\documentclass[twoside]{article}
\usepackage{amsfonts, amsmath} % used for R in Real numbers
\pagestyle{myheadings}
  
\markboth{ $W^{1,p}$ estimates for quasilinear parabolic equations }
{Ireneo Peral \& Fernando Soria}

\begin{document}
\setcounter{page}{121}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
2001-Luminy conference on Quasilinear Elliptic and Parabolic Equations 
and Systems,\newline 
Electronic Journal of Differential Equations, 
Conference 08, 2002, pp 121--131. \newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
 \vspace{\bigskipamount} \\
%
 A note on $W^{1,p}$ estimates for quasilinear parabolic equations
%
\thanks{ {\em Mathematics Subject Classifications:} 35K10, 35K55, 42B25.
\hfil\break\indent
{\em Key words:} semilinear parabolic equations, gradient estimates, Calderon-Zygmund theory.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Published October 22, 2002. \hfil\break\indent
I. Peral was supported by grant BFM2001-0183 from M.C.Y.T. Spain.
\hfil\break\indent
F. Soria was supported by grant BFM2001-0189 from M.C.Y.T. 
} }

\date{}
\author{Ireneo Peral \& Fernando Soria} 
\maketitle

\begin{abstract} 
  This work deals with the study of the $W^{1,p}$ regularity for 
  the solutions to parabolic equations in divergence form. 
  An argument by perturbation based in real analysis is used.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}{Lemma}
\numberwithin{equation}{section}

\newtheorem{Theorem}{Theorem}[section]
\newtheorem{Definition}[Theorem]{Definition}
\newtheorem{Lemma}[Theorem]{Lemma}
\newtheorem{Corollary}[Theorem]{Corollary}
\newtheorem{Example}[Theorem]{Example}

\section{Introduction}\label{sec:s1}

In this paper we study interior $W^{1,p}$
estimates for solutions to quasilinear parabolic equations in
divergence form, namely, solutions to the equation
\begin{equation}\label{eq:madre}
u_t-\mathop{\rm div} a(x,t,\nabla u)=0,\quad x\in\Omega,\quad t>0,
\end{equation}
where $a:\Omega\times(0,\infty) \times {\mathbb R}^N \rightarrow \mathbb R^N$.
We assume that $a(x,t,\xi)$ is a Caratheodory function
(in the sense that it is measurable in $(x,t)$ and continuous with
respect to $\xi$ for each $x$) and that satisfies the
following conditions:
\begin{description}
\item{(a1)} $a(x,t, 0)=0$
\item{(a2)} $\langle a(x,t,\xi)-a(x,t,\eta),(\xi-\eta)\rangle\geq
\gamma|\xi|^2$.
\item{(a3)} $|a(x,t,\eta)|\leq \Gamma|\eta| $,
\end{description}
where $\gamma$ and $\Gamma$ are positive constants.

We will work under the following hypotheses.

\noindent\textbf{(H1)} (\textit{Reference operator.})
For fixed $a_0$ satisfying
(a1)--(a3), we consider the corresponding parabolic equation
\begin{equation}\label{eq:ref}
w_t-\mathop{\rm div}(a_0(\nabla w))=0
\end{equation}
If $u$ is a weak solution to \eqref{eq:ref} (see Definition 3.2 below)
then there exists $\gamma>0$ such that
\begin{equation}\label{eq:regref}
\sup_{R'}|\nabla_x
u(x,t)|^2\leq \gamma \frac 1{|R|}\int_R |\nabla_x u(x,t)|^2dx\,dt,
\end{equation}
for all {\it parabolic rectangles}
$$
R=\big \{(x,t):|x_i-x_i^0|<\rho,\;i=1,\dots N,\;t_0-\rho^2<t<t_0\big\}
\subset\Omega,
$$
where
$$
R'=\big\{(x,t):|x_i-x_i^0|<\frac{\rho}{2},\;i=1,\dots N,\;
t_0-(\frac{\rho}{2})^2<t<t_0\big\}.
$$

\noindent\textbf{(H2)} (\textit{Approximation property.})
 The vector field
$a(x,t,\xi)$ is close to $a_0(\xi)$ in the following sense:
\begin{equation}\label{eq:aprox}
|a(x,t,\xi)-a_0(\xi)|\le \epsilon \,|\xi|,
\end{equation}
uniformly in $x$ and $t$.

The simplest example for $a_0(\xi)$ is just $a_0(\xi)=\xi$, which corresponds to the heat
equation. A classical example for $a$ is $a(x,t,\xi)=A(x,t)\xi$ with
$A(x,t)$ a bounded $N\times N$ matrix such that
$$
\|A(x,t)-I_{N\times N}\|_\infty\le \epsilon.
$$
Even in the linear case, the results presented here seem to be new.
We also point out that more general situations can be considered
by the method that we develop. For simplicity, we restrict ourselves to
the {\it quadratic growth} case.
With these hypotheses we will be able to show some sort
of parabolic Meyers type inequalities. More precisely we have the
following main result.

\begin{Theorem}\label{th:main}
Let $a$ be a vector field satisfying (a1)--(a3).
 Assume that (H1) holds. Given $p>2$ there
exists $\epsilon_0>0$ such that if for some $0<\epsilon
<\epsilon_0$ (H2) holds, then any weak solution to
$$
u_t-\mathop{\rm div} (a(x,t,\nabla_x u))=0,
$$
satisfies that $|\nabla_xu|\in L^q_{\rm loc}$, $2<q<p$.
\end{Theorem}

The idea is to estimate the level sets of $\nabla_x u$
and obtain the required growth of their measure to have the integrability
property. To do that, we will use a form of the Calder\'{o}n-Zygmund
covering result.
For the elliptic case see \cite{CP}.

\section{Preliminary results}\label{s2}

In this section we present some tools that will be used
along the paper. First of all we will prove the corresponding {\it
Calder\'{o}n-Zygmund} covering result and explain the properties of
the maximal operator which naturally arises in the parabolic
setting under study.
To be systematic we will give a general {\it Covering Lemma} that
includes the particular case that we will need.


\subsection*{The rectangle collection}

For every $k\in \mathbb {Z}$, let ${\mathcal B}_k$ denote a
collection of rectangles satisfying the following properties:

\begin{description}
\item $i)$ All the rectangles of ${\mathcal B}_k$ have the same
side lengths.
\item $ii)$ Any two distinct rectangles of ${\mathcal B}_k$ have
disjoints interiors.
\item $iii)$ $\bigcup\{R\in {\mathcal B}_k\}=\mathbb R^N$
\item $iv)$ If $R\in {\mathcal B}_k$ then $R=\bigcup\{R'\in {\mathcal B}_{k-1}|\, R'\subset R\}$
\item $v)$ If $\delta_k$ denotes the length of the diagonal of
any $R\in {\mathcal B}_k$ then $\delta_k\ge 2\delta_{k+1}$
\end{description}
Observe that given $R\in {\mathcal B}_k$ there exists a unique
rectangle in ${\mathcal B}_{k-1}$, its {\it predecessor} which we
denote by $R^*$ so that $R\subset R^*$.

With these properties in hand we have the following observations.

\begin{Lemma}\label{lema1}
Fix $R_0\in {\mathcal B}_{k_0}$ and $0<\delta<1$.
Assume that $A\subset R_0$ and that $0<|A|<\delta|R_0|$. Then,
there exists a sequence $\{R_d\}\subset
\bigcup\limits_{k=k_0}^\infty{\mathcal B}_k$ such that:
\begin{enumerate}
\item $|A\setminus \bigcup\limits_d R_d|=0$,
\item $|A\bigcap R_d|\ge \delta |R_d|$,
\item If $R_d\subset R$ for some $R\in\bigcup\limits_{k=k_0}^\infty{\mathcal
B}_k$, with $R\subset R_0$, then $|A\bigcap R|\le \delta |R|$ (in
particular $|A\bigcap R^*|\le \delta |R^*|$).
\end{enumerate}
\end{Lemma}

\paragraph{Proof.} The proof goes via the usual
stopping-time argument. Observe that $R_0$ does not satisfies $2)$
by hypothesis. We select all the rectangles in ${\mathcal B}_{k_0+1}$
for which $2)$ holds.

If $R'\subset R_0$, $R'\in {\mathcal B}_{k_0+1}$ has not been selected
we consider the partition of $R'$ by rectangles of ${\mathcal
B}_{k_0+2}$ and we take those satisfying $2)$.

In this way and by induction, we obtain a maximal sequence
$\{R_d\}$ satisfying $2)$ and $3)$. We call ${\mathcal B_A}=\bigcup
R_d$.

To see why $\{R_d\}$ satisfies $1)$ too, we observe that if $x\in
R_0\setminus \bigcup\{\partial R: R\in{\mathcal B_A}\}$ then there
exists a unique sequence $\{R_k(x)\}$ so that
$$ x\in R_k(x)\in{\mathcal B}_k,\,\, \forall k\ge k_0.
$$
Moreover, since $\mathop{\rm diam} R_k(x)=\delta_k\downarrow 0$ as $k\uparrow \infty$,
we obtain that from the Lebesgue differentiation theorem,
$$ \lim_{k\to\infty}\frac{|A\bigcap R_k(x)|}{|R_k(x)|}=1, \quad a.e.\,
x\in A.
$$
 In particular this says that $A\subset \bigcup R_d$
a.e. and, therefore, $1)$ holds.
\hfill$\square$

Observe that the sequence $\{R_d\}$ in Lemma \ref{lema1}, is given by maximal
rectangles in $R_0$ for which property $2)$ holds. We will call
this sequence the {\it Calder\'{o}n-Zygmund} covering of $A$.
Note that given a rectangle $R_k$ in a Calder\'on-Zygmund
covering of the set $A$, there exists a nested finite family of parabolic
rectangles
$$
{\tilde R}_k^1\supset {\tilde R}_k^2\supset\dots \supset{\tilde R}_k^{r(k)}
\supset R_k,
$$
for which
$$
|{\tilde R}_k^l\cap A|\leq \delta|{\tilde R}_k^l|.
$$
We say that
${\tilde R}_k^{r(k)}$ is the {\it predecessor} of $R_k$ and for
simplicity of notation we will write ${\tilde R}_k^{r(k)}\equiv
\bar R_k$.

 We give some applications of Lemma \ref{lema1} that are
relevant for this work.


\begin{Lemma}\label{lema2}
The maximal operator $${\mathcal M}f(x)=\sup_{x\in
R\in \bigcup {\mathcal B}_k}\frac 1{|R|}\int_R |f|$$ is of weak type
$(1,1)$.
\end{Lemma}

 The proof uses the Vitali covering lemma and the observation that
if $R,R'\in{\mathcal B}=\bigcup {\mathcal B}_k$ and with their interiors
with nonempty intersection, then either $R\subset R'$ or
$R'\subset R$.

\begin{Lemma}\label{lema3}
Fix $R_0\in {\mathcal B}$ and $0<\delta <1$.
Given $A\subset B\subset R_0$ with the properties,
\begin{description}
\item$a)$ $|A|<\delta |R_0|$,
\item$b)$ If $R_d$ is a rectangle in the Calder\'{o}n-Zygmund
decomposition of $A$ (with respect to $R_0$ and $\delta$), then
$R_d^*\subset B$,
\end{description}
we conclude that $|A|\le \delta |B|$
\end{Lemma}

\paragraph{Proof.}
Consider the sequence $\{R_d^*\}$ of predecessors
of the C-Z covering of $A$ with respect to $R_0$ and $\delta$.
Select a subsequence $\{R_{d_l}^*\}$ with disjoint interiors so
that
$$\bigcup R_{d_l}^*=\bigcup R_d^*,
$$
(a maximal subsequence). Then,
$$
|A|=\sum_l|A\bigcap R_{d_l}^*|\le \delta \sum_l
|R_{d_l}^*|=\delta |\bigcup R_{d_l}^*|\le \delta |B|.
$$

\section{Approximations Results}\label{s3}

Denote the open unit cube in $\mathbb
R^{N+1}$ by $Q_0=\{(x,t)\in\mathbb R^{N+1}\,| |x_i|<1,\,
i=1,2,\dots ,N,\, |t|<1\}$ and call $Q_0^+=Q_0\cap \{\, t>0\,\}$ and
$Q_0^-=Q_0\cap \{\,t<0\,\}$. A parabolic rectangle $R$ is then the
image of $Q_0$ through any transformation in $\mathbb R^{N+1}$ of the
form
$$ \phi_\alpha (x,t)=(x_0+\alpha x,t_0+\alpha^2t),\quad
\alpha>0,\quad (x_0,t_0)\in\mathbb R^{N+1}.
$$
Let us consider the following dyadic subdivision:
Given a parabolic rectangle $R$
\begin{description}
\item{$i)$} Divide into $2$ equal parts each spatial side.
\item{$ii)$} Divide into $2^2$ equal parts the temporal side.
\end{description}
\noindent We call this procedure a {\it parabolic subdivision}. It
is obvious that with this procedure we obtain $2^{N+2}$ new
parabolic subrectangles. Associated to this subdivision we have
the corresponding Calder\'on-Zygmund decomposition as in the
previous section.

 Before we continue, some notation is in order. Given a parabolic
rectangle
$$
R=\{(x,t): |x_i-x_i^0|<\rho,\,i=1,\dots N,\,
t_0-\rho^2<t<t_0\}
$$
we define the parabolic boundary of $R$ as
$$
\partial_p R=\{ (x,t) : |x_i-x_i^0|\le\rho,\, t_0-\rho^2=t\}\cup
\{ (x,t) : |x_i-x_i^0|=\rho,\, t_0-\rho^2<t<t_0\}.
$$

\begin{Example} \rm
In the case of the heat equation (the simplest model) we have the
following result by Moser \cite{M}.
 Let $w$ be a measurable function in $Q_0$ such that
 \begin{enumerate}
\item[(i)] $w(x,y)\geq 0$
\item[(ii)] $w_t, \nabla_x w, w\in L^2(Q_0)$
\item[(iii)] $w_t-\Delta w\leq 0$ in ${\mathcal D'}(Q_0)$.
\end{enumerate}
We call such $w$ a {\it positive subsolution} of the
heat equation.

 According to the classical results by Moser in \cite{M} we have
that there exists a positive constant $\gamma $ such that if $w$
is a positive subsolution to the heat equation then
\begin{equation}
\sup_{R'}w^2(x,t)\le \gamma \frac 1{|R|}\int_R w^2(x,t)dx\,dt,
\end{equation}
where
$$ R=\big\{(x,t): |x_i-x_i^0|<\rho,\,i=1,\dots N,\,t_0-\rho^2<t<t_0\big\}
$$
and $R'=\phi_{1/2}(R)$, that is,
$$
R'=\big\{(x,t):|x_i-x_i^0|<\frac{\rho}{2},\,i=1,\dots N,\,t_0
-(\frac{\rho}{2})^2<t<t_0\big\}.
$$
We point out that if $u$ is a solution to the heat equation then
$|\nabla_x u|^2$ is also a positive subsolution to the heat
equation. Hence, by (2),
\begin{equation} \sup_{R'}|\nabla_x
u(x,t)|^2\leq \gamma \frac 1{|R|}\int_R |\nabla_x u(x,t)|^2dx\,dt,
\end{equation}
where $R$ and $R'$ are parabolic rectangles related as
above.
\end{Example}

It is clear that hypothesis $(H1)$ extends the previous Example.

\begin{Definition}\rm
 We say that $u\in L^2(Q_0)$, with
$u_t,\nabla_x u\in L^2(Q_0)$, is a weak solution to the equation
$$
u_t-\mathop{\rm div}_x (a(x,t,\nabla_x u))=0
$$
 if we have
$$
\int_{Q_0} u\psi_tdxdt + \int_{Q_0}\langle a(x,t,\nabla_x u) \nabla
\psi\rangle dx\,dt=0,
$$
 for all $\psi$ in $\mathcal W_0^{1,2}$, the
completion of $\mathcal C_0^\infty(\Omega)$ with respect to the
$L^2$-norm of the function and its gradient.
\end{Definition}


\begin{Lemma}\label{lema35}
Assume that (H1),(H2) hold. Let $u$ be a
weak solution to the equation
\begin{equation}u_t-\mathop{\rm div}{}_x (A(x,t,\nabla_x u)=0,
\end{equation}
 such that for $R$, a parabolic rectangle
contained in $Q_0$, $u$ satisfies
$$ \frac{1}{|R|}\int_R |\nabla_x u|^2dx\,dt\le \mu.
$$
If $v$ is the solution to the problem
\begin{gather*}
 v_t-\mathop{\rm div}{}_x(a_0(\nabla v))=0,\quad (x,t)\in R\\
 v\big|_{\partial_p R}=u,
\end{gather*}
then \begin{enumerate}
\item $\frac{1}{|R|}\int_R |\nabla_x(u-v)|^2 dx\,dt\le \mu\epsilon^2$
\item $\frac{1}{|R|}\int_R |\nabla_x v|^2 dx\,dt\leq\mu(1+\epsilon)^2$
\end{enumerate}
\end{Lemma}


\paragraph{Proof.} Write $R=Q\times (T_1,T_2)$. Observe that
\begin{eqnarray*}
\lefteqn{\gamma \frac{1}{|R|}\int_R |\nabla (u-v)|^2dx\,dt }\\
&\le&\frac{1}{|R|}\int_R \langle a_0(\nabla u)-a_0(\nabla v),
\nabla (u-v)\rangle dx\,dt\\
&\le&\frac{1}{|R|}\int_Q |u(x,T_2)-v(x,T_2)|^2dx\\
&&+\frac{1}{|R|}
\int_R \langle a_0(\nabla u)-a_0(\nabla v), \nabla (u-v)\rangle
dx\,dt\\
&=&\frac{1}{|R|}\int_R \left[(u-v)_t-\left(\mathop{\rm div}{}_x a_0(\nabla u)-
\mathop{\rm div}{}_x a_0(\nabla v)\right)\right](u-v)dx\,dt\\
&=&\frac{1}{|R|}\int_R (u_t-\mathop{\rm div}{}_x a_0(\nabla u))(u-v)dx\,dt\\
&&-\frac{1}{|R|}\int_R (v_t-\mathop{\rm div}{}_x a_0(\nabla v))(u-v)dx\,dt\\
&=&\frac{1}{|R|}\int_R [u_t-\mathop{\rm div}{}_x(a(x,t,\nabla_x u)](u-v)dx\,dt\\
&&+ \frac{1}{|R|}\int_R \langle \nabla_x (u-v),
(a_0(\nabla u)-a(x,t,\nabla_x u))\rangle dx\,dt\\
 &\le&\epsilon \frac{1}{|R|}\Big(\int_R |\nabla_x
(u-v)|^2dx\,dt\Big)^{1/2}\Big(\int_R |\nabla_x
u|^2dx\,dt\Big)^{1/2},
\end{eqnarray*}
where we have used for the last inequality that $u$ is a
solution to equation \eqref{eq:madre}, condition (H2) and Cauchy-Schwarz
inequality.
We thus obtain (1). Now (2) is an easy consequence of (1) since, in
fact,
\begin{eqnarray*}
\lefteqn{\Big(\frac{1}{|R|}\int_R |\nabla_x v|^2
dx\,dt\Big)^{1/2}}\\
&=& \Big(\frac{1}{|R|}\int_R|\nabla_x(u+(v-u))|^2 dx\,dt\Big)^{1/2}\\
&\le&\Big(\frac{1}{|R|}\int_R[ |\nabla_x u|^2 dx\, dt\Big)^{1/2}
+\Big(\frac{1}{|R|}\int_R|\nabla_x (v-u))|^2 dx\,dt\Big)^{1/2}\\
&\le& (1+\epsilon)\mu^{1/2}.
\end{eqnarray*}
\;\hfill$\square$


 We will now introduce the (dyadic) parabolic maximal operator,
defined for $f\in L^1_{\rm loc}$ by
$$M f(x)=\sup_{x\in R}\frac{1}{|R|}\int_R |f(y)|dy,
$$
where the supremum is taken
over all parabolic rectangles $R$ containing $x\in \mathbb R^{N+1}$.
Taking into account that $\mathbb R^{N+1}$ is a homogeneous space
with respect to the seminorm
$$ d(x,t)=\max\{|x_i|,|t|^{1/2},\,i=1,2\dots N\},
$$
more precisely that
$$|\phi_2(R)|\le 2^{N+2}|R|,$$
we have that $M$ is of weak type $(1,1)$ as a consequence of the
usual Besicovich covering lemma. The following result is the main
ingredient in Theorem \ref{th:main} below.

\begin{Lemma}\label{lema36}
Assume that $u$ is a weak solution to
$$u_t-\mathop{\rm div}{}_x( a(x,t,\nabla_x u))=0, \hbox{ in } Q_0.
$$
Then, there exists a constant $C>1$ so that for $0<\delta<1$ fixed, one
can find $\epsilon_0=\epsilon_0(\delta) >0$ such that if $(H2)$
holds with $\epsilon<\epsilon_0$, for all parabolic rectangle
$R_k$ in the Calder\'on-Zygmund $\delta$-covering of
$$\{ (x,t) \in \mathbb R^{N+1}: M(|\nabla_x u|^2)(x,t)> C\mu\},
$$
 then its predecessor satisfies
$$ {\bar R_k}\subset\{ (x,t): M(|\nabla_x u|^2)(x,t)> \mu\}.
$$
In particular, we have
$$ |\{ (x,t) \in \mathbb R^{N+1}: M(|\nabla_x u|^2)(x,t)> C\mu\}|
\le |\{ (x,t): M(|\nabla_x u|^2)(x,t)> \mu\}|.
$$
\end{Lemma}

\paragraph{Proof.} The proof is similar to Lemma 3 in \cite{CP} and
we sketch it here for the sake of completeness.
Since we look for a local result, we can assume that a parabolic
dilation of $R_k$ is contained in $Q_0$, to be more precise, let
us assume, say, that $Q=\phi_4(R_k)\subset Q_0$.
We argue by contradiction. If $R_k$ satisfies the hypothesis,
namely,
$$|R_k\cap\{ x: M(|\nabla_x u|^2)> C\mu\}|> \delta|R_k|
$$
and ${\bar R_k}$ does not satisfy the conclusion, there
exists $(x_0,t_0)\in {\bar R_k}$ for which,
$$ \frac{1}{|R|}\int_R
|\nabla_x u|^2dx\,dt\le \mu,\hbox{ for all parabolic rectangles }\,
R \, \text{with} \, (x_0,t_0)\in R.
$$
We solve the problem
\begin{gather*} v_t-\mathop{\rm div}{}_x(a_0(\nabla v))=0,
\quad (x,t)\in Q_0\\
v\big|_{\partial_p Q_0}=u.
\end{gather*}
Then, according to Lemma \ref{lema35} we get:
\begin{enumerate}
\item $\frac{1}{|Q|}\int_Q |\nabla_x v|^2dx\,dt\le (1+\epsilon)^2\mu$

\item $\frac{1}{|Q|}\int_Q |\nabla_x (u-v)|^2dx\,dt\le \epsilon^2\mu$
\end{enumerate}
 Then the restricted maximal operator,
 $$M^*(|\nabla_x u|^2)(x,t)=\sup\limits_{x\in R,\,R\subset \phi_2(R_k)}
\frac{1}{|R|}\int_R |\nabla_x u|^2dyds,
$$
satisfies
$$ M(|\nabla_x u|^2)(x,t)\le \max\{ M^*(|\nabla_x u|^2)(x,t), 4^{N+2}\mu\}.
$$
Consider $C=\max\{4^{N+2}, 4(1+\epsilon)^2\}$. Then
\begin{eqnarray*}
\lefteqn{|\{(x,t)\in R_k : M(|\nabla_x u|^2)> C\mu\}|}\\
&\le& |\{(x,t)\in R_k : M^*(|\nabla_x u|^2)> C\frac{\mu}{2}\}|\\
&&+|\{(x,t)\in R_k : M^*(|\nabla_x (u-v)|^2)>
C\frac{\mu}{2}\}|\\
&\le&|\{(x,t)\in R_k : M^*(|\nabla_x (u-v)|^2)> C\frac{\mu}{2}\}|.
\end{eqnarray*}
By the (1,1) weak type estimate for the maximal operator we conclude
\begin{eqnarray*}
|\{x\in R_k : M(|\nabla_x u|^2)> C\mu\}|
&\le& |\{x\in R_k : M^*(|\nabla_x (u-v)|^2)> C\frac{\mu}{2}\}|\\
&\le& A\frac{2}{C\mu}\int_{R_k}|\nabla_x (u-v)|^2dx\,dt\\
&\le& A\frac{2}{C\mu}\epsilon^2|R_k|.
\end{eqnarray*}
Taking $\epsilon>0$ so that $A\frac{2}{C\mu}\epsilon^2< \delta$ we reach a
contradiction. \hfill$\square$


\section{Proof of the main result}\label{s5}

As a consequence of the approximation Lemma \ref{lema35} and the behavior of
the level sets of the maximal operator described in Lemma \ref{lema36} we
can formulate the following regularity result.

\begin{Theorem}\label{th37}
Assume that (H1) holds. Given $p>2$ there
exists $\epsilon_0>0$ such that if for some $0<\epsilon
<\epsilon_0$, (H2) holds, then any weak solution to
$$u_t-\mathop{\rm div} (A(x,t)\nabla_x u)=0,
$$
satisfies that $|\nabla_x u|\in L^q_{\rm loc}$, $2<q<p$.
\end{Theorem}

\paragraph{Proof.} For $s>0$, call $\omega(s)=|\{ (x,t):
M(|\nabla_x u|^2(x,t)>s\}|$, the distribution function of the
maximal operator. Take $\delta\in(0,1)$ in such a way that
$C^{q/2}\delta<1$, where $C$ is as in Lemma 3.4. Now, there
exists $\epsilon_0$, such that if $0<\epsilon<\epsilon_0$ and
(H2) holds, then Lemmas 3.4 and 2.3 imply
$$ \omega (C\mu_0)\le \delta \omega(\mu_0).
$$
Hence by recurrence
\begin{equation}
\omega
(C^k\mu_0)\le \delta^k \omega(\mu_0).
\end{equation}
Now $|\nabla_x
u|^q\in L^1$ if, in particular, $ M(|\nabla_x u|^2)\in L^{q/2}$,
and this is equivalent to the convergence of the series
$$
\sum_{k=1}^\infty C^{k(q/2)}\omega(C^k\mu_0).
$$
But, $C^{q/2}\delta<1$ and from estimate (7) we obtain
$$
\sum_{k=1}^\infty C^{k(q/2)}\omega(C^k\mu_0)\le \sum_{k=1}^\infty
(C^{q/2}\delta)^k\omega(\mu_0) <\infty
$$
\;\hfill$\square$

\begin{Corollary} Assume $A(x,t)$ a $N\times N$ matrix which is
continuous in $\Omega$ and such that
$$ \langle A(x,t)\xi,\xi\rangle\ge \gamma |\xi|^2.
$$
Then if $u$ is a weak solution to
$$ u_t-\mathop{\rm div}{}_x(A(x,t)\nabla_x u)=0,
$$
we have $u\in W^{1,p}_{\rm loc}$ for all $1<p<\infty$.
\end{Corollary}

\paragraph{Proof.}
 As reference equation we take the heat equation. The
hypothesis $(H2)$ is obtained easily by an orthogonal change of
variables in $\mathbb{R}^N$ and our assumptions on the continuity of $A$.
\hfill$\square$

By the same method we are able to get estimates for equations that
are close to the {p-Laplacian}. For estimates to
$u_t-\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)$ see \cite{DB}.


\begin{thebibliography}{00} \frenchspacing

\bibitem{CP}
 L. A. Caffarelli and I. Peral,
 \emph{On $W^{1,p}$ Estimates for Elliptic Equations in Divergence Form},
 Comm. Pure App. Math., 51, 1998, 1-21.

\bibitem{DB}
 E. Di Benedetto
 \emph{Degenerate Parabolic Equations},
 Springer-Verlag, 1993

\bibitem{L}
G. M. Lieberman,
\emph{Second Order Parabolic Differential Equations}.
 World Scientific, 1996.

\bibitem{M}
J. Moser, \emph{A Harnack inequality for parabolic
differential equations},
Comm. Pure Appl. Math. 17,
1964, 101--134.

\bibitem{LSU}
 O. A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ural'tseva,
 "Linear and Quasilinear Equations of Parabolic Type"
 Translations of Mathematics Monographs, A.M.S. 1968.

\bibitem{To}
 P. Tolksdorff,
 {\it Regularity for a more general class of quasilinear elliptic
 equations} J. of Diff. Equations 51 (1984), 126-150

 \end{thebibliography}

\noindent\textsc{Ireneo Peral} (e-mail: ireneo.peral@uam.es)\\
\textsc{Fernando Soria} (e-mail: fernando.soria@uam.es)\\[3pt]
Departamento de M\'atematicas U.A.M.\\
28049 Madrid, Spain

\end{document}