\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 204, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/204\hfil Poincar\'e type inequality]
{Poincare inequality and Campanato estimates
for weak solutions of parabolic equations}

\author[J. Aramaki \hfil EJDE-2016/204\hfilneg]
{Junichi Aramaki}

\address{Junichi Aramaki \newline
Division of Science,
Faculty of Science and Engineering,
Tokyo Denki University, \newline
Hatoyama-machi, Saitama 350-0394, Japan}
\email{aramaki@mail.dendai.ac.jp}

\thanks{Submitted December 24, 2015. Published July 28, 2016.}
\subjclass[2010]{35A09, 35K10, 35D35}
\keywords{Poincar\'e type inequality; weak solution; parabolic equation}

\begin{abstract}
 We shall show that the Poincar\'e type inequality holds for the
 weak solution of a parabolic equation. The key is to control the $L^p$
 norm of the first derivative of the weak solution with respect to
 the time variable. The inequality is necessary to get an  estimate in
 the Campanato space $\mathcal{L}^{p,\mu } $ for general parabolic equations.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

Let $B_r(x_0) \subset \mathbb{R}^n $ be a ball with center $x_0$ and the radius
$r>0$. Let $u\in W^{1,p}(B_r(x_0))$ and  define
\[
u_{x_0,r}=\frac{1}{|B_r(x_0)|} \int _{B_r(x_0)}udx.
\]
Then we have the  well known Poincar\'e inequality
\begin{equation}
\int _{B_r(x_0)}|u-u _{x_0,r}|^ p dx
\le C r^p \int _{B_r(x_0)}|\nabla u |^p dx, \label{e1.1}
\end{equation}
where the constant $C$  depends  on $n$ and $p$, but is independent of $r$ and $u$
(see, for example, Chen and Wu \cite[Appendix I Corollary 3.1]{CW}).
This type estimate is used in the Campanato space approach to obtain the
H\"older regularity of weak solutions  for elliptic or quasilinear elliptic
system with degeneracies. For example, for the elliptic system in divergence form,
see Giaquinta \cite{G}, and the regularity of solutions for the quasilinear
elliptic system with degeneracy as $p$-Laplacian   has been treated recently
in Giacomoni et al. \cite{GST1, GST2, GST3} who essentially used the technique
of the Campanato  estimates of the Lieberman \cite[p. 1211]{Li2} and
\cite[p. 45]{G}, using the Poincar\'e inequality \eqref{e1.1}.
See, for example, \cite[Theorem A.1 in the Appendix]{GST3}.

We are interested in the H\"older regularity of weak solutions for
parabolic equations. To apply the Campanato estimate in this case,
it suffices to use the following Poincar\'e type inequality.
To explain precisely,  let  $z_0=(x_0,t_0) \in Q_T= \Omega \times (0,T)$.
If we put
\[
u_{z_0,r} = \frac{1}{|Q_r(z_0)|}\iint _{Q_r(z_0)}u \,dx\,dt
\]
for any cylinder $Q_r(z_0)= B_r(x_0)\times (t_0,t_0+r^2] \subset Q_T$,
the following inequality
\begin{equation}
\iint _{Q_r(z_0)} |u - u _{z_0,r}|^p \,dx\,dt
\le Cr^p \iint _{Q_r(z_0)} |\nabla u |^p \,dx\,dt, \label{e1.2}
\end{equation}
where $\nabla u $ is the gradient of $u$ with respect to the space variable
$x$ does not hold for a general function $u (x,t) \in L^p(0,T;W^{1,p}(Q_r(z_0))$.
This is the fundamental difference from the elliptic theory.
However, when $u$ is a weak solution of a parabolic equation, by using the
 equation and combining the Poincar\'e inequality \eqref{e1.1} with respect to
the  space variable, we shall  show that the inequality  \eqref{e1.2} holds.

Such inequality is used in the Campanato space $\mathcal{L}^{p,\mu }$
 estimates for weak solutions of a parabolic equation. In fact, Yin \cite{Yin}
used the inequality for $p=2$ and conducted us to $\mathcal{L}^{2,\mu }$-estimate
for weak solution of  parabolic equations. We are convinced that we can use
the general inequality \eqref{e1.2} for $\mathcal{L}^{p.\mu }$-estimates for
parabolic equations. It will appear in the future work.

\section{Main results}

In this section, we give the main theorem of this paper.
 Let $\Omega $ be a bounded domain in $\mathbb{R}^n$ with $C^1$ boundary
$\partial \Omega $, and define $Q_T=\Omega \times (0,T)$ with $T>0$.
We consider the  parabolic equation
\begin{equation}
u_t -\sum _{i,j=1}^n \bigl( a_{ij}(x,t) u_{x_j}\bigr) _{x_i}=0 \quad
\text{in } Q_T  \label{e2.1}
\end{equation}
where $a_{ij} \in L^{\infty }(Q_T)$ satisfies the ellipticity condition:
there exist constants $a_0$ and $A_0$ with $0<a_0\le A_0 < \infty $ such that
\[
a_0 |\xi |^2 \le \sum _{i,j=1}^n a_{ij}(x,t) \xi _i \xi _j \le A_0 |\xi |^2
\]
for all $(x,t) \in Q_T$ and $\xi \in \mathbb{R}^n$.

We shall consider the weak solution of \eqref{e2.1}.

\begin{definition} \label{def2.1} \rm
Let $1\le p< \infty $. We say $u \in L^p(0,T; W^{1,p}(\Omega ))$ is a weak
solution of \eqref{e2.1} if the following equality holds.
\begin{equation}
\iint _{Q_T} \Bigl(-u v _t + \sum _{i,j=1}^n a_{ij} u_{x_j}v _{x_i}\Bigr)\,dx\,dt
=0 \label{e2.2}
\end{equation}
for all $v \in W^{1,p'}(0,T; W^{1,p'}_0(\Omega ))$ with $v(x,0)=v(x,T)=0$
 where $p'$ is the conjugate exponent of $p$, i.e., $\frac{1}{p}+ \frac{1}{p'}=1$.
\end{definition}

Here $W^{1,p}(\Omega )$ and $W^{1,p}_0(\Omega )$ denote the standard Sobolev spaces.
 We note that since $W^{1,p'}(0,T;W^{1,p'}_0(\Omega ))
\subset C^0([0,T]; W^{1,p'}_0(\Omega ))$ according to the Sobolev embedding theorem,
the values $v(x,0)$ and $v(x,T)$ are meaningful. We  use some standard notation.
We will denote any point in $Q_T$ by $z=(x,t)$, and for
$z_1=(x_1,t_1)$, $z_2=(x_2,t_2)\in Q_T$, the parabolic distance is defined by
\[
\operatorname{dist} (z_1,z_2)= \max \{|x_1-x_2|,|t_1-t_2|^{1/2}\}.
\]
For $r>0$ and $z_0=(x_0,t_0) \in Q_T$, we write
\[
B_r(x_0)= \{ x\in \mathbb{R}^n ; |x-x_0|<r\}, \quad
Q_r(z_0)= B_r(x_0)\times (t_0,t_0+r^2).
\]
We also write the average of a function $u$ on $Q_r(z_0)$ by
\[
u_{z_0,r}= \frac{1}{|Q_r(z_0)|}\iint _{Q_r(z_0)}u \,dx\,dt,
\]
where $|Q_r(z_0)|= r^2 |B_r(x_0)|$ and $|B_r (x_0)|$ is the volume of $B_r(x_0)$.

We are in a position to state the main theorem.

\begin{theorem} \label{thm2.2}
Let $1<p<\infty $, $u\in L^p(0,T;W^{1,p}(\Omega ))$ be a weak solution of
\eqref{e2.1}, and $z_0=(x_0,t_0) \in Q_T$ with  $Q_{2r}(z_0) \subset Q_T$.
Then there exists a constant $C>$ independent of $r$ and $u$ such that
\[
\iint _{Q_r(z_0)}|u-u _{z_0,r}|^p dz \le Cr^p \iint _{Q_{2r}(z_0)}|\nabla u |^p dz,
\]
where $\nabla u $ denotes the gradient of $u$ with respect to the space variable $x$.
\end{theorem}

\section{Proof of Theorem \ref{thm2.2}}

In this section, we use the technique  based on \cite[Lemmas 3 and 4]{S}.
We assume that  $u \in L^p(0,T;W^{1,p}(\Omega ))$ is a weak solution of
\eqref{e2.1} and $z_0=(x_0,t_0)\in Q_T$ and $Q_{2r}(z_0)\subset Q_T$.
By a translation, we assume that $z_0=(0,0)$, and we write $Q_r=Q_r(0,0)$
and $B_r=B_r(0)$ for the  brevity of notation. Choose a smooth cut-off
function $\sigma (x)$ such that
\begin{equation}
\sigma (x) = \begin{cases}
1 & \text{if } |x |\le r, \\
0& \text{if } |x|\ge 2r,
\end{cases} \label{e3.1}
\end{equation}
$0\le \sigma (x) \le 1$, $|\nabla \sigma (x) |\le 2/r$, and
$\sigma (x) = \sigma (|x |)$ is monotone decreasing with respect to $|x |$.
For $0<s\le t < r^2$, let $\chi _{[s,t]}(\tau )$ be the characteristic function
of $[s,t]$ and define
\[
u _r^{\sigma }
= \frac{\iint _{Q_{2r}} u\sigma dz }{\iint _{Q_{2r}}\sigma dz}
=\frac{ \iint _{Q_{2r}}u\sigma dz}{(2r)^2 \int _{B_{2r}} \sigma\,dx },
\]
and
\[
u_{r,t}^{\sigma }= \frac{ \int _{Q_{2r}(t)} u\sigma\,dx }{\int _{Q_{2r}(t)}\sigma\,dx },
\]
where $Q_r(t) = \{ (x,t) ; |x |<r\}$. Define
\[
\phi (x,\tau )=  \sigma (x) \chi _{[s,t]}(\tau )
\operatorname{sign }(u_{r,t}^{\sigma }-u_{r,s}^{\sigma }) |u_{r,t}^{\sigma }
- u _{r,s}^{\sigma } |^{p-1},
\]
where
\[
\operatorname{sign}(a)= \begin{cases}
 1 &\text{for } a> 0 ,\\
0  &\text{for } a=0,\\
-1 &\text{for } a<0. 
\end{cases}
\]
Using the Steklov averaging to approximate $\phi $ and then taking the limit,
we can use $\phi (x,\tau )$ as a test function of \eqref{e2.2}
(cf.  \cite{S} or \cite{Yin}), so formally
\begin{equation}
\iint _{Q_T} \Bigl(-u \phi _{\tau } + \sum _{i,j=1}^n a_{ij}(x,\tau )u_{x_j}
\phi _{x_i} \Bigr)\,dx\,d\tau =0. \label{e3.2}
\end{equation}
We use the following two lemmas. The first is a  weighted Poincar\'e inequality
with respect to the space variable.

\begin{lemma} \label{lem3.1}
Let the function $\sigma $ be as in \eqref{e3.1}, $u(x) \in W^{1,p}(B_{2r})$
for some $1\le p< \infty $ and define
\[
u_{2r,\sigma } = \frac{\int _{B_{2r}}u\sigma\,dx }{\int _{B_{2r}}\sigma\,dx }.
\]
Then we have
\[
\int _{B_{2r}}|u-u_{2r,\sigma } |^p \sigma\,dx
\le C(n,p)r^p \int _{B_{2r}}|\nabla u |^p \sigma\,dx .
\]
\end{lemma}

For a proof of the above lemma, see Lieberman \cite[Lemma 6.12]{Li}.

\begin{lemma} \label{lem3.2}
Let $1\le p < \infty $ and $u \in L^p(Q_r(z_0))$. Then we have
\[
\iint _{Q_r(z_0)} |u - u_{z_0,r}|^p dz \le 2^p \iint _{Q_r(z_0)}|u-L |^pdz
\]
for any $L \in \mathbb{R}$.
\end{lemma}

\begin{proof}
By the triangle inequality, for any $L \in \mathbb{R}$, we have
\begin{align*}
&\Big( \iint _{Q_{r}(z_0)}|u-u _{z_0,r}|^p dz \Big)^{1/p}  \\
&\le \Big( \iint _{Q_{r}(z_0)}|u-L |^p dz \Big)^{1/p}
+ \Big( \iint _{Q_{r}(z_0)}|u _{z_0,r}-L |^p dz \Big)^{1/p}.
\end{align*}
By the definition of $u_{z_0,r}$ and the H\"older inequality,
\begin{align*}
&\iint _{Q_r(z_0)}|u_{z_0,r}-L|^p dz = |u_{z_0,r}-L|^p |Q_r(z_0)|\\
&= \Big|\frac{1}{|Q_r(z_0)|} \iint _{Q_r(z_0)}(u-L)dz \Big|^p |Q_r(z_0)|\\
&\le  |Q_r(z_0)|^{1-p} \Big[\Bigl(  \iint _{Q_r(z_0)}|u-L|^pdz
\Bigr)^{1/p} |Q_r(z_0)|^{1/p'} \Big]  ^p \\
&= |Q_r(z_0)|^{1-p+p/p'}\iint _{Q_r(z_0)} |u-L |^p dz \\
&= \iint _{Q_r(z_0)} |u-L |^p dz.
\end{align*}
Here we used $1-p+p/p'=0$. Thus we get the conclusion.
\end{proof}

We shall estimate each term of \eqref{e3.2}. We have
\begin{align*}
& \iint _{Q_T}-u \phi _{\tau }\,dx\,d\tau \\
& = \iint _{Q_T}u_{\tau } (x,\tau ) \sigma (x)  \chi _{[s,t]}(\tau )\mathrm{sign}(u_{r,t}^{\sigma }-u_{r,s}^{\sigma }) |u_{r,t}^{\sigma }-u_{r,s}^{\sigma }|^{p-1}dxd\tau \\
& =  \int _s^t \int _{\Omega } u_{\tau } (x,\tau ) \sigma (x) dxd\tau \, \mathrm{sign}(u_{r,t}^{\sigma }-u_{r,s}^{\sigma }) |u_{r,t}^{\sigma }-u _{r,s}^{\sigma }|^{p-1} \\
& = \Big[ \int _{Q_{2r}(t) }u(x,t)  \sigma\,dx - \int _{Q_{2r}(s) }u(x,s)
 \sigma\,dx \Big]\\
&\quad \times \operatorname{sign}(u_{r,t}^{\sigma }-u_{r,s}^{\sigma })
 |u_{r,t}^{\sigma }-u _{r,s}^{\sigma }|^{p-1} \\
& = \int _{\Omega }\sigma\,dx \,(u_{r,t}^{\sigma }- u_{r,s}^{\sigma })
 \mathrm{sign} (u_{r,t}^{\sigma }-u_{r,s}^{\sigma }) |u_{r,t}^{\sigma }
 - u_{r,s}^{\sigma }|^{p-1} \\
& = \int _{\Omega } \sigma\,dx\, |u_{r,t}^{\sigma }-u _{r,s}^{\sigma }|^{p} .
\end{align*}
Thus we have
\begin{equation}
\begin{aligned}
\iint _{Q_T} -u \phi _{\tau } \,dx\,d\tau
&= \int _{\Omega }\sigma (x) dx |u_{r,t}^{\sigma } -u _{r,s}^{\sigma }|^p \\
& \ge \int _{B_r} \sigma (x) dx |u_{r,t}^{\sigma } -u _{r,s}^{\sigma }|^p\\
& \ge c_0 r^n |u_{r,t}^{\sigma } -u _{r,s}^{\sigma }|^p 
\end{aligned} \label{e3.3}
\end{equation}
for some $c_0>0$.

On the other hand, we put
\[
I=-\sum _{i,j=1}^n \iint _{Q_T}a_{ij}(x,\tau ) u_{x_j}(x,\tau )
\phi _{x_i}(x,\tau )dxd\tau .
\]
Since $a_{ij} \in L^{\infty }(Q_T)$, we have
\[
|I |\le C \int _s^t \int _{B_{2r}} |\nabla u(x,\tau ) |
|\nabla \sigma (x) ||u_{r,t}^{\sigma }-u _{r,s}^{\sigma }|^{p-1} \,dx\,d\tau .
\]
If we write $|\nabla \sigma (x) |= |\nabla \sigma (x) |^{1/p}
|\nabla \sigma (x) |^{1/p'}$, and apply the H\"older inequality, we have
\begin{align*}
|I |&\le \Big( \int _s^t \int _{B_{2r}}|\nabla u(x,\tau )^p
 |\nabla \sigma (x)|\,dx\,d\tau \Big)^{1/p} \\
&\quad \times \Big( \int _s^t \int _{B_{2r}}|\nabla \sigma (x)  |u_{r,t}^{\sigma }
-u_{r,s}^{\sigma } |^p \,dx\,d\tau \Big)^{1/p'} .
\end{align*}
Here we use the Young inequality (cf. Evans \cite[p. 706]{E}):
\[
ab \le \delta a^{p'}+ (\delta p')^{-p/p'}p^{-1} b^p
\]
for any $a,b\ge 0$ and any $\delta >0$. Using this inequality,  we have
\begin{align*}
|I|&\le C\delta \int _s^t \int _{B_{2r}}|\nabla \sigma (x) |
 |u_{r,t}^{\sigma }-u_{r,s}^{\sigma }|^p \,dx\,d\tau
\\
&\quad + C(p')^{-p/p'}p^{-1} \delta ^{-p/p'}
\int _s^t \int _{B_{2r}}|\nabla u(x,\tau )|^p |\nabla \sigma (x) |\,dx\,d\tau .
\end{align*}
If we put $\varepsilon = r\delta $, using $t-s \le r^2$, we have
\begin{equation}
\begin{aligned}
|I|&\le 2C \varepsilon r^{-2} (t-s) |B_{2r}||u_{r,t}^{\sigma }-u_{r,s}^{\sigma }|^p \\
& \quad + 2C (p')^{-p/p'}p^{-1} \varepsilon ^{-p/p'}r^{-1+ p/p'}
 \int _s^t \int _{B{2r}}|\nabla u(x,\tau )|^p \,dx\, d\tau \\
&\le C' \varepsilon r^{n}  |u_{r,t}^{\sigma }-u_{r,s}^{\sigma }|^p
 + C(p,\varepsilon ) r^{p-2} \int _s^t \int _{B{2r}}|\nabla u(x,\tau )|^p \,dx\, d\tau
\end{aligned}\label{e3.4}
\end{equation}
From \eqref{e3.2}, \eqref{e3.3} and \eqref{e3.4}, if we choose $\varepsilon >0$
small enough, we have
\begin{equation}
|u_{r,t}^{\sigma }-u_{r,s}^{\sigma }|^p
\le C r^{-n+p-2}\int _s^t \int _{B_{2r}}|\nabla u(x,\tau )|^p \,dx \,d\tau .
\label{e3.5}
\end{equation}
By Lemma \ref{lem3.2} and an elementary inequality, we have
\begin{align*}
\iint _{Q_r}|u-u_{z_0,r}|^p \,dx\,dt
&\le 2^p \iint _{Q_r}|u- u_{r}^{\sigma }|^p \,dx\,dt \\
&\le 4^p \iint _{Q_r}|u- u_{r,t}^{\sigma }|^p \,dx\,dt
+ 4^p \iint _{Q_r}|u_{r,t}^{\sigma }- u_r^{\sigma } |^p \,dx\,dt .
\end{align*}
Since $\sigma \ge 0$ and $\sigma (x) =1$ on $B_r$, we have
\[
\iint _{Q_r}|u- u_{r,t}^{\sigma }|^p \,dx\,dt
= \int _0^{r^2} \int _{B_r}|u- u_{r,t}^{\sigma }|^p \,dx\,dt
\le  \iint _{Q_{2r}}|u- u_{r,t}^{\sigma }|^p\sigma  \,dx\,dt
\]
According to Lemma \ref{lem3.1},
\begin{align*}
\int _{B_r} |u(x,t) - u_{r,t}^{\sigma }|^p dx
&= \int _{B_{2r}} \Big|u(x,t) - \frac{\int _{Q_{2r}(t)}u(y,t)
 \sigma (y)dy}{\int _{Q_{2r}(t) }\sigma (y) dy} \Big|^p \sigma (x) dx \\
&\le C(n,p) r^p \int _{B_{2r}}|\nabla u(x,t) |^p \sigma (x)dx .
\end{align*}
Hence we see that
\begin{equation}
\iint _{Q_r} |u-u_{r,t}^{\sigma }|^p \,dx\,dt
\le Cr^p \iint _{Q_{2r}}|\nabla u (x,t)|^p \,dx\,dt. \label{e3.7)}
\end{equation}
Since
\begin{align*}
u _r^{\sigma }
&= \frac{\iint _{Q_{2r}} u\sigma dz }{(2r)^2 \int _{B_{2r}} \sigma\,dx} \\
&= \frac{\int _0^{(2r)^2} \int _{B_{2r}} u(x,s)\sigma (x) \,dx\,ds }{(2r)^2 
 \int _{B_{2r}}\sigma\,dx } \\
&= \frac{ \int _0^{(2r)^2} \int _{Q_{2r}(s)} u(x,s)\sigma (x) \,dx\,ds }{(2r)^2 \int _{Q_{2r}(s) }\sigma\,dx } \\
&= \frac{1}{(2r)^2} \int _0^{(2r)^2}u_{r,s}^{\sigma } ds \\
&= \frac{1}{|Q_{2r}|}\iint _{Q_{2r}}u_{r,s}^{\sigma }\,dy\,ds,
\end{align*}
we have
\[
u_{r,t}^{\sigma } -u_r^{\sigma }
= \frac{1}{|Q_{2r}|} \iint _{Q_{2r}}(u_{r,t}^{\sigma }
- u _{r,s}^{\sigma })\,dy\,ds.
\]
Therefore, by the H\"older inequality, we have
\begin{align*}
|u_{r,t}^{\sigma } -u_r^{\sigma }|^p
&= |Q_{2r}|^{-p} \Big|\iint _{Q_{2r}}(u_{r,t}^{\sigma }- u _{r,s}^{\sigma })\,dy\, ds
 \Big|^p \\
&\le  |Q_{2r}|^{-p} \Big[\Bigl( \iint _{Q_{2r}}|u_{r,t}^{\sigma }
 - u _{r,s}^{\sigma }|^p \,dy\,ds \Bigr)^{1/p} |Q_{2r}|^{1/p'} \Big]^p \\
&= |Q_{2r}|^{-p+p/p'}  \iint _{Q_{2r}}|u_{r,t}^{\sigma }- u _{r,s}^{\sigma }
|^p \,dy\,ds
  \\
&= |Q_{2r}|^{-1} \iint _{Q_{2r}} |u_{r,t}^{\sigma }- u_{r,s}^{\sigma } |^p \,dy\,ds.
\end{align*}
Thus using \eqref{e3.5}, we have
\begin{align*}
&\iint _{Q_r}|u_{r,t}^{\sigma }- u_{r}^{\sigma }|^p \,dx\,dt\\
&\le \iint _{Q_r} |Q_{2r}|^{-1}
 \iint _{Q_{2r}} |u_{r,t}^{\sigma }-u_{r,s}^{\sigma }|^p \,dy\,ds \,dx\,dt \\
&\le \iint _{Q_r} |Q_{2r}|^{-1} \iint _{Q_{2r}} r^{-n+p-2}
\int _s^t \int _{B_{2r}} |\nabla u(x',\tau )|^p dx'd\tau \,dy\,ds \,dx\,dt  \\
&\le |Q_{2r}|^{-1}  \iint _{Q_r}
 \iint _{Q_{2r}} r^{-n+p-2} \int _s^t
 \int _{B_{2r}} |\nabla u(x',\tau )|^p dx'd\tau \,dy\,ds \,dx\,dt  \\
&\le |Q_{2r}|^{-1} |Q_r||Q_{2r}|r^{-n+p-2}
 \int _s^t \int _{B_{2r}} |\nabla u(x',\tau )|^p dx'd\tau \\
&= Cr^p \int _{Q_{2r}} |\nabla u(x',\tau )|^p dx'd\tau .
\end{align*}
Here we used the fact that $|Q_r|= \omega _n r^{n+2}$ where $\omega _n$
is the volume of the unit sphere in $\mathbb{R}^n$. So we get
\[
\iint _{Q_r} |u-u _{z_0,r}|^p \,dx\,dt
\le C r^p \iint _{Q_{2r}}|\nabla u(x,t) |^p \,dx\,dt.
\]
This completes the proof of Theorem \ref{thm2.2}.


\subsection*{Acknowledgments}
We would like to thank the anonymous referee for his or her very kind
advice about an early version of this article.


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\end{thebibliography}

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