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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 86, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/86\hfil Regularity estimates in Besov spaces]
{Regularity estimates in Besov spaces for  initial-value problems
of general parabolic equations}

\author[M. Xu\hfil EJDE-2009/86\hfilneg]
{Ming Xu}

\address{Ming Xu \newline
Department of mathematics, Jinan University, Guangzhou
510632, China}
\email{stxmin@163.com}

\thanks{Submitted December 2, 2008. Published July 10, 2009.}
\subjclass[2000]{35K30, 42B35}
\keywords{Regularity; adapted Besov spaces}

\begin{abstract}
 In this paper we give regularity estimates
 for solutions to  initial-value problem of general
 parabolic equations with data in adapted Besov spaces
 characterized by heat kernels.
\end{abstract}

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

\section{Introduction}

The purpose of this paper is to consider the regularity estimates for
solutions of  parabolic equations with data in adapted Besov
spaces. The general parabolic equation is defined by
\begin{gather}
\partial_t u+Lu=f,\quad t\in [0,T]\\
u|_{t=0}=u_0,
\end{gather}
where $u_0\in \mathcal{D}(\mathbb{R}^n)$,
$f\in C([0,T];\mathcal{D}(\mathbb{R}^n))(T>0)$, the elliptic operator $L$
is defined by
\begin{equation} \label{eq1.1}
L=-\mathop{\rm div} A\nabla
\end{equation}
where $A=(a_{i,j})_{n\times n}$ is a matrix of complex-valued,
measurable functions satisfying the elliptic conditions
\begin{enumerate}
\item $\lambda|\xi|^2\leq
\mathop{\rm Re}\sum_{i,j}a_{i,j}(x)\xi_i\bar{\xi}_j
=\mathop{\rm Re}\langle A\xi,\xi\rangle$;
\item $|\langle A\xi,\eta \rangle|\leq \Lambda|\xi||\eta|$,
\end{enumerate}
where $0<\lambda\leq\Lambda<\infty$ and $\xi,\eta\in \mathbb{C}^n$.

In the classical theory, if $L$ denotes the Laplace operator, the
regularity estimates of the solution of the heat equations with data
in the conventional Besov spaces have been given by using Fourier
analysis methods(see\cite{L}). Naturally there is one question, if
$L$ is defined by \eqref{eq1.1}, what about the regularity estimates of the
solution of the corresponding parabolic equations in Besov spaces?
The purpose of the paper is to answer such question. The main
difficulty here is that the Fourier analysis methods can't be used,
since the heat kernel of $e^{-tL}$ for the divergence elliptic
operator $L$ is not convolutional now. As we know, the regularity
estimates of the solution of the parabolic equations depend on the
bound of the heat semigroup $e^{-tL}$. In \cite{AT}, Auscher and
Tchamitchian studied bounds for the heat kernel $p_t(x,y)$ of the
semigroup $e^{-tL}(t>0)$ for such divergence elliptic operator $L$
as given in \eqref{eq1.1}. These will be described in the next section in
detail. In this paper we firstly define a kind of adapted Besov
space associated to the parabolic equations by using the heat
kernel, then we apply some other harmonic analysis methods in place
of the Fourier transform methods to get our regularity estimates of
the solution. From this point of view, our results are new.

The paper is organized as follows: in section 2, we give two
assumptions and the definition of adapted Besov spaces, then give
the main theorem; in section 3, we give the proof of the main
theorem.

Through the paper, the constant ``C'' and ``c'' may be different
somewhere, but it is not essential.

\section{Two assumptions and main theorem}

In this paper, we use two assumptions

\noindent\textbf{Assumption (a)}
 The holomorphic semigroup $e^{-zL}$,
$|\mathop{\rm arg}(z)|<\pi/2-\theta$ is represented by the kernel
$a_z(x,y)$ which satisfies the so-called Poisson bound; that is,
for all $\nu>\theta$,
$$
|a_z(x,y)|\leq c_\nu h_{|z|}(x,y)
$$
for all $x,y\in\mathbb{R}^n$ and $|\mathop{\rm arg}(z)|<\pi/2-\nu$, where
$h_t(x,y)=\frac{s(|x-y|^2/t)}{|B(x, t^{1/2})|}$, in which $B(x,
t^{1/2})$ denotes any ball with center $x\in\mathbb{R}^n$ and radius
$t^{1/2}>0$, and $s$ is a positive, bounded, decreasing function
satisfying
$$
\lim_{r\rightarrow\infty}r^{n+\kappa}s(r^2)=0,
$$
for some $\kappa>0$. Denote $P_t=e^{-tL}$ and its kernel is
$p_t(x,y)$, and also assume that $p_t(x,y)$ satisfies the
H\"older continuity estimates
\begin{equation} \label{eq2.1}
|p_t(x+h,y)-p_t(x,y)|+|p_t(y,x+h)-p_t(y,x)|\leq
c\big(\frac{|h|}{t^{1/2}+|x-y|}\big)^\mu h_t(x,y),
\end{equation}
where $0<\mu\leq 1$ and $|h|\leq\frac{1}{2}(t^{1/2}+|x-y|)$.

\begin{remark} \label{rmk2.1} \rm
Auscher and  Tchamitchian \cite{AT}, found that if $A$ was
real, symmetric valued, the heat kernel $p_t(x,y)$ of the semigroup
$e^{-tL}(t>0)$ satisfied the upper Gaussian bounds and the
H\"older continuity estimates. More details about the bound
of $e^{-tL}$ for elliptic operators can be found in \cite{AT}. In
fact here some other elliptic operators have the above similar
properties, such as Schr\"odinger operators or degenerate
elliptic operators, related details can be found in \cite{Sh} or
\cite{CR}.
\end{remark}


 In the following, we assume that
$p_t(x,y)$ is the kernel of $P_t=e^{-tL}$ which can be seen as an
approximation to identity. Set $Q_t=tL e^{-tL}$,then it can be
proved that its kernel $v_t(x,y)$ also satisfies the Poisson bound
and H\"older continuity estimates by using the Cauchy formula
with the previous assumption.

Moreover it is easy to see that $e^{-tL}(1)=1$ and $Le^{-tL}(1)=0$.
We also notice that the adjoint operator $L^*$ has the similar
properties to $L$. Set $\tilde{Q}_t=\sqrt{tL}e^{-tL}$ ,which kernel
also satisfies the Poisson bound and H\"older continuity
estimates due to the representation of $L^{1/2}$. Here it's obvious
that $\tilde{Q}_t(1)=0$ and $\tilde{Q}^*_t(1)=0$, where
$\tilde{Q}_t^*$ is the adjoint operator of $\tilde{Q}_t$. Here for
simplicity, we assume that $L$ is a self-adjoint operator.

\noindent\textbf{Assumption (b)}
 The operator $L$ has a bounded
$H_\infty$-calculus in $L^2(\mathbb{R}^n)$. About the definition and
related properties of $H_\infty$-calculus, readers can refer to
\cite{Mc}\cite{AT}.

\begin{remark} \label{rmk2.2} \rm
By the previous assumptions ,the following equality holds in the
sense of the norm of $W^{1,p}(\mathbb{R}^n)$ for $1<p<\infty$
\begin{equation} \label{eq2.2}
L^{1/2}f=\frac{1}{\pi^{-1/2}}\int_0^\infty e^{-tL}Lf\frac{dt}{\sqrt{t}}.
\end{equation}
Moreover if matrix $A$ is real, symmetric valued, the above equality
holds in the norm of $W^{1,p}$. Here we mention that following
representation formula also holds in the sense of the norm of $L^p$
for $1<p<\infty$ on basis of previous two assumptions,
\begin{equation} \label{eq2.3}
I=c\int_0^\infty Q_tQ_t\frac{dt}{t}.
\end{equation}
Related details can be found in \cite{Mc}\cite{AT}.
\end{remark}

Next we give the adapted Besov space for the corresponding parabolic
equations.

\begin{definition} \label{def2.1} \rm
Let $1< p$, $q<\infty$ and $\alpha\in [-\mu,\mu]$. For
$u\in L^p(\mathbb{R}^n)$
$$
\|u\|_{B_p^{\alpha,q}}=\|u\|_{L^p}+\Big\{\int_0^1 s^{-\frac{1}{2}\alpha
q}\|Q_{s}(u)\|_p^q\frac{ds}{s}\Big\}^{1/q}<\infty.
$$
For $T>0$, $1\leq \rho\leq\infty$,
$$
\|u\|_{\tilde{L}_T^\rho(B_p^{\alpha,q} )}
=\|u\|_{L^\rho_T(L^p) }
+\Big\{\int_0^1 s^{-\frac{1}{2}\alpha q}\|Q_{s}(u)\|_{L^\rho_T(L^p)
}^q\frac{ds}{s}\Big\}^{1/q}<\infty,
$$
where for any $u\in L^\rho_T(L^p)(\mathbb{R}^n)$,
$$
\|u\|_{L^\rho_T(L^p) }=\Big\{\int_0^T \|u\|_{L^p}^\rho
dt\Big\}^{1/\rho}<\infty.
$$
\end{definition}

\begin{remark} \label{rmk2.3} \rm
Note that for any $u\in L^\rho_T(W^{k,p})(\mathbb{R}^n)$, similar
definition can be given in the following form
$$
\|u\|_{L^\rho_T(W^{k,p}) }=\{\int_0^T \|u\|_{W^{k,p}}^\rho
dt\}^{1/\rho}<\infty,
$$
where $W^{k,p}(\mathbb{R}^n)$ is Sobolev space and $k\in
\mathbb{Z}^+$. Here for $p,q,\rho=\infty$, definitions for the above
spaces can be given conventionally. Moreover by the coordinate
transform, we have
$$
\|u\|_{B_p^{\alpha,q}}\sim\|u\|_{L^p}+\Big\{\int_0^2
s^{-\frac{1}{2}\alpha q}\|Q_{s}(u)\|_p^q\frac{ds}{s}\Big\}^{1/q}<\infty.
$$
\end{remark}

Now we give the main theorem in the paper.

\begin{theorem} \label{tm2.1}
Suppose that $L$ satisfies assumptions (a) and (b). Let
$\alpha\in(-\mu,\mu)$ $(0<\mu\leq 1)$ and
$1< p,q,\rho<\infty$. Let $u_0\in B_p^{\alpha,q}$ and
$f\in \tilde{L}_T^\rho(B_p^{\alpha-2+\frac{2}{\rho},q})\cap
{L}_T^\infty(L^p)$. Then the initial-value problem of the parabolic
equation has a solution
$u \in \tilde{L}_T^\rho (B_p^{\alpha+\frac{2}{\rho},q})
\cap {L}_T^1(W^{2,p})$ and there
exists a constant $C>0$ depending only on $n$ and such that
$$
\|u\|_{\tilde{L}_T^\rho(B_p^{\alpha+\frac{2}{\rho_1},q})}\leq
C\big((1+T^{1/\rho_1})\|u_0\|_{B_p^{\alpha,q}}+
(1+T^{1+\frac{1}{\rho_1}-\frac{1}{\rho}})
\|f\|_{\tilde{L}_T^\rho(B_p^{\alpha-2+\frac{2}{\rho},q})}\big),
$$
where $\rho_1\in(\rho,+\infty)$ satisfying
$|\alpha+\frac{2}{\rho_1}|\leq \mu$ and
$|\alpha-2+\frac{2}{\rho}|\leq \mu$.
\end{theorem}

\section{The proof of main theorem}

Before we prove the main theorem, we need the following lemmas.

\begin{lemma} \label{lem3.1}
Let $k_{s,t}(x,y)$ be the kernel of $P_t(Q_s)$. If $t\geq s$, there
exists a constant $c>0$ such that
\begin{equation} \label{eq}
|k_{s,t}(x,y)|\leq c\big(\sqrt{s/t}\big)^\mu h_s(x,y).
\end{equation}
If $t\leq s$, there exists a constant $c>0$ such that
\begin{equation} \label{eq3.2}
|k_{s,t}(x,y)|\leq ch_t(x,y).
\end{equation}
\end{lemma}

\begin{proof}
The proof is similar to \cite[Lemma B.1]{FJ}, which  uses
the vanishing conditions of the kernel of $Q_s$ and H\"older
continuity conditions.
\end{proof}

\begin{remark} \label{rmk3.1} \rm
Here the kernels of $Q_t(Q_s)$ and $Q_t(\tilde{Q}_s)$ have better
properties due to the vanishing moment conditions; that is, let
$K_{s,t}(x,y)$ be the kernel of $Q_t(Q_s)$, then if $t\geq s$, there
exists a constant $c>0$ such that
\begin{equation} \label{eq3.3}
|K_{s,t}(x,y)|\leq c\big(\sqrt{s/t}\big)^\mu h_s(x,y);
\end{equation}
Also if $t\leq s$, there exists a constant $c>0$ such that
\begin{equation} \label{eq3.4}
|K_{s,t}(x,y)|\leq c\big(\sqrt{t/s}\big)^\mu h_t(x,y).
\end{equation}
\end{remark}

\begin{lemma} \label{lem3.2}
Let $1< p$, $q<\infty$ and $\alpha\in (-\mu,\mu)$. For
$u\in L^p(\mathbb{R}^n)$,
$$
\|u\|_{B_p^{\alpha,q}}\sim \|u\|_{L^p}+\Big\{\int_0^1
s^{-\frac{1}{2}\alpha
q}\|\tilde{Q}_{s}(u)\|_p^q\frac{ds}{s}\Big\}^{1/q}<\infty.
$$
\end{lemma}

\begin{proof}
Firstly we prove there exists a constant $c>0$ such that
$$
\|u\|_{B_p^{\alpha,q}}\leq c(\|u\|_{L^p}+\Big\{\int_0^1
s^{-\frac{1}{2}\alpha
q}\|\tilde{Q}_{s}(u)\|_p^q\frac{ds}{s}\Big\}^{1/q})<\infty.
$$
Here we need to verify that
\begin{equation} \label{eq3.5}
\Big\{\int_0^1 s^{-\frac{1}{2}\alpha
q}\|Q_{s}(u)\|_p^q\frac{ds}{s}\Big\}^{1/q}
\leq c\Big(\|u\|_{L^p}+\Big\{\int_0^1 s^{-\frac{1}{2}\alpha
q}\|\tilde{Q}_{s}(u)\|_p^q\frac{ds}{s}\Big\}^{1/q}\Big).
\end{equation}
Note that the following identity holds in $L^p$ for $1<p<\infty$,
$$
I=\int_0^\infty Q_t\frac{dt}{t}=c\int_0^\infty Q_{2t}\frac{dt}{t}.
$$
Also note that $Q_{2t}=\tilde{Q}_t\tilde{Q}_t$, then
$$
I= c\int_0^\infty \tilde{Q}_{t}\tilde{Q}_t\frac{dt}{t}.
$$
Next
\begin{align*}
\Big\{\int_0^1 s^{-\frac{1}{2}\alpha
q}\|Q_{s}(u)\|_p^q\frac{ds}{s}\Big\}^{1/q}
&=\Big\{\int_0^1 s^{-\frac{1}{2}\alpha q}\|\int_0^\infty
Q_s\tilde{Q}_t\tilde{Q}_t(u)\frac{dt}{t}\|_p^q\frac{ds}{s}\Big\}^{1/q}\\
&=\Big\{\int_0^1 s^{-\frac{1}{2}\alpha q}\|\int_0^s
Q_s\tilde{Q}_t\tilde{Q}_t(u)\frac{dt}{t}\|_p^q\frac{ds}{s}\Big\}^{1/q}\\
&\quad +\Big\{\int_0^1 s^{-\frac{1}{2}\alpha q}\|\int_s^\infty
Q_s\tilde{Q}_t\tilde{Q}_t(u)\frac{dt}{t}\|_p^q\frac{ds}{s}\Big\}^{1/q}\\
&= I+II.
\end{align*}
Using Lemma 3.1, remark 3.1 and standard harmonic analysis
technique, we can obtain
$$
I\leq c\Big\{\int_0^1\big(\int_t^1
s^{-\frac{1}{2}\alpha}(\sqrt{\frac{t}{s}})^\mu
\|\tilde{Q}_{t}(u)\|_p\frac{ds}{s}\Big)^q\frac{dt}{t}\Big\}^{1/q}
\leq c\Big\{\int_0^1 t^{-\frac{1}{2}\alpha
q}\|\tilde{Q}_{t}(u)\|_p^q\frac{dt}{t}\Big\}^{1/q} .
$$
Similarly we can also get
\begin{align*}
II&\leq c\Big\{\int_0^1\Big(\int_0^t
s^{-\frac{1}{2}\alpha}(\sqrt{s/t})^\mu
\|\tilde{Q}_{t}(u)\|_p\frac{ds}{s}\Big)^q\frac{dt}{t}\Big\}^{1/q} \\
&\leq c\Big\{\int_0^1 t^{-\frac{1}{2}\alpha
q}\|\tilde{Q}_{t}(u)\|_p^q\frac{dt}{t}\Big\}^{1/q} .
\end{align*}
Thus we have end the proof of \eqref{eq3.5}. The proof of the reverse
inequality of \eqref{eq3.5} depends on \eqref{eq2.3} and Lemma 3.1, which proof is
similar to the above one. We omit the details, the proof is end.
\end{proof}

Now we present the proof of main theorem.


\begin{proof}[Proof of Theorem 2.1]
Similar to the classical theory of
parabolic equations, by using the contraction mapping theorem(Here
readers can refer to the Chapter 4 in \cite{P}), for $u_0\in L^p$
and $f\in {L}^\infty(L^p)$ for $1<p<\infty$, there exists a solution
$u\in {L}_T^1(W^{2,p})$ for \eqref{eq1.1} and
\begin{equation} \label{eq3.6}
u(t,x)=e^{-tL}(u_0)(x)+\int_0^t e^{(\tau-t)L}(f)(\tau,x)d\tau.
\end{equation}
More precisely, by using the bound of $e^{-tL}$ and \eqref{eq3.6}, for
$1<\rho<\rho_1<\infty$, we have
\begin{align*}
\|u\|_{L^p}
&\leq\|e^{-tL}(u_0)(\cdot)\|_{L^p}+\int_0^t
\|e^{(\tau-t)L}(f)(\tau,\cdot)\|_{L^p}d\tau\\
&\leq c(\|u_0\|_{L^p}+T^{1-1/\rho}\|f\|_{{L}^\rho_T(L^p)}),
\end{align*}
then
\begin{equation} \label{eq3.7}
\|u\|_{{L}_T^{\rho_1}(L^p)}\leq
c(T^{1/\rho_1}\|u_0\|_{L^p}+T^{1+\frac{1}{\rho_1}-\frac{1}{\rho}}
\|f\|_{{L}^\rho_T(L^p)}).
\end{equation}
Now apply \eqref{eq3.6} to $Q_{2s}(u)$, then
$$
Q_{2s}(u)=e^{-tL}(Q_{2s}(u_0))+\int_0^t e^{(\tau-t)L}(Q_{2s}(f))
d\tau.
$$
Since $Q_{2s}=\tilde{Q}_s\tilde{Q}_s$, then for $1<p<\infty$
$$
\|Q_{2s}(u)\|_{L^p}\leq
c(\|e^{-tL}\tilde{Q}_s\tilde{Q}_s(u_0)\|_{L^p}+\int_0^t
\|e^{(\tau-t)L}\tilde{Q}_s\tilde{Q}_{s}(f)\|_{L^p} d\tau).
$$
Next by using Lemma 3.1, we have
\begin{align*}
\|Q_{2s}(u)\|_{L_T^{\rho_1}(L^p)}
&\leq c\Big(\Big\{\int_0^T
\min\big(\big(\sqrt{s/t}\big)^\mu,1\big)^{\rho_1}dt\Big\}^{1/\rho_1}\|
\tilde{Q}_s(u_0)\|_{L^p}\\
&\quad +\|\int_0^t
\min\big(\big(\sqrt{\frac{s}{t-\tau}}\big)^\mu,1\big)
\|\tilde{Q}_{s}(f)\|_{L^p}d\tau\|_{L^{\rho_1}_T}
 \Big).
\end{align*}
For the second term of the above inequality, we use the young
inequality, then by some simple calculus,we have
$$
\|Q_{2s}(u)\|_{L_T^{\rho_1}(L^p)}\leq c( s^{\frac{1}{2\rho_1}}\|
\tilde{Q}_s(u_0)\|_{L^p} +s^{\frac{1}{2\rho_2}}
\|\tilde{Q}_{s}(f)\|_{L_T^{\rho}(L^p)}
 ),
$$
where $\frac{1}{\rho_2}=1+\frac{1}{\rho_1}-\frac{1}{\rho}$. By using
Definition 2.2 and Lemma 3.2 and  \eqref{eq3.7},
we obtain the inequality
$$
\|u\|_{\tilde{L}_T^\rho(B_p^{\alpha+\frac{2}{\rho_1},q})}\leq
C\big((1+T^{1/\rho_1})\|u_0\|_{B_p^{\alpha,q}}+
(1+T^{1+\frac{1}{\rho_1}-\frac{1}{\rho}})\|f\|_{\tilde{L}_T^\rho(B_p^{\alpha-2+\frac{2}{\rho},q})}\big),
$$
This completes the proof of the theorem.
\end{proof}

\begin{remark} \label{rmk3.2} \rm
For higher regularity and the cases of critical indexes for Besov
spaces, the method in the paper doesn't work because some
difficulties arise in the process. We will consider these cases
later. For the weighted case, we notice that recently  Cruz-Uribe
and  Rios (\cite{CR}) studied the boundedness of the semigroup
$e^{-tL_\omega}(t>0)$ for $\omega\in A_2$, where the elliptic
operator was defined by
\begin{equation} \label{eq3.8}
L_\omega=-\omega^{-1}\mathop{\rm div}A\nabla
\end{equation}
where $A=(a_{i,j})_{n\times n}$ was a matrix of complex-valued,
measurable functions satisfying some degenerate elliptic conditions.
They pointed that if $A$ is real and symmetric valued, the heat
kernel $p_t(x,y)$ of the semigroup $e^{-tL_\omega}(t>0)$ also
satisfies Gaussian upper bounds. We think that regularity results in
adapted weighted Besov spaces for solutions of the corresponding
parabolic equations can also be obtained by using similar methods to
the ones described in this paper.
\end{remark}

\subsection*{Acknowledgements}
The author would like thank anonymous
the referee for his/her good suggestions on the improvement
of this paper.

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\end{document}