{equation}{section}


\section{Introduction}\label{s1}

Let $\Omega$  be  a bounded $C^{1,1}$ domain in $\mathbb{R}^n$, $n\geq 1$, and	denote by
$Q_T=\Omega\times (0,T)$
 a cylinder in $\mathbb{R}^{n+1}_+=\mathbb{R}^n\times\mathbb{R}_+$.	 Set
$S_T=\partial\Omega\times(0,T)$ for  the lateral boundary of $Q_T$
and  denote by $x=(x',t)=(x_1,\ldots,x_n,t)$  a point in $\mathbb{R}^{n+1}$. We
consider the following regular oblique derivative problem for the uniformly
parabolic operator $\mathcal{P}$   with discontinuous coefficients
\begin{equation}\label{*}
\begin{gathered}
\mathcal{P} u= u_t- \sum_{i,j=1}^n
	 a^{ij}(x)D_{ij}u =f(x)\quad\text{in }	Q_T,\\
   \mathcal{I} u= u(x',0)=0\quad \text{on } \Omega,\\
\mathcal{B} u=	 \sum_{i=1}^n \ell^i(x)D_iu=0\quad\text{on }  S_T.
\end{gathered}
\end{equation}
There is a vast number of existence results in H\"older  and Sobolev spaces
for initial-boundary value problems for linear elliptic and parabolic
operators with H\"older continuous coefficients  $ a^{ij}$  (see \cite{GT},
\cite{LSU}, \cite{Lb}).
In our considerations, we suppose the coefficients $a^{ij}$ belong to the
Sarason class of functions with {\it vanishing mean oscillation\/} VMO
(cf. \cite{S}).

Recall that the class VMO  consists of functions with {\it bounded mean
oscillation\/}	 $BMO$	 (cf. \cite{JN})  whose integral oscillation over
balls
shrinking to a point converges uniformly to zero. The interest to the space
VMO  is
 due mainly to the fact   that it contains as a proper subspace the
bounded uniformly continuous functions.
This
  ensures the extension of the $L^p$-theory for
operators with {\it continuous\/} coefficients
to the case of {\it discontinuous\/} ones. Moreover,	the Sobolev spaces
$W^{1,n}(\Omega)$    and $W^{\theta, n/\theta}(\Omega)$, $0<\theta<1$	 are
also contained in $VMO$, which makes the discontinuity of the coefficients
$a^{ij}\in VMO$   more general than those studied  before  either  for
elliptic
($a^{ij}\in W^{1,n}(\Omega)$, \cite{M})  or parabolic operators ($D_xa^{ij}\in
L^{n+2}$, $D_t a^{ij}\in L^{(n+2)/2}$, \cite{GL1}, \cite{GL2}).


In two innovative  articles, Chiarenza, Frasca and Longo (\cite{CFL1},
\cite{CFL2})  modify the classical methods for obtaining  $L^p$-estimates of
solutions to Dirichlet boundary problem for linear elliptic equations. This
allow  them to move from  $a^{ij}(x)\in C^0(\bar\Omega)$  into $a^{ij}(x)\in
VMO.$	 Roughly speaking, the approach goes back to Calder\'on and Zygmund
(see \cite{CZ1}, \cite{CZ2})  and makes use of an explicit representation
formula for $D^2u$    in terms of singular integrals and their commutators
with
variable Calder\'on-Zygmund type kernel. We refer the reader to the survey
\cite{C}, where an excellent presentation of the state-of-art	and relations
with another similar results for linear and quasilinear (cf. \cite{P})
elliptic
operators can be found.

Later on, the articles \cite{BC}   and \cite{Sf1}  consider unique solvability
in the Sobolev spaces $W^{2,1}_p$, $p\in(1,\infty)$   of the Cauchy-Dirichlet
and oblique derivative problem for the operator $\mathcal{P}$, while \cite{Sf2}
presents  existence results for initial-boundary value problems for
 quasilinear parabolic equations  of the type
$u_t-a^{ij}(x,t,u)D_{ij}u=f(x,t,u,Du)$. An up-to-date	overview of
the classical and the modern results regarding elliptic and parabolic
equations
with discontinuous data can be found in the monograph \cite{MPS}.

In the present work we are interested of the Morrey regularity of  solutions
to  \eqref{*}.	Let us note that the parabolic Morrey spaces $L^{p,\lambda}$
are
subspaces of $L^p$  for every $p\in(1,\infty)$	and $\lambda\in(0,n+2)$
($\lambda\in(0,n)$ in the elliptic case). Whence the existence results in
the
Sobolev spaces $W^{2,1}_p$    for elliptic and parabolic operators with
right-hand side $f\in L^p$  still hold if $f\in L^{p,\lambda}$. A natural
question which arises is whether $\mathcal{P} u\in L^{p,\lambda}$ implies	$u\in
W^{2,1}_{p,\lambda}.$	    It is true for elliptic  operators,
 as  it is
 shown in \cite{PRS} (see also \cite{DR}, \cite{DPR}).	 For  parabolic
operators, in  a  difference, there  is  no  results  concerning  the
regularizing
properties of the operator $\mathcal{P}$ in	the Morrey spaces.

The main goal of the present work is to  show that the solution of
\eqref{*} belongs to $W^{2,1}_{p,\lambda}(Q_T)$   assuming the coefficients of
the uniformly parabolic  operator $\mathcal{P}$ to be bounded VMO	functions and
$f\in L^{p,\lambda}(Q_T)$, $p\in(1,\infty)$, $\lambda \in (0,n+2)$.
The crucial point of our investigations   is the  establishment of suitable
integral estimates of singular integral operators and their commutators with
variable {\it
parabolic Calder\'on--Zygmund (PCZ)  kernel.\/}   The expansion of the kernel
into  spherical harmonics allow us to reduce our considerations over integral
operators with constant PCZ kernel which possesses   good enough regularity.
Constructing a diadic partition of the space subordinated to the utilized
parabolic metric (the standard one or that defined in \cite{FR})  we derive
the
desired Morrey estimates (see Section~\ref{s3}). In Section~\ref{s4}, there
are
established analogous $L^{p,\lambda}$ estimates for nonsingular integrals and
commutators making use of a {\it generalized  reflection\/}  similar to the
one   used for constructing of	half space Green  function.
These results are applied  later in Section~\ref{s5}  to derive
$L^{p,\lambda}(Q_T)$ estimates for the second
spatial derivatives of the solution of \eqref{*}. The
$W^{2,1}_{p,\lambda}(Q_T)$
a~priori estimate of the solution is established  analogously as the
$W^{2,1}_p(Q_T)$
estimate obtained in \cite{Sf1}. Finally, the Morrey  regularity  of
solution
implies   H\"older regularity of its gradient (see Corollary~\ref{c3}), which
is   finer than the one known in the case  $\mathcal{P} u\in L^p$
(cf. \cite[Corollary~1]{Sf1}).





\section{Definitions and preliminary results}\label{s2}

Consider the regular oblique derivative  problem
\begin{equation}\label{1}
\begin{gathered}
\mathcal{P} u\equiv u_t-
	 a^{ij}(x)D_{ij}u =f(x)\quad\text{in }	Q_T,\\
\mathcal{I} u\equiv u(x',0)=0          \quad \text{on } \Omega, \\
\mathcal{B} u\equiv   \ell^i(x)D_iu=0	\quad\text{on }  S_T.
\end{gathered}
\end{equation}
where $Q_T=\Omega\times(0,T)$ is a cylinder in $\mathbb{R}^n\times\mathbb{R}_+$, $
n\geq 1$, the
 base $\Omega$	is bounded $C^{1,1}$ domain in $\mathbb{R}^n$  and   $T>0$.
  Set  $S_T=\partial\Omega\times(0,T)$
for  the lateral boundary of $Q_T$.


Throughout the paper the
standard summation convention on repeated upper and lower indices is adopted.
Moreover, we set
$D_iu=\partial u/ \partial x_i$, $ D_{ij}u= \partial^2 u/\partial x_i\partial
x_j$, $u_t=D_t u=\partial u/\partial t$, $Du=(D_1u,\ldots,D_nu)$ means the
spatial gradient of $u$, $D^2u=\{D_{ij}\}_{ij=1}^n$ stands for its Hessian
matrix and
$x=(x',t)=(x_1,\ldots,x_n,t)\in \mathbb{R}^{n+1}$. In our further
considerations we shall use the   notations
$\mathbb{R}^{n+1}=\mathbb{R}^n\times\mathbb{R}$, $\mathbb{R}^{n+1}_+=\mathbb{R}^n\times\mathbb{R}_+$,
and $\mathbb{D}_+^{n+1}=\mathbb{R}^n_+\times\mathbb{R}_+$.


We suppose that $\mathcal{P}$ is a {\it uniformly parabolic operator,\/} i.e.,
there exists  a positive constant $\Lambda$ such that
\begin{equation}\label{2a}
\Lambda^{-1} |\xi|^2\leq a^{ij}(x,t)\xi_i\xi_j\leq \Lambda |\xi|^2, \quad
{\rm a.a.} \ x\in Q_T,\     \forall \xi\in\mathbb{R}^n.
\end{equation}
Besides that, the  requirement the coefficients  matrix  ${\bf
a}=\{a^{ij}\}_{ij=1}^n$  to be symmetric, leads to  essential
boundedness of
$a^{ij}$'s (cf. \cite{Sf1}).


The boundary operator  $\mathcal{B}$	 is prescribed in terms of a
directional derivative with respect to the unit vector field
$\ell(x)=(\ell^1(x),\dots,\ell^n(x),0)$  defined on $S_T$. We suppose that
$\mathcal{B}$
is a  {\it regular oblique derivative operator\/} (cf. \cite{PP}), i.e., the
field $    \ell$ is never tangential to the boundary $S_T$:
\begin{equation}\label{3a}
    \ell(x  )\cdot	 \nu (x')  =\ell^i(x  ) \nu_i(x')>0\quad  {\rm
on}\  S_T\,, \ell^i   \in {\rm Lip}(\bar S_T).
\end{equation}
Here ${\rm Lip}(\bar S_T)$ is the class of  functions which are uniformly
Lipschitz
continuous on $\bar S_T$ and
$     \nu(x') =(\nu_1(x'),\dots,\nu_n(x'))$  stands for the unit
inner normal to $\partial\Omega$.



Denote by  $\mathcal{P}_0$ a linear parabolic operator with constant  coefficients
$a^{ij}_0$  that  satisfy \eqref{2a}. It is well known from the linear
theory (cf. \cite{LSU}) that the fundamental solution  of the operator $\mathcal{P}_0$
is given by the formula
\begin{equation}\label{fsol0}
\Gamma^0 (y)=\left\{\begin{array}{ll}
\frac{1}{(4\pi \tau)^{n/2}\sqrt{{\rm det}\  {\bf a}_0}}
\exp\Big\{-\frac{A_0^{ij}y_iy_j}{4\tau}\Big\} & {\rm for}\quad \tau>0,\\
0 & {\rm for}\quad  \tau<0,
 \end{array}\right.
\end{equation}
where ${\bf a}_0=\{a^{ij}_0\}$	is the matrix of the coefficients of $\mathcal{P}_0$
and
${\bf  A}_0=\{A^{ij}_0\}={\bf a}_0^{-1}$	is its inverse matrix.
Hereafter  we
denote
by $\Gamma_i^0$ and $\Gamma^0_{ij}$ the derivatives $\partial
\Gamma^0/\partial
y_i$ and
$\partial^2\Gamma^0/\partial y_i\partial y_j$.	In the problem under
consideration \eqref{1}, the coefficients of the operator $\mathcal{P}$ depend on $x$.
To express this dependence in the fundamental solution we define
\begin{equation}\label{fsol}
\Gamma (x;y)=\left\{\begin{array}{ll}
\frac{1}{(4\pi \tau)^{n/2}\sqrt{{\rm det}\  {\bf a}(x)}}
\exp\Big\{-\frac{A^{ij}(x)y_iy_j}{4\tau}\Big\} & {\rm for}\quad \tau>0,\\
0 & {\rm for}\quad  \tau<0,
 \end{array}\right.
\end{equation}
where ${\bf a}(x)=\{a^{ij}(x)\}$  is the matrix of the coefficients of
$\mathcal{P}$
and
${\bf  A}(x)=\{A^{ij}(x)\}={\bf a}(x)^{-1}$  is its inverse matrix. The
derivatives
 $\Gamma_i$ and $\Gamma_{ij}$ are taken with respect to the second variable
$y$.




For the goal of our further considerations, besides   the standard parabolic
metric
$\widetilde{\rho} (x)=\max(|x'|,|t|^{1/2})$, $|x'|=\big(\sum_{i=1}^{n}
x_i^2\big)^{1/2}$
we are going to  use  the one introduced by
Fabes and Rivi\'ere  in  \cite{FR}
\begin{equation}\label{rho}
\rho(x)=\sqrt{\frac{|x'|^2+\sqrt{|x'|^4+4t^2}}{2}},\quad
d(x,y) =\rho(x-y).
\end{equation}
A  ball with respect to the metric $d$	centered at zero and of radius $r$
is simply an  ellipsoid
$$
\mathcal{E}_r(0)=\left\{x\in\mathbb{R}^{n+1}\colon\quad \frac{|x'|^2}{r^2}
+\frac{t^2}{r^4} <1 \right\}.
$$
Obviously, the unit sphere with respect to this metric coincides with the unit
sphere in $\mathbb{R}^{n+1}$, i.e.
 $$\partial\mathcal{E}_1(0)\equiv \Sigma_{n+1}=\Big\{x\in \mathbb{R}^{n+1}\colon
|x|=\Big(\sum_{i=1}^n x_i^2+t^2\Big)^{1/2}=1\Big\}.
$$
Let $I$  be a {\it parabolic cylinder\/}  centered at some point $x$  and with
radius $r$, that is
  $I=I_r(x)=\{y\in \mathbb{R}^{n+1}\colon |x'-y'|<r, |t-\tau|<r^2  \}$.
 It is easy to see that for any ellipsoid  $\mathcal{E}_r$
there exist cylinders
$\underline I$ and  $\overline I$ with	measures comparable to	 $r^{n+2}$
and
such
that	     $\underline I\subset\mathcal{E}_r\subset\overline I$.
Obviously, that relation  gives an equivalence of the both metrics and the
introduced by them topologies.
Later we shall	use  this equivalence without
making	reference to, except if  it is necessary.



It is worth noting that   \eqref{rho}  has been employed in the study of
singular
integral operators with Calder\'on-Zygmund kernels  of mixed homogeneity (see
\cite{FR}).

\begin{definition}\label{d1}  A function $k(x)$ is said to be  a
 parabolic
Calder\'on-Zygmund  (PCZ) kernel  in the space $\mathbb{R}^{n+1}$
 if

i) $k$ is smooth on $\mathbb{R}^{n+1}\setminus \{0\}$;


ii)  $k (rx',r^2t)=r^{-(n+2)} k(x',t)\ $  for each $r>0$;

iii)  $ \int_{\rho(x)=r} k(x) d\sigma_{x} =0\ $  for
each
$r>0$.
\end{definition}
\begin{definition}\label{d2}  We say that a function $k(x;y)$,
$x\in\mathbb{R}^{n+1}$,
$y\in\mathbb{R}^{n+1}\setminus \{0\}$ is a {\it variable PCZ kernel\/},
 if:

i) $k(x;\cdot)$ is  a PCZ kernel (in the sense of Definition~\ref{d1}) for
a.a.
$x\in\mathbb{R}^{n+1}$;


ii) $ \sup_{\rho(y)=1}
\left|\Big(\frac{\partial}{\partial y }
\Big)^\beta k (x;y)\right|\leq C(\beta)\ \ $ for every multiindex
$\beta$, independently of $x$.
\end{definition}







For the sake of the completeness we shall recall here  the definitions and
some
properties of the spaces  we are going to use.
\begin{definition}\label{dBMO}
 We say that the measurable and locally integrable function $f$
belongs to $BMO$ if the seminorm
\begin{equation}\label{BMO}
\|f\|_\ast =\sup_{I} \frac{1}{|{I}|}\int_{I} |f(y)-f_{I}|dy
\end{equation}
is finite. Here
  $I$ ranges over all parabolic cylinders in $\mathbb{R}^{n+1}$ with radius
$r$, and centered at some point $x$
and
$f_{I}= \frac{1}{|I|}\int_{I}
f(y) dy$. Then $\|f\|_\ast$   is a norm in $BMO$  modulo constant
functions under which $BMO$  is a Banach space.
\end{definition}
\begin{definition}\label{dVMO}
  Let $f\in BMO$, $r_0>0$      and denote
\begin{equation}\label{gamma}
\gamma_f(r_0) =\sup_{I_r} \frac{1}{|I_r|}\int_{I_r}
|f(y)-f_{I_r}|dy.
\end{equation}
 We say that $f\in VMO$  if  $\gamma_f(r_0)\rightarrow 0$ as $r_0\to
0$ where the supremum is taken over all parabolic cylinders $I_r$ centered at
some point
$x$ with radius $r\leq	r_0$. The quantity
$\gamma_f(r_0)$  is referred to as  VMO modulus of $f$.
\end{definition}
The spaces $BMO(Q_T)$  and $VMO(Q_T)$	can be defined by taking $I\cap Q_T$
and
$I_r\cap Q_T$  instead of  $I$ and $I_r$ in the definitions of
$\|f\|_\ast$  and $\gamma_f(r_0)$.


 Having a
function $f$   defined in $Q_T$   and belonging to $ BMO(Q_T)$, it is possible
to extend it to the whole $\mathbb{R}^{n+1}$ and the $BMO$  norm of the extension
could
be estimated by the $BMO$ norm of the original function. If in addition
 $f\in VMO(Q_T)$, then we may extend it preserving its VMO-modulus,
as it follows by the results of  Jones \cite{J} and Acquistapace
\cite[Proposition~1.3]{A}.
The next theorem  offers
several alternative descriptions of $VMO$.
\begin{theorem}{\rm (\cite[Theorem~1]{S})}\label{tSar}
For $f\in BMO$,   the following conditions are equivalent:
\begin{itemize}
\item[(i)] $f$	is in $VMO;$
\item[(ii)] $f$  is in the $BMO$-closure of  the space of bounded uniformly
continuous functions;
\item[(iii)]   $\lim_{y\to 0}\|f(x-y)-f(x) \|_\ast=0;$
\end{itemize}
\end{theorem}

If $f$ is a uniformly continuous function, then its VMO-modulus
$\gamma_f(r)$
coincides with the  modulus of continuity $\omega_f(r)$.
Moreover,
for a given $f\in VMO$ we can find a sequence $\{
f_k\}\in
L^\infty\cap C^\infty(\mathbb{R}^{n+1} )$  of functions with $\gamma_{f_k}(r)\equiv
\omega_{f_k}(r)$, such that
$f_k\rightarrow f$ in VMO as $k\to \infty$ and $\gamma_{f_k}(r)\leq \gamma_f
(r)
$ for all  integer numbers $k$. In what follows  we use these
results without explicit reference.


The problem  \eqref{1}	has been  already  studied in the
framework of the Sobolev
spaces $W^{2,1}_p(Q_T)$, $p\in(1,\infty)$  (cf. \cite{Sf1}, \cite{MPS}).
Precisely, assuming \eqref{2a}, \eqref{3a}  and $a^{ij}\in VMO(Q_T)$, it is
proved
that for any $f\in L^p(Q_T)$, $p\in (1,\infty)$,  there exists a unique
weakly
differentiable function $u$
belonging to $L^p(Q_T)$ with all its derivatives
$D^r_tD^s_x u$, $0\leq 2r+s\leq 2$, such that	$u$  satisfies the equation in
\eqref{1}  almost everywhere in $Q_T$  and the boundary conditions holds in
trace sense.

Our goal here is to obtain finer regularity of that solution supposing that
$\mathcal{P} u$
belongs to the Morrey  space $L^{p,\lambda}$.
\begin{definition}\label{dPMS}
We say that a measurable  function $f\in L^p_{\rm loc}$  belongs to the
parabolic Morrey space $L^{p,\lambda}$ if
for any  $p\in(1,+\infty)$ and $\lambda\in(0,n+2)$  the following norm is
finite
\begin{equation}\label{PMS-norm}
\|f\|_{p,\lambda}=\left(\sup_{r>0}\frac{1}{r^\lambda}
\int_{I}|f(y)|^pdy \right)^{1/p}
\end{equation}
where $I$  ranges over	all parabolic cylinders in $\mathbb{R}^{n+1}$ of  radius $r$.
\end{definition}
To define  the	space $L^{p,\lambda}(Q_T)$, we insist the norm
\begin{equation}\label{PMS1-norm}
\|f\|_{p,\lambda;Q_T}=\left(\sup_{r>0}\frac{1}{r^\lambda}
\int_{Q_T\cap I}|f(y)|^pdy \right)^{1/p}
\end{equation}
to be finite.
\begin{definition}\label{solution}
We say that the function $u$ lies in the Morrey space  $
W^{2,1}_{p,\lambda}(Q_T)$, $1<p<\infty$, $0<\lambda<n+2$, if it is weakly
differentiable	and belongs to
$L^{p,\lambda}(Q_T)$, along with all its derivatives $D^r_tD^s_x u$, $0\leq
2r+s\leq 2$.
 Then the following norm is finite
$$
\|u\|_{W^{2,1}_{p,\lambda}(Q_T)}=\|u\|_{p,\lambda;Q_T} +\|D^2
u\|_{p,\lambda;Q_T} +\|D_t u\|_{p,\lambda;Q_T}.
$$
\end{definition}





For a given measurable function  $f\in L^1_{loc}$  we define  the {\it
Hardy-Littlewood maximal operator\/}
 \begin{equation}\label{maxf}
Mf(x)=\sup_{I}\frac{1}{|I|}\int_{I}|f(y)|dy
\quad\text{for a.a. } x\in\mathbb{R}^{n+1},
\end{equation}
where the supremum is taken over all parabolic	cylinders  $I$	centered at
the
point $x$.

A variant of it is the {\it sharp maximal operator\/}
\begin{equation}\label{sharpf}
f^{\#}(x)=\sup_{I}\frac{1}{|I|}\int_{I}|f(y)-f_I|dy
 \quad\text{for a.a. } x\in\mathbb{R}^{n+1}.
\end{equation}


The following lemmas give  $L^{p,\lambda}$  estimates for $f$, $Mf $  and
$f^{\#}$. Their $L^p$  variants are proved in \cite{Br}. Analogous
$L^{p,\lambda}$  estimates, but in the space $\mathbb{R}^n$    endowed with the
Euclidean
metric	can be found in \cite{CF}  and \cite{DPR}. To prove the
$L^{p,\lambda}$
estimates below, we follow the
same lines of reasoning  as in the paper cited above, making use of the
parabolic
metrics $\widetilde{\rho}$  or $\rho$ and corresponding to them diadic	partition of
the
space:
 $$
\mathbb{R}^{n+1}=2I+\bigcup_{k=1}^\infty   2^{k+1}I\setminus 2^{k}I
$$
where $I$   is either parabolic cylinder or ellipsoid centered at some point
$x\in\mathbb{R}^{n+1}$ with radius $r$. We note that $2^kI$  means parabolic cylinder
(ellipsoid)  with the same center and radius $2^kr$.
\begin{lemma}{\rm (Maximal inequality)}\label{l1}
Let $f\in L^{p,\lambda}$, $p\in(1,\infty)$, $\lambda\in(0,n+2)$.
Then there exists a constant $C$ independent of $f$ such that
\begin{equation}\label{equ2}
||Mf||_{p,\lambda}\leq	C \|f\|_{p,\lambda}.
\end{equation}
\end{lemma}
\begin{lemma}{\rm  (Sharp inequality)}\label{l2}
Let $f$ be the same as in Lemma~\ref{l1}. Then
there exists a constant $C$ independent of $f$ such that
\begin{equation}\label{equ3}
\|f\|_{p,\lambda}\leq  C \|f^{\#}\|_{p,\lambda}
\end{equation}
\end{lemma}


Analogous estimates are valid also in $\mathbb{D}^{n+1}_+$  where the corresponding
diadic partition of the space has the form
$$
\mathbb{D}^{n+1}_+=2I_+ +\bigcup_{k=1}^\infty	 2^{k+1}I_+\setminus 2^{k}I_+
$$
where $I_+=I\cap\{x_n>0, t>0\}$  and $I$  is a	parabolic cylinder.
 Then
$$
\|Mf\|_{p,\lambda;\mathbb{D}^{n+1}_+}\leq
C\|f\|_{p,\lambda;\mathbb{D}^{n+1}_+},\quad
\|f\|_{p,\lambda;\mathbb{D}^{n+1}_+}\leq C\|f^{\#}\|_{p,\lambda;\mathbb{D}^{n+1}_+}.
$$


\begin{lemma}{\rm (John-Nirenberg type	lemma)}\label{l3}
Let $1<p<\infty$, $a\in BMO $ and $I$	be a parabolic cylinder. Then
$$
\left(\frac{1}{|I|} \int_{I}\left|a(y)-a_I\right|^pdy
\right)^{1/p}\leq C(p)\|a\|_\ast.
$$
\end{lemma}
\begin{lemma}{\rm \cite[Lemma~2.10]{BC}}\label{l33}
 Let $a\in BMO$. Then, for any	positive  integer  $j$	  and  parabolic
cylinder $I$
 $$
 |a_{2^jI}-a_I| \leq C(n)j\|a\|_\ast.
 $$
\end{lemma}
The next lemma gives an important property of the Calder\'on-Zygmund kernels.
\begin{lemma}\cite[Pointwise H\"ormander condition]{BC}\label{l4}
Let $k$  be a PCZ kernel. Then for any parabolic cylinder $I_0$   of center
$x_0$ one has
$$
|k(x-y)-k(x_0-y)| \leq C(k) \frac{\rho(x_0-x)}{\rho(x_0-y)^{n+3}}
$$
for any $x\in I_0$  and $y\not\in 2I_0$.
\end{lemma}







\section{Singular integral estimates in Morrey spaces}\label{s3}

Let $k(x;y)$  be a  variable PCZ kernel. For
any functions
$f\in L^{p,\lambda}$, $p\in(1,\infty)$, $\lambda\in(0,n+2)$	 and $a\in
L^\infty$
define a  singular integral operator
$\mathcal{K} f$ and its commutator $\mathcal{C}[a,f]$ by
\begin{align}
\label{lim1}
\mathcal{K} f(x)&=\lim_{\varepsilon \to 0} \int_{\rho(x-y)>\varepsilon }
k(x;x-y)f(y)dy=\lim_{\varepsilon \to 0} \mathcal{K}_\varepsilon f(x)\\
\nonumber
\mathcal{C}[a,f](x)&=\lim_{\varepsilon \to 0} \int_{\rho(x-y)>\varepsilon}
k(x;x-y)[a(y)-a(x)]f(y)dy\\
\label{lim2}
&=\lim_{\varepsilon \to 0} \mathcal{C}_\varepsilon
f(x)=\mathcal{K}(af)(x) -a(x)\mathcal{K} f(x).
\end{align}


The aim of this section is to derive  $L^{p,\lambda}$ a~priori estimates for
the
singular operators  $\mathcal{K} f$ and $\mathcal{C}[a,f]$. For this goal we are
going to
exploit
the well known technique, based on an expansion into spherical harmonics (cf.
 \cite{CZ1}, \cite{CZ2}, \cite{CFL1}, \cite{BC}).

Any homogeneous polynomial $p(x)$, $x\in \mathbb{R}^N$ of degree $m$, solution of
Laplaces  equation	  $\Delta u=0$,
is called  {\it $N$-dimensional solid harmonic\/} of degree $m$. Its
restriction to the unit
sphere $\Sigma_N$  is called   {\it $N$-dimensional  spherical
harmonic\/} of degree m.

Denote by $Y_m$ the space of
 $(n+1)$-dimensional spherical	harmonics of
 degree $m.$	It is a  finite-dimensional  space with ${\rm dim\,}Y_m=g_m$
where
\begin{equation}\label{Y2}
   g_m=\binom{m+n}{n}
-\binom{m+n-2}{n} \leq C(n) m^{n-1}
\end{equation}
and  the second binomial coefficient is equal to $0$  when $m=0,1$, i.e.,
$g_0=1$, $g_1=n+1$. Further, let  $\{ Y_{sm}(x)\}_{s=1}^{g_m}$	be an {\it
orthonormal  base\/} of $Y_m$. Then $\{ Y_{sm}(x)\}_{s=1,m=0}^{g_m,\
\infty}
$ is a {\it complete orthonormal  base\/} in  $L^2(\Sigma_{n+1})$  and
\begin{equation}\label{Y1}
\sup_{x\in\Sigma_{n+1}} \left|\left(\frac{\partial}{\partial x}
\right)^\beta  Y_{sm}(x)  \right| \leq C(n) m^{|\beta|+(n-1)/2},\quad
m=1,2,\ldots.
\end{equation}

If, for instance, $\phi\in
C^\infty(\Sigma_{n+1})$
then $\phi(x)\sim \sum_{s,m} b_{sm} Y_{sm}(x) $  is the Fourier series
expansion of $\phi(x)$ with respect to $\{Y_{sm}(x)\}$, where
\begin{equation}\label{Y3}
b_{sm}=\int_{\Sigma_{n+1}}\phi ( y) Y_{sm}( y)d\sigma,\quad
|b_{sm}|\leq C(l) m^{-2l}\sup_{\overset{|\beta|=2l}{y\in\Sigma_{n+1}}}
\left|\Big(\frac{\partial}{\partial y}\Big)^\beta\phi(y) \right|
\end{equation}
for every $l>1 $  and the notation $\sum_{s,m}$  stands for
$\sum_{m=0}^\infty\sum_{s=1}^{g_m}$.

We are in a position now to formulate our result concerning singular
operators.
\begin{theorem}\label{th2}	Let $k(x;y)$  be a variable PCZ kernel. Then for any
 $f\in L^{p,\lambda},$	$p\in (1,\infty)$, $\lambda\in(0,n+2)$	and $a\in
L^\infty$
   the integrals $\mathcal{K} f$  and  $\mathcal{C}[a,f]$  there exist
and \begin{equation}\label{K4b}
\lim_{\varepsilon\to 0} \|\mathcal{K}_\varepsilon f-\mathcal{K} f\|_{p,\lambda}=
\lim_{\varepsilon\to 0}  \|\mathcal{C}_\varepsilon [a,f]-\mathcal{C}
[a,f]\|_{p,\lambda}=0.
\end{equation}
Furthermore,
there exists a constant $C=C(n,p,k)$, independent of $f$,  such that
 \begin{equation}\label{Kf}
\|\mathcal{K} f\|_{p,\lambda} \leq C\|f\|_{p,\lambda},\quad
\|\mathcal{C}[a,f] \|_{p,\lambda}\leq C\|a\|_\ast \|f\|_{p,\lambda}.
\end{equation}
\end{theorem}

\paragraph{Proof}
By  density  arguments it is enough to prove the theorem for $f\in
C_0^\infty(\mathbb{R}^{n+1})$.

Let $x,y\in \mathbb{R}^{n+1}$  and $\bar y=\frac{y}{\rho(y)}\in \Sigma_{n+1}.$	Having
in mind the homogeneity properties of the variable PCZ kernel, we can write
$$
\rho(y)^{n+2}k(x;y)= k(x;\bar y)=\sum_{s,m}
b_{sm}(x) Y_{sm}(\bar y).
$$
  Hence $k(x;y)=\rho(y)^{-(n+2)}\sum_{s,m}b_{sm}(x) Y_{sm}(\bar y)$.

>From the  Definition~\ref{d2}~$ii)$
  and the estimate \eqref{Y3}  it follows
\begin{equation}\label{Y4}
\|b_{sm}\|_\infty\leq C(l,\beta)m^{-2l}.
\end{equation}
Let we note that the function $k(x,\bar y)$  is $C^\infty$   with respect to
$\bar y$ and hence it is equal to its series expansion. So we
 consider the
 integrals
\begin{align}
\label{Kepsf}
\mathcal{K}_\varepsilon f(x)&=\int_{\rho(x-y)>
\varepsilon} \sum_{s,m}b_{sm}(x)  \mathcal{H}_{sm}(x-y) f(y)dy,\\
\label{Cepsf}
\mathcal{C}_\varepsilon [a,f](x)&=\int_{\rho(x-y)>
\varepsilon}\sum_{s,m}b_{sm}(x) \mathcal{H}_{sm}(x-y) [a(y) - a(x)]f(y)dy.
\end{align}
where $\mathcal{H}_{sm}(x-y)$ stands for the kernel
$Y_{sm}(\overline{x-y})\rho(x-y)^{-(n+2)}$.
We note that  the series
\begin{align*}
&\big|\sum_{s,m} b_{sm}(x)\mathcal{H}_{sm}(x-y)f(y) \big|\leq
|f(y)|\varepsilon^{-(n+2)}\sum_{s,m} \|b_{sm}\|_\infty\|Y_{sm}\|_\infty\\
&\leq C\varepsilon^{-(n+2)} |f(y)| \sum_{m=1}^\infty
m^{-2l}m^{n-1}m^{(n-1)/2}
\end{align*}
 converges for $l>(3n-1)/4$. Hence, by the dominated convergence
theorem, we can write
\begin{equation}\label{Ksumf}
\mathcal{K}_\varepsilon f(x) = \sum_{s,m}b_{sm}(x) \int_{\rho(x-y)>\varepsilon}
\mathcal{H}_{sm}(x-y) f(y) dy.
\end{equation}
Identical arguments are valid also for the commutator, so
$$
\mathcal{C}_\varepsilon [a,f](x) = \sum_{s,m}b_{sm}(x)
\int_{\rho(x-y)>\varepsilon} \mathcal{H}_{sm}(x-y)[a(x)-a(y)] f(y) dy.
$$




It is easy to check that $\mathcal{H}_{sm}(x)$	is
 PCZ kernel in the sense of Definition~\ref{d1}.
 Moreover,
$$
\sup_{ x\in\Sigma_{n+1}} \left|\nabla_{x'} \mathcal{H}_{sm}(x) \right|
 \leq
C(n) m^{(n+1)/2}
$$
according to \eqref{Y1}.
Later on, for all $x\in \mathbb{R}^{n+1}$  we have estimates also for the derivatives
of $\mathcal{H}_{sm}(x)$, that is
\begin{align*}
D_i\mathcal{H}_{sm}(x)=&
D_i\big(Y_{sm}\left(\frac{x}{\rho(x)}\right)\rho(x)^{-(n+2)} \big)\\
=&D_iY_{sm}\big(\frac{x}{\rho(x)}\big)\frac{1}{\rho(x)}
\rho(x)^{-(n+2)}
-(n+2)\rho(x)^{-(n+3)}D_i\rho(x) Y_{sm}\big(\frac{x}{\rho(x)}\big).
\end{align*}
Hence
\begin{equation}\label{Hx}
|D_i\mathcal{H}_{sm}(x)| \leq
C(n) m^{(n+1)/2}\rho(x)^{-(n+3)}.
\end{equation}
The derivative with respect to $t$ is calculated analogously
\begin{equation}\label{Ht}
|D_t\mathcal{H}_{sm}(x)|
\leq
C(n)m^{(n+1)/2}\rho(x)^{-(n+4)}.
\end{equation}
Now it is easy to see that  $\mathcal{H}_{sm}(x)$	satisfies  H\"ormander type
condition.
\begin{lemma}\label{l5}
 Let $I_0$  be a cylinder centered at  $x_0$   with radius $r$.
Consider  $x\in I_0$  and $y\not\in 2I_0$. Then the PCZ kernel
$\mathcal{H}_{sm}(x)$ satisfies
\begin{equation}\label{PHE}
\left|\mathcal{H}_{sm}(x-y)- \mathcal{H}_{sm}(x_0-y)	\right| \leq C(n) m^{(n+1)/2}
\frac{\rho(x_0-x)}{\rho(x_0-y)^{n+3}}.
\end{equation}
\end{lemma}
The proof is analogous to that of Lemma~\ref{l4}   making use of  \eqref{Hx}
and \eqref{Ht}.
\begin{lemma}\label{l6} Let $f$ and $a$  be the same as above. Then
  the singular integrals
\begin{align*}
\mathcal{K}_{sm}f(x)&=P.V.\int_{\mathbb{R}^{n+1}}\mathcal{H}_{sm}(x-y)
 f(y)dy, \\
\mathcal{C}_{sm}[a,f](x)&=P.V.\int_{\mathbb{R}^{n+1}}
\mathcal{H}_{sm}(x-y)[a(y)-a(x)] f(y) dy
\end{align*}
 satisfy  the estimates
 \begin{align}\label{Ksharp}
 (\mathcal{K}_{sm}f)^{\#}(x)&\leq C m^{\frac{n+1}{2}}
\big(M(|f|^p)(x)\big)^{\frac{1}{p}},\\
 \label{Csharp}
 (\mathcal{C}_{sm}[a,f])^{\#}(x)&\leq C
\|a\|_\ast\Big\{\big(M(|\mathcal{K}_{sm}f|^p)(x)\big)^{\frac{1}{p}}
  +  m^{\frac{n+1}{2}}
\big(M(|f|^p)(x)\big)^{\frac{1}{p}} \Big\},
\end{align}
where the constants depend on $n$, $p$, $\lambda$ but not on $f$.
\end{lemma}

\paragraph{Proof}
For arbitrary $x_0\in \mathbb{R}^{n+1}$ and  cylinder $I$
centered at $x_0$ with radius $r$, we construct the  operator
\begin{align*}
J&=\frac{1}{|I|} \int_I
\left|\mathcal{K}_{sm}f(y)  -(\mathcal{K}_{sm}f)_I\right| dx\\
&=
 \frac{1}{|I|} \int_I
\left|\mathcal{K}_{sm}f(y) -\mathcal{K}_{sm2r}f(x_0)
+ \mathcal{K}_{sm2r}f(x_0) -(\mathcal{K}_{sm}f)_I\right| dx \\
& \leq	\frac{2}{|I|} \int_I
\left|\mathcal{K}_{sm}f(y) -\mathcal{K}_{sm2r}f(x_0)\right| dx,
\end{align*}
where
$\mathcal{K}_{sm2r}f(x_0)=\int_{\rho(y-x_0)>2r}\mathcal{H}_{sm}(x_0-y)f
( y)dy.
$
We define $(2I)^c=\mathbb{R}^{n+1}\setminus 2I$ and write
$f(x)=f(x)\chi_{2I}(x)+f(x)\chi_{(2I)^c}(x)=f_1(x)+f_2(x)$. Hence
\begin{align*}
(\mathcal{K}_{sm}f )^{\#}(x_0)\leq& \frac{C}{|I|} \int_I
|\mathcal{K}_{sm}f_1(y)| dy\\
 & + \frac{C}{|I|} \int_I
|\mathcal{K}_{sm}f_2(y) -\mathcal{K}_{sm2r}f(x_0)|  dy=J_1+J_2.
\end{align*}
Thus,
\begin{align*}
J_1&\leq \frac{C}{|I|}\left(\int_I1dy \right)^{1/p'}\left(\int_I
|\mathcal{K}_{sm}f_1(y)|^pdy \right)^{1/p}=\frac{C}{|I|^{1/p}}
\|\mathcal{K}_{sm}f_1\|_p\\
&\leq \frac{C}{|I|^{1/p}} \|f_1\|_p  \leq C \big(M(|f|^p)(x_0)\big)^{1/p}
\end{align*}
after applying	\cite[Theorem 1]{FR} and  taking
the supremum with respect to $I$. The second integral gives
 \begin{align*}
J_2&\leq \frac{C}{|I|} \int_I\left(\  \int_{(2I)^c} \left|
\mathcal{H}_{sm}(y-\xi) -\mathcal{H}_{sm}(x_0-\xi)	 \right||f(\xi)| d\xi\right)
dy\\
&\leq Cm^{(n+1)/2} \frac{1}{|I|} \int_I
\left( \int_{(2I)^c} \frac{\rho(x_0-y)}{\rho(x_0-\xi)^{n+3}} |f(\xi)|
d\xi\right) dy\\
&\leq  Cm^{(n+1)/2} r \sum_{k=1}^\infty
\int_{2^{k+1}I\setminus 2^kI} \frac{|f(\xi)|}{\rho(x_0-\xi)^{n+3}}
d\xi\\
 &\leq C m^{(n+1)/2} r \sum_{k=1}^\infty
\frac{1}{(2^k r)^{n+3}} \left( \int_{2^{k+1}I} 1d\xi\right)^{1/p'}
 \left(
\int_{2^{k+1}I} |f(\xi)|^pd\xi \right)^{1/p}\\
&\leq  C  m^{(n+1)/2} \frac{1}{r^{n+2}} \sum_{k=1}^\infty
\frac{1}{(2^{k(n+3)}}|2^{k+1}I|
 \left(\  \frac{1}{|2^{k+1}I|}
\int_{2^{k+1}I} |f(\xi)|^pd\xi \right)^{1/p}\\
&\leq C m^{(n+1)/2} \big(M(|f|^p)(x_0)\big)^{1/p},
\end{align*}
where we have applied Lemma~\ref{l5} for the cylinder $I$  centered at
 $x_0$	and containing $y$ while $\xi\in (2I)^c$. The final inequality has
been
reached after taking the supremum with respect to $I$.
Since $x_0$   was chosen arbitrary, the  estimate  \eqref{Ksharp}  holds true
for any
$x\in \mathbb{R}^{n+1}$.



As it concerns	the commutator we shall employ	the
idea of Stromberg (see \cite{T}) which consists of expressing
$\mathcal{C}_{sm}[a,f]$
as a sum of  integral operators and estimating	their sharp functions.
Precisely
 \begin{align*}
\mathcal{C}_{sm}[a,f](x)&=   \mathcal{K}_{sm}(a -a_I)f(x)-(a(x)-a_I)\mathcal{
K}_{sm}f(x)\\
&=\mathcal{K}_{sm}(a   -a_I)f\chi_{2I}(x) + \mathcal{K}_{sm}(a
-a_I)f\chi_{2I^c}(x)
-(a(x)-a_I)\mathcal{K}_{sm}f(x) \\
&=A(x)+B(x) +C(x).
\end{align*}
Now for any $p>1$  and $q\in (1,p)$  we have
\begin{align*}
G_1=&\frac{1}{|I|}\int_{I} |A(x)-A_{I}| dx
\leq\frac{2}{|I|} \int_I |\mathcal{K}_{sm}(a   -a_I)f\chi_{2I}(x)|dx\\
 &\leq \frac{C}{|I|}\left( \int_{I}
|\mathcal{K}_{sm}(a   -a_{I})f\chi_{2I}(x)|^qdx \right)^{1/q} \left(  \int_I
1^{q'}dx \right)^{1/q'}\\
&\leq |I|^{-1/q}  \left(  \int_{2I}|f(y)|^pdy \right)^{1/p}
\left( \int_{2I}|a(y)-a_I|^{pq/(p-q)} dy\right)^{(p-q)/pq},
\end{align*}
as follows from
 \cite[Theorem~1]{FR}. Further the John-Nirenberg type lemma
and Lemma~\ref{l33} applied  to the second integral yield
\begin{align*}
\int_{2I}|a(y)-a_I|^{pq/(p-q)} dy & \leq
\int_{2I}|a(y)-a_{2I}|^{pq/(p-q)} dy   +
\int_{2I}|a_{2I}-a_I|^{pq/(p-q)} dy\\
&\leq C(p,q) \|a\|_\ast^{pq/(p-q)}
\end{align*}
and hence
$$
G_1\leq C \|a\|_\ast \left(\frac{1}{|2I|}\int_{2I}|f(y)|^pdy
\right)^{1/p}\leq C\|a\|_\ast \big(M(|f|^p)(x_0)\big)^{1/p}.
 $$
To estimate the sharp function of $B(x)$ we proceed analogously as we did for
 $\mathcal{K}_{sm}f$
$$
G_2=\frac{1}{|I|} \int_I |B(x)-B_I|dx \leq
 \frac{2}{|I|}\int_I |B(x)-B(x_0)| dx.
$$
The integrand above  satisfies

\begin{align*}
&|B(x)-B(x_0)| =|\mathcal{K}_{sm}(a-a_I)f \chi_{(2I)^c}(x)-\mathcal{K}_{sm}(a-a_I)f
\chi_{(2I)^c}(x_0)| \\
&\leq  \int_{(2I)^c}\left|  \mathcal{H}_{sm}(x-y)-
\mathcal{H}_{sm}(x_0-y)\right|	|a(y)-a_I||f(y)| dy\\
&\leq C(n)m^{(n+1)/2}\rho(x_0-x) \int_{(2I)^c}
\frac{|a(y)-a_I||f(y)|}{\rho(x_0-y)^{n+3}}dy\\
&\leq C(n)m^{(n+1)/2} r \left( \int_{(2I)^c}
\frac{|f(y)|^p}{\rho(x_0-y)^{n+3}}dy\right)^{1/p}
 \left( \int_{(2I)^c}
\frac{|a(y)-a_I|^{p'}}{\rho(x_0-y)^{n+3}}dy\right)^{1/p'},
\end{align*}
 $\frac{1}{p}+\frac{1}{p'}=1$, as  consequence from  the H\"ormander
pointwise estimate  since $x\in I$ and	 $y\in (2I)^c$.
The first integral above is estimated directly
\begin{align*}
\int_{(2I)^c}\frac{|f(y)|^p}{\rho(x_0-y)^{n+3}}dy
&=\sum_{k=1}^\infty
\int_{2^{k+1}I\setminus 2^kI}
\frac{|f(y)|^p}{\rho(x_0-y)^{n+3}}dy\\
\leq& \sum_{k=1}^\infty \frac{(2^{k+1} r    )^{n+2}}{(2^k r    )^{n+3}}
\frac{1}{|2^{k+1}I|}\int_{2^{k+1}I}|f(y)|^p dy
\leq \frac{2^{n+2}}{r}M(|f|^p)(x_0),
\end{align*}
while  the second one is estimated by the help of Lemmas~\ref{l3}
and~\ref{l33}
\begin{align*} &\int_{(2I)^c}
\frac{|a(y)-a_I|^{p'}}{\rho(x_0-y)^{n+3}}dy=\sum_{k=1}^\infty
\int_{2^{k+1}I\setminus 2^k I}
\frac{|a(y)-a_I|^{p'}}{\rho(x_0-y)^{n+3}}dy\\
&\leq \sum_{k=1}^\infty \frac{1}{(2^k r    )^{n+3}}
\int_{2^{k+1}I}|a(y)-a_I|^{p'}dy\\
&\leq  \sum_{k=1}^\infty \frac{1}{(2^k r )^{n+3}}
 \left(\int_{2^{k+1}I}|a(y)-a_{2^{k+1}I}|^{p'}dy+
\int_{2^{k+1}I}|a_{2^{k+1}I}-a_I|^{p'}dy \right)\\
&\leq \frac{C(n,p')}{r}\|a\|^{p'}_\ast \sum_{k=1}^\infty
\frac{(k+1)^{p'}}{2^k}
\leq  \frac{C(n,p')}{r}\|a\|^{p'}_\ast.
\end{align*}
Hence
$$
G_2\leq C(n,p) m^{(n+1)/2}   \|a\|_\ast \big(M(|f|^p)(x_0)\big)^{1/p}.
$$
Finally,
\begin{align*}
G_3&=\frac{1}{|I|} \int_I |C(x)-C_I| dx
\leq   \frac{2}{|I|} \int_I |a(x)-a_I| |\mathcal{K}_{sm}f(x)|dx\\
&\leq 2\left(\	\frac{1}{|I|}\int_I |a(x)-a_I|^{p'}dx\right)^{1/p'}
\left(\  \frac{1}{|I|}\int_I |\mathcal{K}_{sm}f(x)|^{p}dx\right)^{1/p}\\
&\leq C(p) \|a\|_\ast \big(M(|\mathcal{K}_{sm}f|^p)(x_0)\big)^{1/p}.
\end{align*}
Combining $G_1$, $G_2$, $G_3$, taking the supremum with respect to $I$	 and
having in mind that the point was chosen arbitrary we get \eqref{Csharp}.
\quad$\Box$\smallskip


The above lemma and  the sharp inequality yield 	$L^{p,\lambda}$
estimates   for the integral operator $\mathcal{K}_{sm}f$ and its commutator.
\begin{lemma}\label{l6a}
Let $f$, $a$, $\mathcal{K}_{sm}f$	and $\mathcal{C}_{sm}[a,f]$  be as above. Then
\begin{align}\label{Kpl}
&\|\mathcal{K}_{sm}f\|_{p,\lambda} \leq C m^{(n+1)/2}\|f\|_{p,\lambda}\\
\label{Cpl}
&\|\mathcal{C}_{sm}[a,f]\|_{p,\lambda} \leq C
m^{(n+1)/2}\|a\|_\ast\|f\|_{p,\lambda}
\end{align}
where the constants depend on $n$, $p$, $\lambda$  but not on $f$.
\end{lemma}

\paragraph{Proof}  We are going to  estimate	the $L^{p,\lambda}$
norms
of the	sharp functions of  the  corresponding	operators  in  order  to
employ the
sharp inequality (Lemma~\ref{l2}). Let we note that
\eqref{Ksharp} holds true for any $q\in(1,p).$	Therefore, the maximal
inequality (Lemma~\ref{l1})  asserts
  \begin{align*}
\int_I|(\mathcal{K}_{sm}f)^{\#}(x)|^p dx
&\leq Cm^{(n+1)p/2}
\int_I |M(|f|^p)|^{p/q}(x)dx\\
&\leq Cm^{(n+1)p/2} r^\lambda \|M(|f|^q)\|^{p/q}_{p/q,\lambda},
\leq Cm^{(n+1)p/2} r^\lambda \||f|^q\|^{p/q}_{p/q,\lambda}\\
&\leq Cm^{(n+1)p/2} r^\lambda\|f\|^p_{p,\lambda}.
\end{align*}
Dividing of $r^\lambda$  and taking the supremum with respect to $r$  we get
$$
\|(\mathcal{K}_{sm}f)^{\#}  \|_{p,\lambda}\leq C(n,p) m^{(n+1)/2}
\|f\|_{p,\lambda}
$$
and the assertion follows from	Lemma~\ref{l2}.

The $L^{p,\lambda}$  estimate for the commutator follows analogously. After
using \eqref{Csharp}	for $q\in(1,p)$  and applying the maximal inequality
we
arrive to
$$
\int_I |(\mathcal{C}_{sm}[a,f])^{\#}(x)|^p dx \leq C(n,p)\|a\|^p_\ast r^\lambda
\Big\{\|\mathcal{K}_{sm}f\|^p_{p,\lambda} +
m^{(n+1)p/2} \|f\|^p_{p,\lambda}\Big\}
$$
and by the help of
 \eqref{Kpl}  we get
$$
\|(\mathcal{C}_{sm}[a,f])^{\#}\|_{p,\lambda}\leq C(n,p)\|a\|_\ast
m^{(n+1)/2} \|f\|_{p,\lambda}
$$
which leads to the assertion  after applying   Lemma~\ref{l2}.
\quad$\Box$\smallskip

\begin{lemma}\label{l7}
  Denote by $\mathcal{K}_{sm\varepsilon}f$   and
$\mathcal{C}_{sm\varepsilon}[a,f]$  the nonsingular integral operators with
constant
PCZ
kernel
$$
\mathcal{H}_{sm\varepsilon}(x-y)=\begin{cases}
\mathcal{H}_{sm}(x-y)&\text{for }\rho(x-y)\geq \varepsilon\\
0&\text{for }\rho(x-y)< \varepsilon.
\end{cases}
$$
Then for any functions $a$ and	$f$  as above, we have
\begin{align}\label{Kpl1}
&\|\mathcal{K}_{sm\varepsilon}f\|_{p,\lambda} \leq C(n,p)
m^{(n+1)/2}\|f\|_{p,\lambda}\\
\label{Cpl1}
&\|\mathcal{C}_{sm\varepsilon}[a,f]\|_{p,\lambda} \leq C(n,p)
m^{(n+1)/2}\|a\|_\ast\|f\|_{p,\lambda}.
\end{align}
Moreover
\begin{equation}\label{Ksmlim}
\lim_{\varepsilon\to 0}\|\mathcal{K}_{sm\varepsilon}f-\mathcal{
K}_{sm}f\|_{p,\lambda}=
\lim_{\varepsilon\to 0}\|\mathcal{C}_{sm\varepsilon}[a,f]-\mathcal{
C}_{sm}[a,f]\|_{p,\lambda}=0.
\end{equation}
\end{lemma}

\paragraph{Proof}
Fix  an ellipsoid  $\mathcal{E}_\varepsilon\equiv\mathcal{E}_\varepsilon(x_0)=\{y\in
\mathbb{R}^{n+1}
\colon \rho(x_0-y)<\varepsilon\}$  and set for $\mathcal{E}_{\varepsilon/2}$ the
ellipsoid centered at the same point with radius $\varepsilon/2$. Hence
\begin{align*}
\mathcal{K}_{sm\varepsilon}f(x_0)&=
\frac{C}{\varepsilon^{n+2}}\int_{\mathcal{E}_{\varepsilon/2}}
|\mathcal{K}_{sm\varepsilon}f(x_0)|dy\\
&\leq	    \frac{C}{\varepsilon^{n+2}} \int_{\mathcal{E}_{\varepsilon/2}}
|\mathcal{K}_{sm}f(y)|dy +
\frac{C}{\varepsilon^{n+2}} \int_{\mathcal{E}_{\varepsilon/2}}
|\mathcal{K}_{sm\varepsilon}f(x_0)- \mathcal{K}_{sm}f(y)|dy.
\end{align*}
The density $f$  could be written as a sum of the kind
$$
f(x)=f(x)\chi_{\mathcal{E}_{\varepsilon}}(x) + f(x) \chi_{\mathcal{
E}^c_{\varepsilon}}(x)=
f_1(x)+f_2(x)
$$
which allows us to write $\mathcal{K}_{sm}f=\mathcal{K}_{sm}f_1+\mathcal{K}_{sm}f_2$
and hence
\begin{align*}
\mathcal{K}_{sm\varepsilon} f(x_0)=& \frac{C}{\varepsilon^{n+2}}
\int_{\mathcal{E}_{\varepsilon/2}}  \big|\mathcal{K}_{sm}f_1(y) \big| dy\\
&+ \frac{C}{\varepsilon^{n+2}} \int_{\mathcal{E}_{\varepsilon/2}} \big|
\mathcal{K}_{sm}f_2(y) -\mathcal{K}_{sm\varepsilon}f(x_0) \big| dy = E_1 +E_2.
\end{align*}
The first integral is analogous to $J_1$  from Lemma~\ref{l6}  and hence
$$
E_1\leq C\big(M(|f|^q)(x_0) \big)^{1/q}
$$
for any $q\in(1,p)$.

The second integral is analogous to $J_2$  and hence
$$
E_2\leq Cm^{(n+2)/2}\big(M(|f|^q)(x_0)\big)^{1/q}
$$
for any $q\in (1,p)$.

Since the point $x_0$  was chosen arbitrary, the estimates hold true for any
$x\in \mathbb{R}^{n+1}$. Using the same arguments as in the proof of Lemma~\ref{l6a}
we get the desired estimates \eqref{Kpl1}  and \eqref{Cpl1}.

It is known from \cite{FR}   and \cite{BC}  that  the limits
$$
\lim_{\varepsilon\to 0} \mathcal{K}_{sm\varepsilon} f(x)=\mathcal{K}_{sm}f(x),\quad
\lim_{\varepsilon\to 0} \mathcal{C}_{sm\varepsilon}[a, f](x)=\mathcal{
C}_{sm}[a,f](x),
$$
there exist in $L^p$.
  This allows us to assert that  taking $\varepsilon\to 0$  in
\eqref{Kpl1}  and \eqref{Cpl1}	we get exactly \eqref{Kpl}  and \eqref{Cpl},
respectively, and the assertion \eqref{Ksmlim}	follows.
\quad$\Box$\smallskip


Now, after giving the proofs of several helpful results, we shall turn
back to the proof of Theorem~\ref{th2}.

First of all,	 the spherical expansion of the kernel	leads to expansions of
the
nonsingular integrals $\mathcal{K}_\varepsilon f$
and $\mathcal{C}_{\varepsilon}[a,f]$, that is
\begin{gather}\label{Kepsf2}
\mathcal{K}_\varepsilon f(x) = \sum_{s,m} b_{sm}(x) \mathcal{K}_{sm\varepsilon}
f(x),\\
\label{Cepsf2}
\mathcal{C}_\varepsilon [a,f](x) = \sum_{s,m} b_{sm}(x) \mathcal{C}_{sm\varepsilon}
[a,f](x)
\end{gather}
In Lemma~\ref{l7}  we show that  $\mathcal{K}_{sm\varepsilon}f$ and
$\mathcal{C}_{sm\varepsilon}[a,f]$   are bounded in $L^{p,\lambda}$ uniformly
with
respect to $\varepsilon$. Moreover, the series
$$
  \sum_{s,m}
\|b_{sm}\mathcal{K}_{sm\varepsilon} f\|_{p,\lambda}\leq
C	 \|f\|_{p,\lambda}  \sum_{m=1}^\infty  m^{-2l+(n+1)/2+(n-1)},
$$
$$
  \sum_{s,m}
\|b_{sm}\mathcal{C}_{sm\varepsilon}[a,f]\|_{p,\lambda}\leq
C \|a\|_\ast \|f\|_{p,\lambda}	\sum_{m=1}^\infty  m^{-2l+(n+1)/2+(n-1)}
$$
are totally convergent in $L^{p,\lambda}$, uniformly in $\varepsilon$
for
  $l>(3n+1)/4$. Whence
  $$
  \|\mathcal{K}_\varepsilon f\|_{p,\lambda}\leq \|f\|_{p,\lambda},
  \quad \|\mathcal{C}_{\varepsilon}[a,f]\|_{p,\lambda}
  \leq C\|a\|_\ast \|f\|_{p,\lambda}
   $$
   with $C=C(n,p,\lambda,k)$.
Setting
 $$
\mathcal{K} f(x)  =\sum_{s,m} b_{sm}(x)\mathcal{K}_{sm}
f(x),\quad
\mathcal{C}[a,f](x) =\sum_{s,m} b_{sm}(x)\mathcal{
C}_{sm}[a,f] (x), $$
we obtain  through Lemma~\ref{l6a}
$$
\|\mathcal{K} f\|_{p,\lambda}\leq
 C\|f\|_{p,\lambda},\quad
\|\mathcal{C} [a,f]\|_{p,\lambda}
\leq C \|a\|_\ast \|f\|_{p,\lambda}.
$$
Finally, the dominated convergence theorem, applied in $L^{p,\lambda}$
to the series expansions \eqref{Kepsf2} and  \eqref{Cepsf2}  gives
\begin{align*}
\lim_{\varepsilon\to 0} \mathcal{K}_\varepsilon f(x)
=&\lim_{\varepsilon\to 0}
\sum_{s,m}b_{sm}(x)\mathcal{K}_{sm\varepsilon} f(x)
= \sum_{s,m}b_{sm}(x) \lim_{\varepsilon\to 0} \mathcal{K}_{sm\varepsilon}
f(x)\\
=&\sum_{s,m} b_{sm}(x)\mathcal{K}_{sm}f(x)=\mathcal{K} f(x),
\end{align*}
$$\lim_{\varepsilon\to 0} \mathcal{C}_\varepsilon [a,f](x)=
 \sum_{s,m}b_{sm}(x) \lim_{\varepsilon\to 0} \mathcal{C}_{sm\varepsilon}
[a,f](x)=
\mathcal{C}[a,f](x).
$$
This completes the proof of Theorem~\ref{th2}.
\quad$\Box$\smallskip

\begin{theorem}\label{th3}
Let  $Q$  be a cylinder in $\mathbb{R}^{n+1}$ and     $k(x;y)$	 be a variable PCZ
kernel defined in $Q$. Let $f\in L^{p,\lambda}(Q)$, $1<p<\infty$,
$0<\lambda<n+2$  and $a\in  BMO(Q)$. Then $\mathcal{K} f$  and $\mathcal{C}[a,f]$
belong to $L^{p,\lambda}(Q)$  and
\begin{gather}\label{K6}
\lim_{\varepsilon\to 0}  \|\mathcal{K}_\varepsilon f-\mathcal{K}
f\|_{p,\lambda;Q}=0\\
\label{C6}
\lim_{\varepsilon\to 0}   \|\mathcal{C}_\varepsilon [a,f]-\mathcal{C}
[a,f]\|_{p,\lambda;Q}=0
\end{gather}
uniformly with respect to $\varepsilon$. Moreover, the following estimates
hold true
\begin{gather}
\label{K7}
\|\mathcal{K} f\|_{p,\lambda;Q}\leq C\|f\|_{p,\lambda;Q}\\
\label{C7}
\|\mathcal{C} [a,f]\|_{p,\lambda;Q} \leq C\|a\|_\ast \|f\|_{p,\lambda;Q}
\end{gather}
where the constants depend on $n, p, \lambda$	and the kernel $k$.
\end{theorem}

\paragraph{Proof}  Define  the functions
$$
\bar   k(x;y) = \begin{cases}
k(x;y) &\text{for } x\in Q,\  y\in \mathbb{R}^{n+1}\setminus \{0\},\\
0  & \text{elsewhere},
\end{cases}
$$
$$
\bar f(x)=\begin{cases}
f(x) &\text{for }	x\in Q,\\
0  &\text{for }  x\not\in Q.
\end{cases}
$$
Define the operator $\bar{\mathcal{K}}\bar f(x)=\int_{\mathbb{R}^{n+1}}\bar k(x,x-y)\bar
f(y)dy$.
So we can consider $\mathcal{K} f$  as a restriction of $\bar{\mathcal{K}}\bar{f}$
on $Q$ which means
$$
 \|\mathcal{K}_\varepsilon f-\mathcal{K} f\|_{p,\lambda;Q} \leq
 \|\bar{\mathcal{K}}_\varepsilon \bar{f}-\bar{\mathcal{K}}
\bar{f}\|_{p,\lambda}.
 $$
Then  \eqref{K6} follows from \eqref{K4b}.

  The  extension theorems for $BMO$  functions	(cf.
\cite{J}, \cite{A}) allow  to define $\bar{a}\in BMO(\mathbb{R}^{n+1})$
such that $\bar a\vert_Q =a$. Arguments similar to
those already applied  for $\mathcal{K} f$ lead to \eqref{C6}.
Finally
$$
\|\mathcal{K} f\|_{p,\lambda;Q}\leq \|\bar{\mathcal{K}}\bar{f}	\|_{p,\lambda}\leq C
\|\bar{f}\|_{p,\lambda}= C\|f\|_{p,\lambda;Q}.
$$
The estimate \eqref{C7}  follows in the same manner.
 \quad$\Box$\smallskip

\begin{theorem}\label{th4}
Let $k(x;y)$ be a variable PCZ kernel and $a\in VMO\cap L^\infty(Q)$
with  VMO   modulus $\gamma_a(r)$  defined by \eqref{gamma}.	Then for any
$\varepsilon>0$  there exists a positive number
$ r_0= r_0(\varepsilon,\gamma_a)$ such that for any  $r\in (0, r_0)$
and any parabolic cylinder  $I_r\subset Q$   one has
\begin{equation}\label{Ceps}
\| \mathcal{C}[a,f]\|_{p,\lambda;I_r} \leq C\varepsilon \|f\|_{p,\lambda;I_r}
\end{equation}
for any function $f\in L^{p,\lambda}(I_r)$.
\end{theorem}


\paragraph{Proof}
 As it was pointed out above, by the equivalence of the topologies induced by
the standard parabolic metric and the metric \eqref{rho}, we may prove the
theorem in ellipsoids instead of parabolic cylinders. For this goal we
consider
$\mathcal{E}_r$ centered at $x_0$ and  of radius $r$
$$
\mathcal{E}_r=\left\{ x\in \mathbb{R}^{n+1}\colon \frac{(x'-x'_0)^2}{r^2}+
\frac{(t-t_0)^2}{r^4}<1\right\} \text{and}\quad  \mathcal{
E}^c_r=\mathbb{R}^{n+1}\setminus
\mathcal{E}_r.
$$
 From the properties of $BMO$	and VMO  functions
(see Theorem~\ref{tSar}), it  follows that for any $\varepsilon>0$  there
exists
a number $r_0(\varepsilon,\gamma_a)$  and a  continuous and uniformly bounded
function $g$   with modulus of continuity $\omega_g(r_0)<\varepsilon/2$   and
 $\|a-g\|_\ast<\varepsilon/2$.
Fix an ellipsoid
$\mathcal{E}_r\subset Q,$	 such that $r\in (0,r_0)$
and construct a function
$$
h(x)=\begin{cases}
 g(x)&\text{for }  x\in\mathcal{E}_r,\\
 g\Big(
x'_0+r\frac{x'-x'_0}{\rho(x-x_0)},t_0+r^2\frac{t-t_0}{\rho(x_0-x)^2}
\Big) &\text{for } x\in \mathcal{E}_r^c
\end{cases}
$$
which  is uniformly continuous in $\mathbb{R}^{n+1}$ and its
oscillation in $\mathcal{E}_r^c$ equals the oscillation of
$g$  over  the surface $\partial\mathcal{E}_r.$  Then the oscillation of $h$
in
$\mathbb{R}^{n+1}$  is no greater than	 the oscillation of $g$  in $\mathcal{E}_{r_0}$.
Then
$$
\|\mathcal{C}[a,f]\|_{p,\lambda;\mathcal{E}_r} \leq \|\mathcal{C}[a-g,f]
\|_{p,\lambda;\mathcal{E}_r}+
\|\mathcal{C}[g,f] \|_{p,\lambda;\mathcal{E}_r}.
$$
The first norm could be estimated according to \eqref{C7}.
For the second one  we	employ	the constructed above
function.
Whence
\begin{align*}
\|\mathcal{C}[a,f]\|_{p,\lambda;\mathcal{E}_r} &\leq C \left(\|a-g\|_\ast +\|h
\|_\ast  \right)
\|f\|_{p,\lambda;\mathcal{E}_r}\\
&\leq	C\left(\|a-g\|_\ast
+\omega_g(r_0)\right)
\|f\|_{p,\lambda;\mathcal{E}_r}\leq C\varepsilon \|f\|_{p,\lambda;\mathcal{E}_r}.
\end{align*}


\section{Nonsingular integral estimates in  Morrey spaces}\label{s4}

Suppose now that the coefficients of the operator $\mathcal{P}$ are defined in
$\mathbb{D}^{n+1}_+$	and construct a {\it generalized reflection\/}	$T$  in the
next
manner.
Denote by ${\bf a}^n(y)$ the last row of the matrix ${\bf a}=\{a^{ij}\}$
and define
\begin{equation}\label{GR}
T(x',t;y',t) =x'-2x_n\frac{{\bf a}^n(y',t)}{a^{nn}(y',t)},\quad
T(x)=T(x',t;x',t),
\end{equation}
for any $x',y'\in \mathbb{R}^n_+$ and any fixed $t\in \mathbb{R}_+$. Obviously, $T$
maps $\mathbb{R}^n_+$ into $\mathbb{R}^n_-$ and $k(x,T(x)-y)$ turns out to be nonsingular
kernel
for any $x,y\in \mathbb{D}^{n+1}_+$.
The following $L^{p,\lambda}$  estimates concerns integrals with
kernels like that.

\begin{theorem}\label{th6}
Let  $f\in L^{p,\lambda}(\mathbb{D}^{n+1}_+)$, $a\in L^\infty (\mathbb{D}^{n+1}_+)$ and
\begin{gather*}
\widetilde{\mathcal{K}} f(x) =\int_{\mathbb{D}^{n+1}_+} k (x,T(x)-y)f(y) dy\\
\widetilde{\mathcal{C}} [a,f](x)
=\int_{\mathbb{D}^{n+1}_+} k (x,T(x)-y)[a(y)-a(x)]f(y) dy
\end{gather*}
be  integral operators with nonsingular kernels. Then there exist
 constants depending on $n$, $p$ and  $\lambda$,  such that
\begin{gather} \label{tlK}
\|\widetilde{\mathcal{K}} f \|_{p,\lambda;\mathbb{D}^{n+1}_+}
\leq C\|f\|_{p,\lambda;\mathbb{D}^{n+1}_+}\\
 \label{tlC}
\|\widetilde{ \mathcal{C}} [a,f] \|_{p,\lambda;\mathbb{D}^{n+1}_+}
\leq C\|a\|_\ast\| f\|_{p,\lambda;\mathbb{D}^{n+1}_+}.
\end{gather}
\end{theorem}


\paragraph{Proof}
 For all $x=(x_1,\ldots,x_n,t), x_n>0$ we define $\tilde
x=(x_1,\ldots,-x_n,t)$
for any  $t\in\mathbb{R}_+$. Then
there exist  two positive constants $C_1$  and $C_2$  depending on $n$	and
$\Lambda$, such that
$$C_1\rho(\tilde x - y) \leq \rho(T(x)-y) \leq C_2 \rho(\tilde x -y)
$$
for every $x,y\in \mathbb{D}^{n+1}_+$ (cf. \cite{CFL1}, \cite{BC}). We consider
again the expansion
in spherical harmonics of  $k(x;T(x)-y)$.
For the numbers  $s$  and $m$  as in \eqref{Y2}  and
\eqref{Y1}, we have
$$
k (x,T(x)-y) =\sum_{s,m}b_{sm}(x)\frac{Y_{sm}(\overline{T(x)- y})
}{\rho(T(x)-y)^{n+2}}=\sum_{s,m}b_{sm}(x)\mathcal{H}_{sm}( T(x)- y).
 $$
 Hence
$$
\widetilde{    \mathcal{K}} f(x) =\sum_{s,m}b_{sm}(x)\int_{\mathbb{D}^{n+1}_+} \mathcal{H}_{sm}
(T(x)-y)f(y) dy=\sum_{s,m} b_{sm}(x) \widetilde{    \mathcal{K}}_{sm} f(x).
$$
>From the properties of the spherical harmonics \eqref{Y2} and \eqref{Y1}, it
follows
 $$
|\mathcal{H}_{sm}(T(x)-y)|\leq C \frac{m^{(n-1)/2}}{\rho(T(x)-y)^{n+2}} \leq C
\frac{m^{(n-1)/2}}{\rho(\tilde x-y)^{n+2}}.
$$
Consider the operator
\begin{equation}\label{Rf}
Rf(x)=\int_{\mathbb{D}^{n+1}_+}   \frac{f(y)}{\rho(\tilde x -y)^{n+2}}
dy
\end{equation}
for  which  $|\widetilde{\mathcal{K}}_{sm}f|\leq C(n,\Lambda)m^{(n-1)/2}|Rf|$.
We choose
$x_0=(x'', 0,t), x''\in \mathbb{R}^{n-1}, t\in \mathbb{R}_+$  and consider a cylinder
$I=I_r(x_0)$ centered at $x_0$ with  radius
$r.$	As usual $2^kI$  means $I_{2^kr}(x_0)$ for any
integer  $k$, $I_+=I \cap \{x_n>0, t>0\}$ and $I_-=I\cap \{x_n<0, t>0\}$.
Then we can write $f(x)$  as
$$
f(x)=f(x)\chi_{2I_+}(x) +\sum_{k=1}^\infty f(x) \chi_{2^{k+1}I_+\setminus
2^kI_+}(x) =\sum_{k=0}^\infty  f_k(x).
$$
It follows from \cite[Lemma~3.3]{BC}  that,
\begin{align*}
\|Rf_0\|_{p,I_+}^p &\leq \|Rf_0\|_{p,\mathbb{D}^{n+1}_+}^p \leq C(n,p)
\|f_0\|^p_{p,\mathbb{D}^{n+1}_+}\\
&=C \int_{2I_+} |f(y)|^p dy \leq C(n,p,\lambda) r^\lambda
\|f\|^p_{p,\lambda;\mathbb{D}^{n+1}_+}.
\end{align*}
Later on,
$$
|Rf_k(x)|\leq \int_{\mathbb{D}^{n+1}_+} \frac{|f_k(y)|}{\rho(\tilde x
-y)^{n+2}}
dy
$$
where $\rho(\tilde x -y)\geq \rho(x-y)\geq r(2^k-1)\geq 2^{k-1}r$  since $x\in
I_+$, $\tilde x\in I_-$  and $y\in 2^{k+1}I_+\setminus 2^k I_+$. Hence
\begin{align*}
|R f_k(x)|^p&\leq \left( \int_{\mathbb{D}^{n+1}_+}
\frac{|f(y)|\chi_{2^{k+1}I_+\setminus
2^kI_+}(y)}{\rho(\tilde x-y)^{n+2}} dy	\right)^p\\
&\leq
\frac{1}{(2^{k-1}r)^{p(n+2)}} \left( \int_{2^{k+1}I_+}|f(y)| dy
\right)^p\\
&\leq	\frac{1}{(2^{k-1}r)^{p(n+2)}} \left( \int_{2^{k+1}I_+}1dy
\right)^{p/p'}  \left( \int_{2^{k+1}I_+}|f(y)|^p dy \right)\\
&\leq C(n,p,\lambda)2^{k(\lambda-(n+2))}
r^{\lambda-(n+2)} \|f\|^p_{p,\lambda;\mathbb{D}^{n+1}_+}
\end{align*}
and
$$
\int_{I_+} |Rf(y)|^pdy \leq C	r^\lambda
\sum_{k=0}^\infty 2^{k(\lambda-(n+2))} \|f\|^p_{p,\lambda;\mathbb{D}^{n+1}_+}.
$$
The series  above  is convergent since $\lambda<n+2$ and therefore
\begin{equation}\label{Rf1}
\|Rf\|_{p,\lambda;\mathbb{D}^{n+1}_+}\leq
C(n,p,\lambda)\|f\|_{p,\lambda;\mathbb{D}^{n+1}_+}.
\end{equation}
>From the expansion of $\widetilde{ \mathcal{K}} f$	it  follows
\begin{align*}
\|\widetilde{\mathcal{K}} f\|_{p,\lambda;\mathbb{D}^{n+1}_+} &\leq \sum_{s,m}
\|b_{sm}\|_\infty
\|\widetilde{\mathcal{K}}_{sm}f\|_{p,\lambda;\mathbb{D}^{n+1}_+}\\
&\leq C(n,p,\lambda,\Lambda)\|f\|_{p,\lambda;\mathbb{D}^{n+1}_+} \sum_{m=1}^\infty
m^{-2l+(n-1)/2+(n-1)}
\end{align*}
for any $l>0.$	So the series  converges for a suitable choices of $l$
which proves \eqref{tlK}.

The commutator could be written  as a sum of integral operators analogously as
it was already done for $\widetilde{\mathcal{K}} f$
$$
\widetilde{ \mathcal{C}}[a,f](x)=\sum_{s,m} b_{sm}(x)\widetilde{ \mathcal{C}}_{sm}[a,f](x). $$
 Moreover, using similar arguments as  those for $\widetilde{\mathcal{K}} f$, we get
$$
|\widetilde{	\mathcal{C}}_{sm}[a,f]| \leq Cm^{(n-1)/2} \int_{\mathbb{D}^{n+1}_+}
\frac{|a(y)-a(x)|}{\rho(\widetilde{x}-y)^{n+2}} |f(y)| dy.$$  Denote by
$$
R_af(x)=\int_{\mathbb{D}^{n+1}_+}  \frac{|a(y)-a(x)|}{\rho(\tilde
x-y)^{n+2}} |f(y)| dy.
 $$
 From \cite[Theorem~2.1]{B}  we have that for any $q\in(1,p)$ there exists a
constant $C(q)$  such that
$$
|(R_af)^\#(x_0)|\leq C\|a\|_\ast \left\{\big(M(R|f|)^q(x_0)\big)^{1/q}+
\big(M(|f|^q)(x_0)\big)^{1/q} \right\}
$$
for every $x_0\in \mathbb{D}^{n+1}_+. $   Let $x_0=(x'',0,0)$,  $r>0$, $f\in
L^{p,\lambda}$. Hence
\begin{align*}
\int_{I_+} |(R_af)^\#(y)|^pdy	\leq& C\|a\|^p_\ast
\Big\{\int_{I_+}\big(M(R|f|)^q(y)\big)^{p/q}dy\\
&+\int_{I_+}\big(M(|f|^q)(y)\big)^{p/q}dy \Big\}=C\|a\|^p_\ast \{
J_1+J_2\}.
\end{align*}
It is easy to see that
\begin{align*}
(R|f|(x))^q  &=\left( \int_{\mathbb{D}^{n+1}_+}\frac{|f(y)|}{\rho(\tilde
x-y)^{n+2}} dy\right)^q\\
&\leq \left( \int_{\mathbb{D}^{n+1}_+}\frac{dy}{\rho(\tilde x-y)^{n+2}}
\right)^{q/q'}
\left( \int_{\mathbb{D}^{n+1}_+}\frac{|f(y)|^q}{\rho(\tilde x-y)^{n+2}}
dy\right)\leq C R(|f|^q)(x),
\end{align*}
whence
\begin{align*}
 J_1&\leq \int_{I_+} \left| M(R(|f|^q))(x) \right|^{p/q}dx \leq
r^\lambda
\| M(R(|f|^q))\|^{p/q}_{p/q,\lambda;\mathbb{D}^{n+1}_+}\\
&\leq
 r^\lambda\| R(|f|^q) \|^{p/q}_{p/q,\lambda;\mathbb{D}^{n+1}_+} \\
&\leq r^\lambda \||f|^q\|^{p/q}_{p/q,\lambda;\mathbb{D}^{n+1}_+} \leq r^\lambda
\|f\|^p_{p,\lambda;\mathbb{D}^{n+1}_+}
\end{align*}
as follows from Lemma~\ref{l1} and  \eqref{Rf1}. Analogous arguments allow us
to estimate $J_2$.
Using the sharp inequality, we get
$$
\|\widetilde{\mathcal{C}}_{sm}[a,f]\|_{p,\lambda;\mathbb{D}^{n+1}_+}	\leq C
m^{(n-1)/2}\|a\|_\ast\|f\|_{p,\lambda;\mathbb{D}^{n+1}_+}.
 $$
 The representation of the commutator $\widetilde{    \mathcal{C}}[a,f]$ as  Fourier
series
gives
 $$
 \|\widetilde{\mathcal{C}}[a,f]\|_{p,\lambda;\mathbb{D}^{n+1}_+}	\leq
 C\|a\|_\ast \|f\|_{p,\lambda;\mathbb{D}^{n+1}_+} \sum_{m=1}^\infty
m^{(n-1)/2-2l+(n-1)}
 $$
 and the series  converges for $l>(3n-1)/4$  which proves \eqref{tlC}.
\quad$\Box$\smallskip

\begin{corollary}\label{c2}
Let $I_r$ be a parabolic cylinder in $\mathbb{R}^{n+1}_+$, $a\in VMO\cap
L^\infty(I_r)$
with	VMO modulus $\gamma_a(r)$.
 Then for every  $\varepsilon>0$   there exists a positive number
$r_0(\varepsilon,\gamma_a)$, such that for every  $f\in L^{p,\lambda}(I_r)$,
$r<r_0$   is fulfilled
$$
\|\widetilde{\mathcal{C}}[a,f] \|_{p,\lambda;I_r}   \leq C(p,\lambda,\Lambda)
\varepsilon \|f\|_{p,\lambda;I_r}.
 $$
\end{corollary}
 The proof is analogous to that of Theorem~\ref{th4}.


\section{A priori estimates, existence and uniqueness}\label{s5}

\begin{theorem}\label{th5}
Suppose $a^{ij}\in VMO(Q_T)$ and conditions  \eqref{2a}, \eqref{3a}   to
be fulfilled. Then for every $f\in L^{p,\lambda}(Q_T)$, $1<p<\infty$,
$0<\lambda<n+2$, the problem	\eqref{1}  has a unique solution $u\in
W^{2,1}_{p,\lambda}(Q_T)$. Moreover, it satisfies
\begin{equation}\label{upl}
\|u\|_{W^{2,1}_{p,\lambda}(Q_T)}\leq C\|f\|_{p,\lambda;Q_T}
\end{equation}
where the constant depends on $n, p, \lambda, \Lambda, T, \partial\Omega$  and
the VMO-moduli of $a^{ij}$.
\end{theorem}

\paragraph{Proof}
We begin with the establishment of the a~priori estimate \eqref{upl}. Let
$u\in
W^{2,1}_p(Q_T)$ be a solution of \eqref{1}. Its existence follows from
\cite{Sf1} having in mind that $L^{p,\lambda}(Q_T)$  is a subspace of
$L^p(Q_T)$ for every $p\in(1,\infty).$	To estimate the  $L^{p,\lambda}$ norms
of the	derivatives $D_{ij}u$ of this solution we
use their representation  inside  the cylinder (cf. \cite{BC})	and near  the
boundary (cf. \cite{Sf1}).

{\it Step 1:  Interior estimate.\/} By density arguments, we consider
$u\in
C_0^\infty(\mathbb{R}^{n+1}_+)$ and  $u(x',0)=0$. For $x\in{\rm supp\,}u$  the
following {\it interior representation formula\/} holds (cf. \cite{BC})
\begin{equation} \label{IF}
 \begin{aligned}
D_{ij}u(x)=&P.V.\int_{\mathbb{R}^{n+1}}\Gamma_{ij}(x;x-y)
\left\{\big(a^{hk}(y)-a^{hk}(x)\big)
D_{hk}u(y) +f(y)\right\}dy\\
&+f(x)\int_{\Sigma_{n+1}}\Gamma_j(x;y)n_id\sigma_y,
\end{aligned}
\end{equation}
where $\Gamma_{ij}$ are the  derivatives of the fundamental solution
\eqref{fsol} with respect  to the second variable, and
$n_i$ is the $i-th$ component of the outer normal of the surface
$\Sigma_{n+1}$.

As it is shown in   \cite{FR},	  $\Gamma^0_{ij}$
  is a constant PCZ kernel.
 Later on, from the boundedness of the fundamental solution (cf.
\cite{LSU})
$$
\sup_{y\in\Sigma_{n+1}}\left| \left(\frac{\partial}{\partial y}
\right)^\beta \Gamma(x;y)\right|\leq C(\beta,\Lambda)
 $$
it  follows that $\Gamma_{ij}(x;y)$   is a variable PCZ kernel.
  Define
\begin{align*}
\mathcal{K}_{ij}f(x)&=P.V.\int_{\mathbb{R}^{n+1}}\Gamma_{ij}(x;x-y)f(y)dy,\\
\mathcal{C}_{ij}[a,f](x)&=P.V.\int_{\mathbb{R}^{n+1} }
\Gamma_{ij}(x;x-y)[a(y)-a(x)]f(y)dy \\
&=\mathcal{K}_{ij}(af)(x)-a(x)(\mathcal{K}_{ij}f)(x).
\end{align*}
Hence for $x\in {\rm supp\,}u$
$$
D_{ij}u(x)=\sum_{h,k=1}^n  \mathcal{C}_{ij}[a^{hk},D_{hk}](x) +\mathcal{K}_{ij}f(x)+
f(x)\int_{\Sigma_{n+1}}\Gamma_j(x;y)n_id\sigma_y.
$$
Consider parabolic cylinder $I$ with radius $r$. From Theorems~\ref{th2}
and~\ref{th4} it
follows
\begin{equation}\label{22}
\|D^2u\|_{p,\lambda;I}	\leq C \left(\gamma_a(r_\alpha)
\|D^2u\|_{p,\lambda;I} +\|f\|_{p,\lambda;I} \right).
\end{equation}
 Choosing
$r$
smaller, if  necessary, such that $C\gamma_a(r)<1$,  we get
$$
\|D^2u\|_{p,\lambda;I}	\leq C\|f\|_{p,\lambda;I} \leq C\|f\|_{p,\lambda;Q_T},
$$
where the constant depends on $n,p,\lambda,\gamma_a(r), \|D\Gamma\|_\infty$.
To estimate $u_t $  we employ  the equation
$$
u_t=a^{ij}(x)D_{ij}u+f(x)
$$
and the boundedness of the coefficients. Hence,
\begin{align*}
\|u_t\|_{p,\lambda;I}& \leq C\|a\|_\infty \|D^2u\|_{p,\lambda;I}
+\|f\|_{p,\lambda;I}\\
&\leq C\|f\|_{p,\lambda;Q_T},
\end{align*}
 where $C=C(n,p,\lambda,\Lambda, \|D\Gamma\|_\infty,
\gamma_a(r),\|a\|_{\infty,Q_T})$, where
$\|a\|_{\infty,Q_T}=\max\|a^{ij}\|_{\infty,Q_T}$
where the maximum is taken over $i,j=1,\ldots,n$.

The estimate of the solution follows  from the representation
$u(x)=\int_0^t u_s(x',s)ds$
  and the Jensen inequality, which give
$$
\|u\|_{p,\lambda;I} \leq Cr^2 \|f\|_{p,\lambda;Q_T}
$$
where the constant depends on the same quantities.
  Combining the estimates above we get that for any parabolic cylinder $I$,
such that $I\cap S_T=\emptyset$,
	we have
\begin{equation}\label{uI}
\|u\|_{W^{2,1}_{p,\lambda}(I)} \leq C \|f\|_{p,\lambda;Q_T}
\end{equation}

 Considering a cylinder $Q'=\Omega'\times(0,T)$ with $\Omega'\Subset\Omega$,
making a
covering of $Q'$  by parabolic cylinders $I_\alpha$, $\alpha\in \mathcal{A}$,
considering a partition of the unit subordinated to this covering,
applying \eqref{uI}  for each $I_\alpha$ and using the
 interpolation inequality to lower order terms we get
\begin{equation}\label{uQ}
\|u\|_{W^{2,1}_{p,\lambda}(Q')}  \leq C
     \big(\|f\|_{p,\lambda;Q_T} +
+\|u\|_{p,\lambda;Q''}\big),
\end{equation}
where $Q''=\Omega''\times(0,T)$   and $\Omega'\Subset\Omega''\Subset\Omega$
and the constant depends on $n$, $p$, $ \lambda$, $\Lambda$, $ T$, $
\|D\Gamma\|_\infty$, $ \gamma_a(r)$, $\|a\|_{\infty,Q_T}. $



{\it Step 2: Boundary estimate.\/}
Let us	suppose   now $u\in C_0^\infty(\mathbb{D}^{n+1}_+)$.
 Define the semycilinder
$$
B_+ =\{x \in \mathbb{D}^{n+1}_+\colon |x'|<R,\ x_n>0,\ 0<t<R^2\}
 $$
 with a base $\Omega_+=\{|x'|<R,\ x_n>0\}$   and  $S_+=\{|x''|<R,\ x_n=0,
0<t<R^2\}$. Consider the problem
\begin{equation}\label{VP}
\begin{gathered}
{ \mathcal{P}} u  \equiv u_t- a^{ij}(x)D_{ij}u =f(x)
\quad\text{a.e. in } B_+,\\
 \mathcal{I} u\equiv u(x',0)=0  \text{on }  \Omega_+,\\
 \mathcal{B} u\equiv \ell^i(x)D_i u=0	\text{on } S_+.
\end{gathered}
\end{equation}
Than
 we have the following {\it
boundary representation formula\/} for the second
spatial derivatives of the solution of \eqref{VP} (cf. \cite{Sf1})
$$D_{ij}u(x)=I_{ij}(x)-J_{ij}(x)+H_{ij}(x)$$
 where
\begin{gather*}
I_{ij}(x)= P.V.
\int_{B_+ }\Gamma_{ij}(x;x-y)F(x;y) dy
+f(x) \int_{\Sigma_{n+1}}\Gamma_j(x;y)n_i
d\sigma_y,\\
  i,j=1,\dots,n\,;\\
J_{ij}(x)=\int_{B_+
}\Gamma_{ij}(x;T(x)-y',t-\tau)F(x;y) dy,\   i,j=1,\dots,n-1;\\
J_{in}(x)=J_{ni}(x)=\int_{B_+}\sum_{l=1}^n\Gamma_{il}(x;T(x)-y',t-\tau)
\left(\frac{\partial T(x)}{\partial x_n} \right)^l F(x;y)dy\\
 i=1,\dots,n-1; \\
J_{nn}(x)=\int_{B_+}\sum_{l,s=1}^n \Gamma_{ls}(x;T(x)-y',t-\tau)
\left( \frac{\partial T(x)}{\partial x_n} \right)^l
\left(\frac{\partial T(x)}{\partial x_n} \right)^s
F(x;y)dy;
 \end{gather*}
\begin{align*}
 H_{ij}(x)=& P.V. \int_{S_+}G_{ij}(x;x''-y'',x_n,t-\tau) g(y'',\tau)
dy''d\tau\\
&+g(x'',t)\int_{\Sigma_n} G_j(x;y'',x_n,\tau) n_i
d\sigma_{(y'',\tau)}.
\end{align*}
In the above expressions $T(x)$ is given by \eqref{GR}	and
\begin{gather*}
\frac{\partial
T(x)}{\partial x_n}=\left(-2\frac{a^{n1}(x)}{a^{nn}(x)},\ldots,-2\frac{a^{n
n-1}(x)}{a^{nn}(x)},-1\right),\\
g(y'',\tau)=\Big[\ell^k(0)-\ell^k(y'',\tau)\Big]D_ku(y'',\tau)- \ell^k(0)
\Big[(\Gamma_k\ast F)\Big|_{y_n=0}\Big](y'',\tau),\\
F(x;y)=f(y)+[a^{hk}(y)-a^{hk}(x)]D_{hk}u(y)
\end{gather*}
and $G=\Gamma \mathcal{Q}$  where $\mathcal{Q}$  is a regular bounded function.


We will use in the sequel the following notations
\begin{gather*}
\widetilde{\mathcal{K}}_{ij}f(x)=\int_{\mathbb{D}^{n+1}_+}\Gamma_{ij}(x;T(x)-y',t-\tau)
f (y) dy \\
\widetilde{\mathcal{C}}_{ij}[a,f](x)=\int_{\mathbb{D}^{n+1}_+}\Gamma_{ij}
(x;T(x)-y',t-\tau)[a( y) - a (x) ] f( y) d y.
\end{gather*}
Hence
\begin{gather*}
I_{ij}(x)=\mathcal{K}_{ij}f(x)+\mathcal{C}_{ij}[a^{hk},D_{hk}u](x),\quad
i,j=1,\ldots,n;\\
J_{ij}(x) =\widetilde{	 \mathcal{K}}_{ij}f(x)
+\widetilde{\mathcal{C}}_{ij}[a^{hk},D_{hk}u](x),\quad i,j=1,\ldots,n.
\end{gather*}
Note that the components of the vector $\frac{\partial T(x)}{\partial x_n}$
are
bounded so the integrals $J_{in}$  and $J_{nn}$  can be presented as a sum of
nonsingular integral operators, exactly as $J_{ij}$, $i,j\not= n$.
>From  Theorems~\ref{th2}  and \ref{th6} it  follows
\begin{equation}\label{IJ}
\begin{gathered}
\|I_{ij}\|_{p,\lambda;B_+}\leq C\left(\|f\|_{p,\lambda;B_+}+ \gamma_a(R)\|
D^2u\|_{p,\lambda;B_+} \right), \\
\|J_{ij}\|_{p,\lambda;B_+}\leq C\left(\|f\|_{p,\lambda;B_+}+ \gamma_a(R)\|
D^2u\|_{p,\lambda;B_+} \right).
\end{gathered}
\end{equation}
To estimate the  $L^{p,\lambda}$  norm of $H_{ij}$
we suppose that the vector
field $\ell$  is extended in $B_+$  preserving its
Lipschitz regularity. This automatically leads	also to extension in $B_+$ of
the
function
$g$
\begin{equation}\label{gg}
g(x)=\Big[\ell^k(0)-\ell^k(x)\Big]D_ku(x)- \ell^k(0)	(\Gamma_k\ast
F)(x).
\end{equation}

Moreover, since   $G$	is a product of the fundamental solution
$\Gamma$
and a regular  function $\mathcal{Q}$, the derivatives	$G_{ij}$  possess
properties
similar to that of $\Gamma_{ij}$.
Now  using  \cite[Theorem~1]{Sf1}   we	write
$$
G_{ij}\ast_2 g =  P.V.
\int_{S_+}G_{ij}(x;x''-y'',x_n,t-\tau)g(y'',\tau)
dy''d\tau
$$
\begin{align*}
\int_{B_+\cap I}  |G_{ij}\ast_2 g|^p dx& \leq C
\Big(\int_{B_+\cap I}|g|^pdx + \int_{B_+\cap I}|Dg|^p
dx\Big)\\
& \leq C  r^\lambda
\Big(\frac{1}{r^\lambda}\int_{B_+\cap I}|g|^pdx +\frac{1}{r^\lambda}
\int_{B_+\cap I}|Dg|^p dx\Big),
\end{align*}
where $I$  is a parabolic cylinder with radius $r$. Taking the supremum with
respect to $r$	we get
$$
 \|G_{ij}\ast_2 g\|_{p,\lambda;B_+}\leq C\left(\|g\|_{p,\lambda;B_+}
+\|Dg\|_{p,\lambda;B_+} \right).
$$
The  second integral  in $H_{ij}$  is a   product of $g$
and
 bounded surface integral, hence
$$
\|H_{ij}\|^p_{p,B_+\cap I}  \leq
Cr^\lambda\left(\frac{1}{r^\lambda}\|g\|^p_{p,B_+\cap I}
+\frac{1}{r^\lambda}\|Dg\|^p_{p,B_+\cap I}\right).
$$
Taking the supremum with respect to $r$  we get
$$
\|H_{ij}\|_{p,\lambda;B_+}  \leq C\big(\|g\|_{p,\lambda;B_+}
+\|Dg\|_{p,\lambda;B_+}\big).
$$

An immediate consequence of \eqref{gg}	 is the bound
$$
\|g\|_{p,\lambda;B_+}\leq \|[\ell^k(0)-\ell^k(y)  ]D_k u(y)
\|_{p,\lambda;B_+} +C\|\Gamma_k\ast F\|_{p,\lambda;B_+}.
 $$
 Denoting by $\|\ell\|_{{\rm Lip}(S_T)}$ the Lipschitz constant of
$\ell$, we have
 $$
 \|[\ell^k(0)-\ell^k(y) ]D_ku(y)\|_{p,\lambda;B_+}\leq
CR^2\|\ell\|_{{\rm Lip}(S_T)}\|D u\|_{p,\lambda;B_+}.
  $$
  Later,
$$\|\Gamma_k\ast F \|_{p,\lambda;B_+}\leq \|\Gamma_k\ast f\|_{p,\lambda;B_+}
+\| \Gamma_k\ast[a^{hk}(\cdot)- a_{hk}(x)]D_{hk}u(\cdot)
\|_{p,\lambda;B_+}.
$$
The convolution $\Gamma_k\ast f$  can be considered  as   Riesz
potential  \cite[Lemma~7.12]{GT}  and the estimate  is achieved as in
\cite[Theorem~1]{Sf1} $$
\int_{B_+\cap I}|\Gamma_k\ast f|^pdx \leq
CR^p \int_{B_+\cap I}| f|^pdx
\leq CR^{p} r^\lambda \|f\|_{p,\lambda;B_+}.
$$
Taking again the supremum with respect to $r,$	we get
$$
\|\Gamma_k\ast f \|_{p,\lambda;B_+}\leq C R \|f\|_{p,\lambda;B_+}.
$$

It is known  from the properties of the fundamental solution that $\Gamma_k\in
L^1_{\rm loc}$	 and  it behaves like $\rho(x)^{-(n+1)}$.
Multiplying and dividing  by $\rho(x-y)$  we can apply	the theorems
for integral operators with singular kernels. Note that $\rho(x-y)^{-(n+2)}$
is a non-negative  measurable  function and  we can apply
\cite[Theorem~0.1]{B}.
By the same technique as above, one gets
\begin{gather*}
|\Gamma_k\ast [a^{hk}(\cdot)-a^{hk}(x)]D_{hk}u(\cdot)|\leq
CR\int_{B_+}
\frac{|a^{hk}(y)-a^{hk}(x)||D_{hk}u(y)|}{\rho(x-y)^{n+2}}dy,\\
\|\Gamma_k\ast F\|_{p,\lambda;B_+} \leq
CR\left(\|f\|_{p,\lambda;B_+}+\gamma_a(R)\|D^2 u\|_{p,\lambda;B_+}
\right),\\
 \|g\|_{p,\lambda;B_+}\leq
C\big( R^2\|D u\|_{p,\lambda;B_+}+R\|f\|_{p,\lambda;B_+}+R\gamma_a(R)\|D^2
u\|_{p,\lambda;B_+} \big).
\end{gather*}
Further, the Rademacher theorem asserts existence   almost everywhere
of the derivatives $D_h \ell^k\in L^\infty(Q_T)$. Thus,
$$
D_h g(x)=-D_h\ell^k(x) D_ku(x) +
[\ell^k(0) - \ell^k(x)]D_{kh}u -\ell^k(0)(\Gamma_{kh}\ast F).
$$
The $L^p$ norm of the last term  is estimated according to
Theorem~\ref{th2} while the others two are treated as above. Hence
$$
\|D g\|_{p,\lambda;B_+}\leq C\left(\|D
u\|_{p,\lambda;B_+}+R^2\|D^2u\|_{p,B_+}
+ \|f\|_{p,\lambda;B_+} + \gamma_a(R)\|D^2 u\|_{p,\lambda;B_+}\right).
$$
Finally, applying the Gagliardo-Nirenberg  interpolation inequality
to
$\|Du\|_{p,B_+}$, we obtain
$$
\|Du\|_{p,\lambda;B_+} \leq C\left(\frac{1}{\varepsilon} \|u\|_{p,\lambda;B_+}
+\varepsilon\|D^2u\|_{p,\lambda;B_+} \right).
$$
Choosing $\varepsilon=R(R+1)$  for $R$	 small enough we get
$$
\|H_{ij}\|_{p,\lambda;B_+} \leq C\left(\frac{1}{R}\|u\|_{p,\lambda;B_+}
+  \|f\|_{p,\lambda;B_+}+(R+ \gamma_a(R))\|D^2u\|_{p,\lambda;B_+}
\right).
$$
 Combining the last inequality with \eqref{IJ}, we get
 $$
\|D^2u\|_{p,\lambda;B_+} \leq C\left(\frac{1}{R}\|u\|_{p,\lambda;B_+}
+  \|f\|_{p,\lambda;B_+}+\big(R+ \gamma_a(R)\big)\|D^2u\|_{p,\lambda;B_+}
\right),
 $$
 whence, taking $R$  small enough (recall $\gamma_a(R)\to0$ as $R\to0$), one
obtains  $$
\|D^2 u\|_{p,\lambda;B_+}\leq C\left(\|f\|_{p,\lambda;B_+}
+\frac{1}{R}\|u\|_{p,\lambda;B_+} \right). $$
Expressing $u_t$ from the equation we get
\begin{equation}\label{ut}
\|u_t\|_{p,\lambda;B_+}\leq
 C\left(\|f\|_{p,\lambda;B_+} +\frac{1}{R}\|u\|_{p,\lambda;B_+}\right).
\end{equation}
Now, writing  $u(x)=\int_0^t u_s(x',s)ds$  and making use of Jensen's
inequality and	\eqref{ut} we obtain
 $$\|u\|_{p,\lambda;B_+} \leq C R^2\|u_t\|_{p,\lambda;B_+}\leq	C\left(R^2
\|f\|_{p,\lambda;B_+}+R\|u\|_{p,\lambda;B_+}\right).
$$
Hence, choosing $R$  smaller, if necessary, we get $\|u\|_{p,\lambda;B_+}
\leq
C\|f\|_{p,\lambda;B_+}$ and therefore
\begin{equation}\label{LB}
\|u\|_{W^{2,1}_{p,\lambda}(B_+)} \leq C\|f\|_{p,\lambda;B_+}\leq
C\|f\|_{p,\lambda;Q_T} \end{equation}
for each solution to the problem  \eqref{VP}.
Making a covering $\{B_\alpha\}$, $\alpha\in\mathcal{A}$ of the boundary  $S_T$
such that  $Q_T\setminus Q'\subset \bigcup_{\alpha\in\mathcal{A}} B_\alpha$,
considering
a partition of the unit subordinated to this covering and applying the
estimate
\eqref{LB}   for each $B_\alpha$  we get
\begin{equation}\label{uBQ}
\|u\|_{W^{2,1}_{p,\lambda}(Q_T\setminus Q')} \leq C (n,p,\lambda,\Lambda, T,
\|D\Gamma\|_\infty, \gamma_a(r),\|a\|_\infty)\|f\|_{p,\lambda;Q_T} .
\end{equation}
The estimate \eqref{upl}  follows from \eqref{uQ}  and \eqref{uBQ}.


{\it Step 3: Existence and uniqueness.\/}
The uniqueness of the solution $u\in W^{2,1}_{p,\lambda}(Q_T)$ of the problem
under consideration follows trivially from the a~priori estimate  \eqref{upl}.


The   existence of the solution  can be proved by
the {\it  method of continuity,\/}	as it is done in \cite{Sf1}. For this
goal we connect the solvability of \eqref{1}  with the solvability in
$W^{2,1}_{p,\lambda}(Q_T)$ of the problem
\begin{equation}\label{Hu}
\begin{gathered}
\mathcal{H} u\equiv u_t-
 \Delta u =f(x) \quad{\rm a.e.\ in }\quad  Q_T,\\
\mathcal{I} u\equiv u(x',0)=0          \quad {\rm on }\quad \Omega, \\
\mathcal{B} u\equiv   \ell^i(x)D_iu=0	\quad{\rm on }\quad  S_T.
\end{gathered}
\end{equation}
Obviously, for any $f\in L^{p,\lambda}(Q_T)$ the above problem is
uniquely solvable in $W^{2,1}_p(Q_T)$  (cf. \cite{LSU}). In
the representation formula of the solution  of \eqref{Hu}
  the commutators
disappear, so  it is not difficult to establish the appropriate
$L^{p,\lambda}(Q_T)$ estimates which ensure solvability in
$W^{2,1}_{p,\lambda}(Q_T)$ of \eqref{Hu}.
  \quad$\Box$\smallskip

 Let us recall that when $u\in W^{2,1}_p$    its
derivatives $D_iu$  are   H\"older continuous functions for $p>n+2$
(see \cite{LSU}, \cite[Corollary~1]{Sf1}). It is worth noting that the Morrey
regularity of the solution implies a H\"older regularity of $D_iu$ for values
of
$p$ smaller than $n+2$.
\begin{corollary}\label{c3}
Suppose   $a^{ij}\in VMO(Q_T)$, $f\in	L^{p,\lambda}(Q_T)$, $p\in(1,\infty)$,
$\lambda\in(0,n+2)$ and  the conditions  \eqref{2a} and  \eqref{3a} to be
fulfilled. Let $u\in W^{2,1}_{p,\lambda}(Q_T)$	be a solution of the problem
\eqref{1}. Then

$i)$\quad $u\in C^{0,\alpha}(\overline Q_T)$	with
$\alpha=\frac{1}{n+1}+\frac{\lambda-(n+2)}{p}$ for $p>(n+1)(n+2-\lambda);$

$ii)$\quad  $Du\in  C^{0,\beta}(\overline Q_T)$  with $\beta=1+\frac{\lambda}{p}-
\frac{n+2}{p}$	for  $\lambda>\max\{ 0,n+2-p\}$.
 \end{corollary}

\paragraph{Proof}
The assertion $i)$   follows directly from  Theorem~\ref{th5}	  and
\cite[Theorem~4.1]{DP}.

The second assertion could be achieved by the parabolic  Poincar\'e inequality
\cite[Lemma~2.2]{Cn}
\begin{align*}
\int_{Q_T\cap I} \big| Du-(Du)_{Q_T\cap I}\big|^p dxdt
&\leq r^p \int_{Q_T\cap I}\left(|u_t|^p+|D^2u|^p \right) dxdt\\
&\leq C r^{p+\lambda} \|u\|_{W^{2,1}_{p,\lambda}(Q_T)}.
\end{align*}
This implies that the gradient $Du$  belongs to the space of Campanato
$\mathcal{L}^{p,p+\lambda}(Q_T).$	It is well known (cf. \cite[Theorem~3.1]{DP},
\cite[$\S~3.3.2$]{MPS})    that for $p+\lambda\in (n+2,n+2+p)$	the space
$\mathcal{L}^{p,p+\lambda}(Q_T)$  coincides with $C^{0,\beta}(\overline Q_T)$ with
$\beta=(p+\lambda-(n+2))/p$.
\quad$\Box$\smallskip


\subsection*{Acknowledgements}
The author is very  indebted to the referee for the valuable remarks which led
to improvement of the paper.


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\end{document}