# A weighted $L_{p}$-theory for second-order elliptic and parabolic partial
differential systems on a half space

Kyeong-Hun Kim$\,{}^{1}$, Kijung Lee$\,{}^{2}$
$\;{}^{1}\,$Department of Mathematics, Korea University
Seoul, 136-701, Korea
$\;{}^{2}\,$Department of Mathematics, Ajou University
Suwon, 443-749, Korea

###### Abstract

In this paper we develop a Fefferman-Stein theorem, a Hardy-Littlewood theorem
and sharp function estimations in weighted Sobolev spaces. We also provide
uniqueness and existence results for second-order elliptic and parabolic
partial differential systems in weighed Sobolev spaces.

Keywords: Fefferman-Stein theorem, Hardy-Littlewood theorem, Weighted Sobolev
spaces, Sharp function estimations, $L_{p}$-theory, Elliptic partial
differential systems, Parabolic partial differential systems.

AMS 2000 subject classifications: 42B37, 35K45, 35J57.

\- Dedicated to 70th birthday of N.V. Krylov

##  1 Introduction

In this article we consider the elliptic system

$\displaystyle\sum_{i,j=1}^{d}\sum_{r=1}^{d_{1}}a^{ij}_{kr}u^{r}_{x^{i}x^{j}}(x)=f^{k}(x),\quad(k=1,2,\cdots,d_{1})$
(1.1)

and the parabolic system

$\displaystyle
u^{k}_{t}(t,x)=\sum_{i,j=1}^{d}\sum_{r=1}^{d_{1}}a^{ij}_{kr}(t)u^{r}_{x^{i}x^{j}}(t,x)+f^{k}(t,x),\quad(k=1,2,\cdots,d_{1})$
(1.2)

defined for $t>0$ and $x\in\mathbb{R}^{d}_{+}$.

In the study of partial differential equations (PDEs) or of partial
differential systems (PDSs) regularity theory play the key role of describing
essential relations between input data and the unknown solutions; the sharper
the theory is, the more understanding of the relations we get.

The primary goals of this article are to introduce some new mathematical tools
and ideas which are useful in the study of systems in $L_{p}$-spaces involving
weights and to provide another nice regularity theory for these systems.

In this article we use weighted Sobolev spaces for the unknown function
$u=(u^{1},\cdots,u^{d_{1}})$ and the inputs $f^{k}$. The need to introduce
weights comes from, for instance, the theory of stochastic partial
differential equations (SPDEs) or stochastic partial differential systems
(SPDSs), where a Hölder space approach does not allow us to obtain results of
reasonable generality and Sobolev spaces without weights are trivially
inappropriate (see [14] for details). To study such stochastic systems one has
to develop a nice regularity theory for the corresponding deterministic
systems in advance. Also Sobolev spaces with weights are very useful in
treating degenerate elliptic and parabolic equations (see, for instance, [16])
and in studying equations defined on non-smooth domains such as domains with
wedges (see, for instance, [5, 16, 18]).

In principle there are three main methods for $L_{p}$-theory: multiplier
theory, Calderón-Zygmund theory and the pointwise estimate using sharp
functions. Multiplier theory fits well when the principal operator is almost
Laplacian and the equation under consideration is defined on the entire space,
and Calderón-Zygmund theory works well when there exists an integral
representation of solutions and the integral is taken over $\mathbb{R}^{n}$
for some $n$. However, these two methods do not fit our case since we are
dealing with weighted $L_{p}$-theories for systems (1.1) and (1.1) defined on
a half space. Thus we use an approach based on pointwise estimates of the
sharp function of second order derivatives, but unlike the standard theory
(for instance, [13]) we need to use the weighted version. The elaboration of
this approach is one of our main results.

We also mention that if $d_{1}=1$ then weighted $L_{p}$-theories for single
equations defined on a half space can be constructed based on integration by
parts without relying on sharp function estimations (see the proof of Lemma
4.8 and Lemma 6.3 of [10]). However it seems that the arguments in the proof
of Lemma 4.8 and Lemma 6.3 of [10] cannot be reproduced for $L_{p}$-theory of
systems unless $p=2$ and some stronger algebraic conditions on $A^{ij}$ are
additionally assumed.

Interestingly, we discovered some very useful tools in the perspective of
linear Partial differential equations/systems theory. Even though, in this
article, we only consider the systems with coefficients independent of $x$,
the sharp function estimates and the tools used to derive them will naturally
lead to many subsequent works studying, for instance, elliptic and parabolic
equations and systems with discontinuous coefficients defined in an arbitrary
domain $U$ of $\mathbb{R}^{d}$. In this context, we refer the readers to very
extensive literature [13] and recent articles [1, 2, 3, 7, 6] (also see the
references therein), where (standard) $L_{p}$-theories are constructed for
single equations with VMO (or small BMO)-coefficients.

The article is organized as follows. In section 2 we prove the Fefferman-Stein
theorem and Hardy Littiewood theorem with our special weights; the proofs are
quite elementary. In section 3 we introduce weighted Sobolve spaces and
formulate our regularity results for the systems, Theorem 3.10 and Theorem
3.13. The _useful_ tools and ideas for proving Theorem 3.10 and Theorem 3.13
are in section 4 and 5; the local estimations and the sharp function
estimations. Finally Theorem 3.10 and Theorem 3.13 are proved in section 6.

As usual $\mathbb{R}^{d}$ stands for the Euclidean space of points
$x=(x^{1},...,x^{d})$ and
$\mathbb{R}^{d}_{+}=\\{x\in\mathbb{R}^{d}:x^{1}>0\\}$. For $i=1,...,d$, multi-
indices $\alpha=(\alpha_{1},...,\alpha_{d})$, $\alpha_{i}\in\\{0,1,2,...\\}$,
and functions $u(x)$ we set

$u_{x^{i}}=\frac{\partial u}{\partial x^{i}}=D_{i}u,\quad
D^{\alpha}u=D_{1}^{\alpha_{1}}\cdot...\cdot
D^{\alpha_{d}}_{d}u,\quad|\alpha|=\alpha_{1}+...+\alpha_{d}.$

By $\delta^{kr}$ we denote the Kronecker delta on the indices $k,r$. If we
write $N=N(\cdots)$, this means that the constant $N$ depends only on what are
in parenthesis.

The authors are sincerely grateful to Ildoo Kim for finding few errors in the
earlier version of this article.

##  2 F-S and H-L theorems in weighted $L_{p}$-spaces

Denote

$\Omega=\mathbb{R}\times\mathbb{R}^{d}_{+}:=\\{(t,x)=(t,x^{1},x^{2},\ldots,x^{d})\;:x^{1}>0\\}.$

Also, by $\mathcal{B}(\mathbb{R}^{d}_{+})$ and $\mathcal{B}(\Omega)$ we denote
the Borel $\sigma$-algebra on $\mathbb{R}^{d}_{+}$ and $\Omega$ respectively.
Fix $\alpha\in(-1,\infty)$ and define the weighted measures

$\nu(dx)=\nu_{\alpha}(dx)=(x^{1})^{\alpha}dx,\quad
d\mu=\mu_{\alpha}(dtdx):=\nu_{\alpha}(dx)dt.$

Then $(\mathbb{R}^{d}_{+},\mathcal{B}(\mathbb{R}^{d}_{+}),\nu)$ and
$(\Omega,\mathcal{B}(\Omega),\mu)$ are measure spaces with
$\nu(\mathbb{R}^{d}_{+})=\mu(\Omega)=\infty$. Let $p\in[1,\infty)$ and
$L_{p}(\Omega,\mu)=L_{p}(\Omega,\mu;\mathbb{R}^{d_{1}})$
($L_{p}(\mathbb{R}^{d}_{+},\nu)$ resp.) be the collection of Borel-measurable
functions $u=(u^{1},\ldots,u^{d_{1}})$ defined on $\Omega$ (on
$\mathbb{R}^{d}_{+}$ resp.) satisfying

$\displaystyle\|u\|^{p}_{L_{p}(\Omega,\mu)}:=\int_{\Omega}|u|^{p}d\mu<\infty,\quad\quad\left(\|u\|^{p}_{L_{p}(\mathbb{R}^{d}_{+},\nu)}:=\int_{\mathbb{R}^{d}_{+}}|u|^{p}\nu(dx)<\infty,\text{respectively}\right).$

Denote

$\mathcal{B}^{0}(\Omega):=\\{C\in\mathcal{B}(\Omega)\;:\;|C|:=\mu(C)<\infty\\},\quad\mathcal{B}^{0}(\mathbb{R}^{d}_{+}):=\\{D\in\mathcal{B}(\mathbb{R}^{d}_{+})\;:\;|D|:=\nu(D)<\infty\\}.$

We say $f\in L_{1,loc}(\Omega,\mu;\mathbb{R}^{d_{1}})$ if $fI_{C}\in
L_{1}(\Omega,\mu)$ for any $C\in\mathcal{B}^{0}(\Omega)$, where $I_{C}$ is the
indicator function of $C$. For $f=(f^{1},\ldots,f^{d_{1}})\in
L_{1}(\Omega,\mu;\mathbb{R}^{d_{1}})$ and $C\in\mathcal{B}^{0}(\Omega)$ we
define

$\displaystyle
f_{C}:=\frac{1}{|C|}\int_{C}fd\mu=-\int_{C}fd\mu=\left(-\int_{C}f^{1}d\mu,\ldots,-\int_{C}f^{d_{1}}d\mu\right).$

Similarly write $h\in L_{1,loc}(\mathbb{R}^{d}_{+},\nu;\mathbb{R}^{d_{1}})$ if
$hI_{D}\in L_{1}(\mathbb{R}^{d}_{+},\nu)$ for any
$D\in\mathcal{B}^{0}(\mathbb{R}^{d}_{+})$, and define

$h_{D}:=\frac{1}{|D|}\int_{D}h\nu(dx)=-\int_{D}h\nu(dx)=\left(-\int_{D}h^{1}\nu(dx),\ldots,-\int_{D}h^{d_{1}}\nu(dx)\right).$

Let $(\mathbb{C}_{n},n\in\mathbb{Z})$ denote the filtration of the partitions
of $\bar{\Omega}$ defined by

$\displaystyle\mathbb{C}_{n}=\Big{\\{}\Big{[}\frac{i_{0}}{4^{n}},\frac{i_{0}+1}{4^{n}}\Big{)}\times\Big{[}\frac{i_{1}}{2^{n}},\frac{i_{1}+1}{2^{n}}\Big{)}\times\cdots\times\Big{[}\frac{i_{d}}{2^{n}},\frac{i_{d}+1}{2^{n}}\Big{)}\;:\;i_{0},i_{2},\ldots,i_{d}\in\mathbb{Z},\;i_{1}\in\\{0\\}\cup\mathbb{N}\Big{\\}},$

and $(\mathbb{D}_{n},n\in\mathbb{Z})$ be the corresponding filtration of the
partitions of $\bar{\mathbb{R}}^{d}_{+}$, that is,

$\displaystyle\mathbb{D}_{n}:=\Big{\\{}\Big{[}\frac{i_{1}}{2^{n}},\frac{i_{1}+1}{2^{n}}\Big{)}\times\cdots\times\Big{[}\frac{i_{d}}{2^{n}},\frac{i_{d}+1}{2^{n}}\Big{)}\;:\;i_{0},i_{2},\ldots,i_{d}\in\mathbb{Z},\;i_{1}\in\\{0\\}\cup\mathbb{N}\Big{\\}}.$

For any $(t,x)\in\Omega$, by $C_{n}(t,x)$ ($D_{n}(x)$ resp.) we denote the
unique cube in $\mathbb{C}_{n}$ (in $\mathbb{D}_{n}$ resp.) containing $(t,x)$
($x$ respectively). Let $\mathbb{L}=\mathbb{L}(\Omega)$ (resp.
$\mathbb{L}(\mathbb{R}^{d}_{+})$) denote the set of
$\mathbb{R}^{d_{1}}$-valued continuous functions with compact support in
$\Omega$ ( in $\mathbb{R}^{d}_{+}$ respectively).

######  Lemma 2.1.

(i) We have $\inf_{C\in\mathbb{C}_{n}}|C|\to\infty$ as $n\to-\infty$ and, for
any $f\in\mathbb{L}(\Omega)$, $\lim_{n\to\infty}f_{C_{n}(t,x)}=f(t,x)$ holds
for any $(t,x)\in\Omega$.

(ii) We have $\inf_{D\in\mathbb{D}_{n}}|D|\to\infty$ as $n\to-\infty$ and, for
any $f\in\mathbb{L}(\mathbb{R}^{d}_{+})$, $\lim_{n\to\infty}f_{D_{n}(x)}=f(x)$
holds for any $x\in\mathbb{R}^{d}_{+}$.

###### Proof.

It is obvious since $f$ is continuous. ∎

######  Lemma 2.2.

(i) For any $C\in\mathbb{C}_{n}$ there exists a unique
$C^{\prime}\in\mathbb{C}_{n-1}$ such that $C\subset C^{\prime}$ and

$\displaystyle\frac{|C^{\prime}|}{|C|}\leq N(\alpha)<\infty.$

(ii) For any $D\in\mathbb{D}_{n}$ there exists a unique
$D^{\prime}\in\mathbb{D}_{n-1}$ such that $D\subset D^{\prime}$ and

$\displaystyle\frac{|D^{\prime}|}{|D|}\leq N(\alpha)<\infty.$

###### Proof.

We only prove (i). Since $\mathbb{C}_{n-1}$ is a partition of $\Omega$, only
one member of it contains $C$; we call it $C^{\prime}$. Let

$\displaystyle
C^{\prime}=\Big{[}\frac{i_{0}}{4^{n-1}},\frac{i_{0}+1}{4^{n-1}}\Big{)}\times\Big{[}\frac{i_{1}}{2^{n-1}},\frac{i_{1}+1}{2^{n-1}}\Big{)}\times\cdots\times\Big{[}\frac{i_{d}}{2^{n-1}},\frac{i_{d}+1}{2^{n-1}}\Big{)}.$

Then we have

$\displaystyle|C^{\prime}|=\mu(C^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{2^{(d+1)(n-1)}}\int^{\frac{i_{1}+1}{2^{n-1}}}_{\frac{i_{1}}{2^{n-1}}}(x^{1})^{\alpha}dx^{1}$
$\displaystyle=$
$\displaystyle\frac{1}{2^{(d+1)(n-1)}}\cdot\frac{1}{\alpha+1}\Bigg{[}\left(\frac{i_{1}+1}{2^{n-1}}\right)^{\alpha+1}-\left(\frac{i_{1}}{2^{n-1}}\right)^{\alpha+1}\Bigg{]}.$

Note that $C$ is one of $4\cdot 2^{d}$ cubes belonging to $\mathbb{C}_{n}$
inside $C^{\prime}$ and by the location of $C$ we have either

$\displaystyle|C|=\frac{1}{2^{(d+1)n}}\cdot\frac{1}{\alpha+1}\Bigg{[}\left(\frac{i_{1}+1}{2^{n-1}}\right)^{\alpha+1}-\left(\frac{i_{1}+1}{2^{n-1}}-\frac{1}{2^{n}}\right)^{\alpha+1}\Bigg{]}$
(2.1)

or

$\displaystyle|C|=\frac{1}{2^{(d+1)n}}\cdot\frac{1}{\alpha+1}\Bigg{[}\left(\frac{i_{1}+1}{2^{n-1}}-\frac{1}{2^{n}}\right)^{\alpha+1}-\left(\frac{i_{1}}{2^{n-1}}\right)^{\alpha+1}\Bigg{]}.$
(2.2)

Case 1: Let $i_{1}\geq 1$ and $\alpha\geq 0$. Denoting

$\displaystyle a=\frac{i_{1}+1}{2^{n-1}},\quad b=\frac{i_{1}}{2^{n-1}},\quad
c=\frac{i_{1}+1}{2^{n-1}}-\frac{1}{2^{n}},\quad\phi(x)=x^{\alpha+1},$

we get

$\displaystyle\frac{|C^{\prime}|}{|C|}$ $\displaystyle=$ $\displaystyle
2^{d+1}\cdot\frac{\phi(a)-\phi(b)}{\phi(a)-\phi(c)}\quad\textrm{or}\quad
2^{d+1}\cdot\frac{\phi(a)-\phi(b)}{\phi(c)-\phi(b)}$ (2.3) $\displaystyle=$
$\displaystyle
2^{d+1}\left(1+\frac{\phi(c)-\phi(b)}{\phi(a)-\phi(c)}\right)\quad\textrm{or}\quad
2^{d+1}\left(1+\frac{\phi(a)-\phi(c)}{\phi(c)-\phi(b)}\right)$
$\displaystyle=$ $\displaystyle
2^{d+1}\left(1+\frac{\phi^{\prime}(\beta)}{\phi^{\prime}(\alpha)}\right)\quad\textrm{or}\quad
2^{d+1}\left(1+\frac{\phi^{\prime}(\alpha)}{\phi^{\prime}(\beta)}\right),$

where $\alpha,\beta$ are some numbers satisfying $b<\beta<c<\alpha<a$; we used
mean value theorem. Since $\alpha+1>1$, the function $\phi$ is convex and
increasing on $(0,\infty)$. Hence, we have

$\displaystyle\frac{\phi^{\prime}(\beta)}{\phi^{\prime}(\alpha)}\leq
1,\quad\quad\frac{\phi^{\prime}(\alpha)}{\phi^{\prime}(\beta)}\leq\frac{\phi^{\prime}(a)}{\phi^{\prime}(b)}=\frac{a^{\alpha}}{b^{\alpha}}=\left(\frac{i_{1}+1}{i_{1}}\right)^{\alpha}\leq
2^{\alpha},$

and therefore

$\displaystyle\frac{|C^{\prime}|}{|C|}\leq 2^{d+1}(1+2^{\alpha})\leq
2^{\alpha+d+2}.$

Case 2: Assume $i_{1}=0$ and $\alpha\geq 0$. By similar but simpler
calculation we obtain

$\displaystyle\frac{|C^{\prime}|}{|C|}\leq 2^{\alpha+d+2}.$

Case 3: Assume $\alpha\in(-1,0)$. If $|C|$ is given as in (2.2), then since
$\phi(x)$ is concave,

$\frac{\left(\frac{i_{1}+1}{2^{n-1}}\right)^{\alpha+1}-\left(\frac{i_{1}}{2^{n-1}}\right)^{\alpha+1}}{\left(\frac{i_{1}+1}{2^{n-1}}-\frac{1}{2^{n}}\right)^{\alpha+1}-\left(\frac{i_{1}}{2^{n-1}}\right)^{\alpha+1}}\leq
2.$

Let $|C|$ be given as in (2.1). If $i_{1}=0$, then

$\frac{\left(\frac{i_{1}+1}{2^{n-1}}\right)^{\alpha+1}-\left(\frac{i_{1}}{2^{n-1}}\right)^{\alpha+1}}{\left(\frac{i_{1}+1}{2^{n-1}}\right)^{\alpha+1}-\left(\frac{i_{1}+1}{2^{n-1}}-\frac{1}{2^{n}}\right)^{\alpha+1}}=\frac{2^{\alpha+1}}{2^{\alpha+1}-1},$

and if $i_{1}\geq 1$ then since $\phi$ is concave and $\phi^{\prime}$ is
positive on $(0,\infty)$

$\frac{\left(\frac{i_{1}+1}{2^{n-1}}\right)^{\alpha+1}-\left(\frac{i_{1}}{2^{n-1}}\right)^{\alpha+1}}{\left(\frac{i_{1}+1}{2^{n-1}}\right)^{\alpha+1}-\left(\frac{i_{1}+1}{2^{n-1}}-\frac{1}{2^{n}}\right)^{\alpha+1}}\leq\frac{2^{-n+1}\phi^{\prime}(\frac{i_{1}}{2^{n-1}})}{2^{-n}\phi^{\prime}(\frac{i_{1}+1}{2^{n-1}})}\leq
2^{1-\alpha}.$

The lemma is proved. ∎

######  Remark 2.3.

(i) By Lemma 2.1, Lemma 2.2 and the outline of Section 3.1, 3.2 of [13] we get
Lemma 2.5, Theorem 2.7 and Theorem 2.8 below for free.

(ii) if $C_{n}\in\mathbb{C}_{n}$ and $C_{m}\in\mathbb{C}_{m}$ with $n\leq m$,
then $C_{n}\cap C_{m}=C_{m}$ or $\emptyset$.

######  Definition 2.4.

We call $\tau=\tau(x)\in\mathbb{Z}\cup\\{\infty\\}$ a _stopping time_ if
$\\{x:\tau(x)=n\\}=\emptyset$ or union of some elements in $\mathbb{C}_{n}$
for each $n\in\mathbb{Z}$.

For $f\in L_{1,loc}(\Omega,\mu;\mathbb{R}^{d_{1}})$, $h\in
L_{1,loc}(\mathbb{R}^{d}_{+},\nu,\mathbb{R}^{d_{1}})$ and $n\in\mathbb{Z}$ we
define

$\displaystyle
f_{|n}(t,x):=\frac{1}{\mu(C_{n}(t,x))}\int_{C_{n}(t,x)}f(s,y)\mu(dsdy)=-\int_{C_{n}(t,x)}f(s,y)\mu(dsdy),$
$\displaystyle
h_{|n}(x):=\frac{1}{\nu(D_{n}(t,x))}\int_{D_{n}(t,x)}h(y)\nu(dy)=-\int_{D_{n}(x)}h(y)\nu(dy),$

and

$\displaystyle
f_{|\tau}(t,x):=f_{|\tau(t,x)}(t,x)\quad\mathrm{if}\;\;\tau(t,x)\neq\infty;\quad
f_{|\tau}(t,x):=f(t,x)\quad\mathrm{if}\;\;\tau(t,x)=\infty.$

######  Lemma 2.5.

Let $\\{\mathbb{C}_{n}\;:\;n\in\mathbb{Z}\\}$ be a filtration of partitions of
$\bar{\Omega}$.

(i) Let $g\in L_{1,loc}(\Omega,\mu;\mathbb{R}^{1})$, $g\geq 0$ and let $\tau$
be a stopping time. Then

$\displaystyle\int_{\Omega}g_{|\tau}(t,x)I_{\tau<\infty}(t,x)\mu(dtdx)$
$\displaystyle=$
$\displaystyle\int_{\Omega}g(t,x)I_{\tau<\infty}(t,x)\mu(dtdx),$
$\displaystyle\int_{\Omega}g_{|\tau}(t,x)\mu(dtdx)$ $\displaystyle=$
$\displaystyle\int_{\Omega}g(t,x)\mu(dtdx).$

(ii) Let $g\in L_{1}(\Omega,\mu;\mathbb{R}^{1})$, $g\geq 0$ and let
$\lambda>0$ be a constant. Then

$\displaystyle\tau(t,x):=\inf\\{n:g_{|n}(t,x)>\lambda\\}\quad\quad(\inf\emptyset:=\infty)$

is a stopping time. Furthermore, we have

$\displaystyle 0\leq g_{|\tau}(t,x)I_{\tau<\infty}\leq
N_{0}\lambda,\quad|\\{(t,x):\tau(t,x)<\infty\\}|\leq\lambda^{-1}\int_{\Omega}g(t,x)I_{\tau<\infty}\mu(dtdx).$

######  Remark 2.6.

$($Riesz-Calderón-Zygmund decomposition$)$ Any $g\in
L_{1}(\Omega,\mu;\mathbb{R}^{1})$ is decomposed by

$\displaystyle g=\xi+\eta,$

where $\xi=g-g_{|\tau}$,
$\eta=g_{|\tau}=g_{|\tau}\;I_{\tau<\infty}+g_{|\tau}\;I_{\tau=\infty}$.
Moreover, we have (i) $\eta\leq N_{0}\lambda$ a.e. (ii)
$|\\{(t,x):\xi(t,x)\neq 0\\}|\leq\lambda^{-1}\|g\|_{L_{1}(\Omega,\mu)}$ (iii)
$\xi_{|\tau}=0$.

Now, for $f\in L_{1,loc}(\Omega,\mu;\mathbb{R}^{d_{1}})$ we define the maximal
function

$\displaystyle\mathcal{M}f(t,x):=\left(\sup_{n<\infty}|f^{1}|_{|n}(t,x),\;\ldots\;,\sup_{n<\infty}|f^{d_{1}}|_{|n}(t,x)\right)$

and the sharp function

$\displaystyle
f^{\\#}(t,x)=\left(\sup_{n<\infty}-\int_{C_{n}(t,x)}|f^{1}(s,y)-f^{1}_{|n}(s,y)|\mu(dsdy),\,\ldots,\,\sup_{n<\infty}-\int_{C_{n}(t,x)}|f^{d_{1}}(s,y)-f^{d_{1}}_{|n}(s,y)|\mu(dsdy)\right).$

We define $\mathcal{M}h(x)$ and $h^{\\#}(x)$ similarly for functions
$h=h(x)\in L_{1,loc}(\mathbb{R}^{d}_{+},\nu;\mathbb{R}^{d_{1}})$.

######  Theorem 2.7.

Let $p\in(1,\infty)$. Then for any $f\in L_{p}(\Omega,\mu;\mathbb{R}^{d_{1}})$
and $h\in L_{p}(\mathbb{R}^{d}_{+},\nu;\mathbb{R}^{d_{1}})$, we have

$\displaystyle\|\mathcal{M}f\|_{L_{p}(\Omega,\mu;\mathbb{R}^{d_{1}})}\leq
N\|f\|_{L_{p}(\Omega,\mu;\mathbb{R}^{d_{1}})},\quad\|\mathcal{M}h\|_{L_{p}(\mathbb{R}^{d}_{+},\nu;\mathbb{R}^{d_{1}})}\leq
N\|h\|_{L_{p}(\mathbb{R}^{d}_{+},\nu;\mathbb{R}^{d_{1}})}$

where $N=N(\theta,p,d,d_{1})$.

######  Theorem 2.8.

Let $p\in(1,\infty)$. Then for any $f\in L_{p}(\Omega,\mu;\mathbb{R}^{d_{1}})$
and $h\in L_{p}(\mathbb{R}^{d}_{+},\nu;\mathbb{R}^{d_{1}})$ we have

$\displaystyle\|f\|_{L_{p}(\Omega,\mu;\mathbb{R}^{d_{1}})}\leq
N\|f^{\\#}\|_{L_{p}(\Omega,\mu;\mathbb{R}^{d_{1}})},\quad\|h\|_{L_{p}(\mathbb{R}^{d}_{+},\nu;\mathbb{R}^{d_{1}})}\leq
N\|h^{\\#}\|_{L_{p}(\mathbb{R}^{d}_{+},\nu;\mathbb{R}^{d_{1}})}$

where $N=N(\theta,p,d,d_{1})$.

We investigate the relation between our maximal and sharp functions and more
general ones. Let $B^{\prime}_{r}(x^{\prime})$ denote the open ball in
$\mathbb{R}^{d-1}$ of radius $r$ with center $x^{\prime}$. For
$x=(x^{1},x^{\prime})\in\mathbb{R}^{d}_{+}$ and $t\in\mathbb{R}$, denote

$B_{r}(x)=B_{r}(x^{1},x^{\prime})=(x^{1}-r,x^{1}+r)\times
B^{\prime}_{r}(x^{\prime}),\quad Q_{r}(t,x):=(t,t+r^{2})\times B_{r}(x)$

and $\mathbb{Q}$ be the collection of all such open sets
$Q_{r}(t,x)\subset\Omega$. For $f\in L_{1,loc}(\Omega,\mu:\mathbb{R}^{d_{1}})$
we define

$\displaystyle
f^{i}_{Q}=-\int_{Q}f^{i}\;d\mu,\quad\mathbb{M}f^{i}(t,x)=\sup_{(t,x)\in
Q}-\int_{Q}f^{i}d\mu,\quad(f^{i})^{\sharp}(t,x)=\sup_{(t,x)\in
Q}-\int_{Q}|f^{i}-f^{i}_{Q}|d\mu,\quad i=1,\ldots,d_{1},$

where the supremum is taken for all $Q\in\mathbb{Q}$ containing $(t,x)$.
Denote

$\mathbb{M}f:=(\mathbb{M}f^{1},\ldots,\mathbb{M}f^{d_{1}}),\quad
f^{\sharp}:=((f^{1})^{\sharp},\ldots,(f^{d_{1}})^{\sharp}).$

For functions $h\in L_{1,loc}(\mathbb{R}^{d}_{+},\nu,\mathbb{R}^{d_{1}})$, the
functions $\mathbb{M}h(x)$ and $(h)^{\sharp}(x)$ are defined similarly.

######  Lemma 2.9.

For a scalar function $g=g(t,x)$ and $h=h(x)$ we have

$\displaystyle g^{\\#}(t,x)\leq N\;g^{\sharp}(t,x),\quad\quad h^{\\#}(x)\leq
N\;h^{\sharp}(x)$

where $N=N(\theta,p,d)$.

###### Proof.

We only prove the first assertion. For $(t,x)\in\Omega$, denote the
corresponding unique cube $C_{n}(t,x)\in\mathbb{C}_{n}$ by

$\displaystyle\Big{[}\frac{i_{0}}{4^{n}},\frac{i_{0}+1}{4^{n}}\Big{)}\times\Big{[}\frac{i_{1}}{2^{n}},\frac{i_{1}+1}{2^{n}}\Big{)}\times\cdots\times\Big{[}\frac{i_{d}}{2^{n}},\frac{i_{d}+1}{2^{n}}\Big{)}$

where $i_{0},i_{2},\ldots,i_{d}\in\mathbb{Z}$ and
$i_{1}\in\\{0\\}\cup\mathbb{N}$. We define
$Q_{(n)}(t,x):=Q_{\frac{d}{2^{n}}}(t^{*},x^{*})$ with
$t^{*}=\frac{i_{0}}{4^{n}}$ and
$x^{*}=(\frac{i_{1}+d}{2^{n}},\frac{i_{2}}{2^{n}},\ldots,\frac{i_{d}}{2^{n}})$.
We have $(t,x)\in C_{n}(t,x)\subset\overline{Q_{(n)}(t,x)}$ and

$\displaystyle\frac{|Q_{(n)}(t,x)|}{|C_{n}(t,x)|}=N(d)\cdot\frac{(i_{1}+2d)^{\alpha+1}-i_{1}^{\alpha+1}}{(i_{1}+1)^{\alpha+1}-i_{1}^{\alpha+1}}$
(2.4)

by simple calculation. If $i_{1}=0$, (2.4) is $N(d)(2d)^{\alpha+1}$; if
$i_{1}\geq 1$ and $\alpha\geq 0$ then (2.4) is less than or equal to

$\displaystyle N(d)\cdot(2d)\left(\frac{i_{1}+2d}{i_{1}}\right)^{\alpha}\leq
N(d)\cdot(2d)\cdot(1+2d)^{\alpha},$

by mean value theorem. If $\alpha\in(-1,0)$ then we use the concavity of
$x^{\alpha+1}$ to prove that (2.4) is less then $N(d)(2d)^{\alpha+1}$. The
lemma is proved. ∎

Lemma 2.9 and Theorem 2.8 imply the following version of Fefferman-Stein
theorem:

######  Theorem 2.10.

$($Fefferman-Stein$)$ Let $p\in(1,\infty)$. Then for any $f\in
L_{p}(\Omega,\mu;\mathbb{R}^{d_{1}})$ and $h\in
L_{p}(\mathbb{R}^{d}_{+},\nu,\mathbb{R}^{d_{1}})$, we have

$\displaystyle\|f\|_{L_{p}(\Omega,\mu;\mathbb{R}^{d_{1}})}\leq
N\|f^{\sharp}\|_{L_{p}(\Omega,\mu;\mathbb{R}^{d_{1}})},\quad\quad\|h\|_{L_{p}(\mathbb{R}^{d}_{+},\nu;\mathbb{R}^{d_{1}})}\leq
N\|h^{\sharp}\|_{L_{p}(\mathbb{R}^{d}_{+},\nu;\mathbb{R}^{d_{1}})}$

where $N=N(\theta,p,d,d_{1})$.

The following lemma will be used in the proof of Theorem 2.12 below.

######  Lemma 2.11.

Let $\alpha>-1$ and $\phi(x)=x^{\alpha+1}$ on $x>0$. Then for any $x>0$ and
$r>0$ we have

$\displaystyle\frac{\phi(x+2r)-\phi(x+r)}{\phi(x+r)-\phi(x)}\;\leq\;2^{\alpha+1}.$

###### Proof.

If $\alpha\in(-1,0]$ the claim is obvious since $\phi$ is concave.

Assume $\alpha>0$, fix $r>0$, and define

$\displaystyle f(x):=\frac{\phi(x+2r)-\phi(x+r)}{\phi(x+r)-\phi(x)}.$

We show that $f^{\prime}(x)\leq 0$ for $x>0$ so that $f(x)\leq
f(0)=2^{\alpha+1}-1$; note that $f(0)$ does not depend on $r$. A simple
calculation shows

$\displaystyle
f^{\prime}(x)=r(\alpha+1)\cdot\frac{2(x+2r)^{\alpha}x^{\alpha}-(x+2r)^{\alpha}(x+r)^{\alpha}-(x+r)^{\alpha}x^{\alpha}}{((x+r)^{\alpha+1}-x^{\alpha+1})^{2}}.$
(2.5)

The numerator in (2.5) is

$\displaystyle 2\cdot
x^{\alpha}(x+r)^{\alpha}(x+2r)^{\alpha}\cdot\left[(x+r)^{-\alpha}-\frac{x^{-\alpha}+(x+2r)^{-\alpha}}{2}\right].$
(2.6)

Since the function $x^{-\alpha}$ is convex and $x+r$ is the midpoint of $x$
and $x+2r$, the square bracket in (2.6) is non-positive and so is
$f^{\prime}(x)$. The lemma is proved. ∎

######  Theorem 2.12.

$($Hardy-Littlewood$)$ Let $p\in(1,\infty)$. Then for $f\in
L_{p}(\Omega,\mu;\mathbb{R}^{d_{1}})$ and $h\in
L_{p}(\mathbb{R}^{d}_{+},\nu,\mathbb{R}^{d_{1}})$ we have

$\displaystyle\|\mathbb{M}f\|_{L_{p}(\Omega,\mu;\mathbb{R}^{d_{1}})}\leq
N\|f\|_{L_{p}(\Omega,\mu;\mathbb{R}^{d_{1}})},\quad\|\mathbb{M}h\|_{L_{p}(\mathbb{R}^{d}_{+},\nu;\mathbb{R}^{d_{1}})}\leq
N\|h\|_{L_{p}(\mathbb{R}^{d}_{+},\nu;\mathbb{R}^{d_{1}})}.$

###### Proof.

Again we only proof the first assertion. We follow the outline for the proof
of Theorem 3.3.2 which does not involve a weight in the norm. Without loss of
generality we assume $d_{1}=1$ and $g:=f\geq 0$.

For $\lambda>0$, denote $A(\lambda):=\\{(t,x):\mathbb{M}g(t,x)>\lambda\\}$.
Then since $\mathbb{M}g$ is lower semi-continuous, $A(\lambda)$ is open. To
prove the theorem it is enough to show that for any $\lambda>0$ and compact
set $K\subset A(\lambda)$

$\displaystyle|K|\leq\frac{N}{\lambda}\int_{\Omega}I_{A(\lambda)}(t,x)g(t,x)\mu(dtdx),$

where $N=N(\theta,p,d)$. For the details see the proof of Theorem 3.3.2 of
[13].

For any $(t,x)\in K$ there exists $Q$ containing $(t,x)$ such that
$\int_{Q}gd\mu>\lambda|Q|$. Also, we observe that $Q\subset A(\lambda)$ and
there exists a finite cover $\\{Q_{1},\ldots,Q_{n}\\}$ of $K$ such that

$\displaystyle\int_{Q_{i}}gd\mu>\lambda|Q_{i}|.$

For $Q=(t-\frac{1}{2}r^{2},t+\frac{1}{2}r^{2})\times(x^{1}-r,x^{1}+r)\times
B^{\prime}_{r}(x^{\prime})\in\mathbb{Q}$, denote

$3Q:=\big{(}t-\frac{3}{2}r^{2},t+\frac{3}{2}r^{2}\big{)}\times(x^{1}-3r,x^{1}+3r)\times
B^{\prime}_{3r}(x^{\prime}).$

When $Q$ is close to the boundary of $\Omega$, $3Q$ may not be in $\Omega$.
Hence, we define

$\displaystyle Q^{*}=3Q\cap\Omega.$

Using a Vitali covering argument one can find the disjoint subset
$\\{\tilde{Q}_{1},\ldots,\tilde{Q}_{k}\\}$ of $\\{Q_{1},\ldots,Q_{n}\\}$
satisfying $K\subset\bigcup^{k}_{j=1}\tilde{Q_{j}}^{*}$ (see the proof of
Theorem 3.3.2 of [13]). To measure $|K|$ we compute the ratio
$\frac{|\tilde{Q_{j}}^{*}|}{|\tilde{Q_{j}}|}$. For
$Q_{j}=(t-\frac{r^{2}}{2},t+\frac{r^{2}}{2})\times(x^{1}-r,x^{1}+r)\times
B^{\prime}_{r}(x^{\prime})$ we have

$\displaystyle\frac{|\tilde{Q_{j}}^{*}|}{|\tilde{Q_{j}}|}=3^{d}\cdot\frac{\phi(x+3r)-\phi((x-3r)\vee
0)}{\phi(x+r)-\phi(x-r)},$

where $\phi(x)=x^{\theta-d+p+1}$ and $a\vee b:=\max\\{a,b\\}$. We note

$\displaystyle\frac{\phi(x+3r)-\phi((x-3r)\vee 0)}{\phi(x+r)-\phi(x-r)}$ (2.7)
$\displaystyle=$ $\displaystyle\frac{\phi(x-r)-\phi((x-3r)\vee
0)+\phi(x+r)-\phi(x-r)+\phi(x+3r)-\phi(x+r)}{\phi(x+r)-\phi(x-r)}$
$\displaystyle\leq$ $\displaystyle
2+\frac{\phi(x+3r)-\phi(x+r)}{\phi(x+r)-\phi(x-r)},$

where the last inequality is true since $\phi$ is increasing and convex. Now,
Lemma 2.11 with $x-r$, $2r$ instead of $x$, $r$ implies (2.7) is less than or
equal to $2+2^{\alpha+1}$. Hence, we have

$\displaystyle\frac{|\tilde{Q_{j}}^{*}|}{|\tilde{Q_{j}}|}\leq
3^{d}\cdot(2+2^{\alpha+1}),\quad|\tilde{Q_{j}}^{*}|\leq
3^{d}\cdot(2+2^{\alpha+1})|\tilde{Q_{j}}|.$

Thus,

$\displaystyle|K|$ $\displaystyle\leq$
$\displaystyle\sum^{k}_{j=1}|\tilde{Q_{j}}^{*}|\leq
3^{d}\cdot(2+2^{\alpha+1})\sum^{k}_{j=1}|\tilde{Q_{j}}|$ $\displaystyle\leq$
$\displaystyle
3^{d}\cdot(2+2^{\alpha+1})\lambda^{-1}\sum^{k}_{j=1}\int_{\tilde{Q_{j}}}g\,d\mu\leq
3^{d}\cdot(2+2^{\alpha+1})\lambda^{-1}\int_{\Omega}gI_{A(\lambda)}\,d\mu.$

The theorem is proved. ∎

##  3 A weighted $L_{p}$-theory for systems in a half space

Let $C^{\infty}_{0}(\mathbb{R}^{d};\mathbb{R}^{d_{1}})$ denote the set of all
$\mathbb{R}^{d_{1}}$-valued infinitely differentiable functions with compact
support in $\mathbb{R}^{d}$. By $\mathcal{D}$ we denote the space of
$d$-dimensional distributions on
$C^{\infty}_{0}(\mathbb{R}^{d};\mathbb{R}^{d_{1}})$; precisely, for
$u\in\mathcal{D}$ and $\phi\in
C^{\infty}_{0}(\mathbb{R}^{d};\mathbb{R}^{d_{1}})$ we define
$(u,\phi)\in\mathbb{R}^{d_{1}}$ with components
$(u,\phi)^{k}=(u^{k},\phi^{k})$, $k=1,\ldots,d_{1}$; each $u^{k}$ is a usual
scalar-valued distribution.

For $p\in(1,\infty)$ we define
$L_{p}=L_{p}(\mathbb{R}^{d};\mathbb{R}^{d_{1}})$ as the space of all
$\mathbb{R}^{d_{1}}$-valued functions $u=(u^{1},\ldots,u^{d_{1}})$ satisfying

$\|u\|^{p}_{L_{p}}:=\sum^{d_{1}}_{k=1}\|u^{k}\|^{p}_{L_{p}}<\infty.$

Denote $x=(x^{1},\ldots,x^{d})$. In this paper we define

$\|u_{x}\|^{p}_{L_{p}}=\sum_{i=1}^{d}\|u_{x^{i}}\|^{p}_{L_{p}},\quad\|u_{xx}\|^{p}_{L_{p}}=\sum_{i,j=1}^{d}\|u_{x^{i}x^{j}}\|^{p}_{L_{p}},\quad\textrm{etc.}$

For any $\gamma\in\mathbb{R}$, define the space of Bessel potential
$H^{\gamma}_{p}=H^{\gamma}_{p}(\mathbb{R};\mathbb{R}^{d_{1}})$ as the space of
all distributions $u$ on $\mathbb{R}^{d}$ such that $(1-\Delta)^{\gamma/2}u\in
L_{p}$, where each component is defined by

$((1-\Delta)^{\gamma/2}u)^{k}=(1-\Delta)^{\gamma/2}u^{k}$

and the norm is given by

$\|u\|_{H^{\gamma}_{p}}:=\|(1-\Delta)^{\gamma/2}u\|_{L_{p}}.$

Then $H^{\gamma}_{p}$ is a Banach space with the given norm and
$C^{\infty}_{0}(\mathbb{R}^{d};\mathbb{R}^{d_{1}})$ is dense in
$H^{\gamma}_{p}$ (see [19]). Note that $H^{\gamma}_{p}$ are usual Sobolev
spaces for $\gamma=0,1,2,\ldots$. It is well known that the first order
differentiation operator, $D:H^{\gamma}_{p}\to H^{\gamma-1}_{p}$, is bounded.
On the other hand, if $\text{supp}\,(u)\subset(a,b)$, where
$-\infty<a<b<\infty$, then

$\|u\|_{H^{\gamma}_{p}}\leq c(d,a,b)\|u_{x}\|_{H^{\gamma-1}_{p}}$ (3.1)

(see, for instance, Remark 1.13 in [10]).

Now we introduce the weighted Sobolev spaces taken from [10] and [17]. Take a
nonnegative real-valued function $\zeta(x)=\zeta(x^{1})\in
C^{\infty}_{0}(\mathbb{R}_{+})$ such that

$\sum_{n=-\infty}^{\infty}\zeta(e^{n+s})>c>0,\quad\forall s\in\mathbb{R},$
(3.2)

where $c$ is a constant. Note that any nonnegative function $\zeta$ with
$\zeta>0$ on $[1,e]$ satisfies (3.2). For $\theta\in\mathbb{R}$, let
$H^{\gamma}_{p,\theta}:=H^{\gamma}_{p,\theta}(\mathbb{R}^{d}_{+};\mathbb{R}^{d_{1}})$
denote the set of all $d$-dimensional distributions $u=(u^{1},u^{2},\cdots
u^{d_{1}})$ on $\mathbb{R}^{d}_{+}$ such that

$\|u\|^{p}_{H^{\gamma}_{p,\theta}}:=\sum_{n\in\mathbb{Z}}e^{n\theta}\|\zeta(\cdot)u(e^{n}\cdot)\|^{p}_{H^{\gamma}_{p}}<\infty.$
(3.3)

It is known that for different $\zeta$ satisfying (3.2), we get the same
spaces $H^{\gamma}_{p,\theta}$ with equivalent norms, and for any $\eta\in
C^{\infty}_{0}(\mathbb{R}_{+};\mathbb{R})$,

$\sum_{n=-\infty}^{\infty}e^{n\theta}\|\eta(\cdot)u(e^{n}\cdot)\|^{p}_{H^{\gamma}_{p}}\leq
N\sum_{n=-\infty}^{\infty}e^{n\theta}\|\zeta(\cdot)u(e^{n}\cdot)\|^{p}_{H^{\gamma}_{p}},$
(3.4)

where $N$ depends only on $\gamma,\theta,p,d,d_{1},\eta,\zeta$. Furthermore,
if $\gamma$ is a nonnegative integer, then

$\displaystyle\|u\|^{p}_{H^{\gamma}_{p,\theta}}\sim\sum_{|\beta|\leq\gamma}\int_{\mathbb{R}_{+}^{d}}|(x^{1})^{|\beta|}D^{\beta}u(x)|^{p}(x^{1})^{\theta-d}\,dx.$
(3.5)

Let $M^{\alpha}$ be the operator of multiplying by $(x^{1})^{\alpha}$ and
$M:=M^{1}$. For $\nu\in(0,1]$, denote

$|u|_{C}=\sup_{x\in\mathbb{R}^{d}_{+}}|u(x)|,\quad[u]_{C^{\nu}}=\sup_{x\neq
y}\frac{|u(x)-u(y)|}{|x-y|^{\nu}}.$

Below we collect some other important properties of the spaces
$H^{\gamma}_{p,\theta}$.

######  Lemma 3.1.

$(\cite[cite]{[\@@bibref{}{kr99}{}{}]},\cite[cite]{[\@@bibref{}{kr99-1}{}{}]})$
Let $\gamma,\theta\in\mathbb{R}$ and $p\in(1,\infty)$.

  * (i)

$C^{\infty}_{0}(\mathbb{R}^{d}_{+};\mathbb{R}^{d_{1}})$ is dense in
$H^{\gamma}_{p,\theta}$.

  * (ii)

Assume that $\gamma=m+\nu+d/p$ for some $m=0,1,\cdots$ and $\nu\in(0,1]$. Then
for any $u\in H^{\gamma}_{p,\theta}$ $i\in\\{0,1,\cdots,m\\}$, we have

$|M^{i+\theta/p}D^{i}u|_{C}+[M^{m+\nu+\theta/p}D^{m}u]_{C^{\nu}}\leq
N\|u\|_{H^{\gamma}_{p,\theta}}.$ (3.6)

  * (iii)

Let $\alpha\in\mathbb{R}$. Then $M^{\alpha}H^{\gamma}_{p,\theta+\alpha
p}=H^{\gamma}_{p,\theta}$ and

$\|u\|_{H^{\gamma}_{p,\theta}}\leq
N\|M^{-\alpha}u\|_{H^{\gamma}_{p,\theta+\alpha p}}\leq
N\|u\|_{H^{\gamma}_{p,\theta}}.$

  * (iv)

For any $MD,DM:H^{\gamma}_{p,\theta}\to H^{\gamma-1}_{p,\theta}$ are bounded
linear operators, and

$\|u\|_{H^{\gamma}_{p,\theta}}\leq
N\|u\|_{H^{\gamma-1}_{p,\theta}}+N\|Mu_{x}\|_{H^{\gamma-1}_{p,\theta}}\leq
N\|u\|_{H^{\gamma}_{p,\theta}},$ $\|u\|_{H^{\gamma}_{p,\theta}}\leq
N\|u\|_{H^{\gamma-1}_{p,\theta}}+N\|(Mu)_{x}\|_{H^{\gamma-1}_{p,\theta}}\leq
N\|u\|_{H^{\gamma}_{p,\theta}}.$

Furthermore, if $\theta\neq d-1,d-1+p$, then

$\|u\|_{H^{\gamma}_{p,\theta}}\leq
N\|Mu_{x}\|_{H^{\gamma-1}_{p,\theta}},\quad\|u\|_{H^{\gamma}_{p,\theta}}\leq
N\|(Mu)_{x}\|_{H^{\gamma-1}_{p,\theta}}.$ (3.7)

  * (v)

For $i=0,1$ let
$\kappa\in[0,1],\;p_{i}\in(1,\infty),\;\gamma_{i},\;\theta_{i}\in\mathbb{R}$
and assume the relations

$\gamma=\kappa\gamma_{1}+(1-\kappa)\gamma_{0},\quad\frac{1}{p}=\frac{\kappa}{p_{1}}+\frac{1-\kappa}{p_{0}},\quad\frac{\theta}{p}=\frac{\theta_{1}\kappa}{p_{1}}+\frac{\theta_{0}(1-\kappa)}{p_{0}}.$

Then

$\|u\|_{H^{\gamma}_{p,\theta}}\leq
N\|u\|^{\kappa}_{H^{\gamma_{1}}_{p_{1},\theta_{1}}}\|u\|^{1-\kappa}_{H^{\gamma_{0}}_{p_{0},\theta_{0}}}.$

######  Remark 3.2.

Let $\theta\in(d-1,d-1+p)$ and $n$ be a nonnegative integer. By Lemma 3.1
$(iii),(iv)$

$\|M^{-n}v\|_{H^{\gamma}_{p,\theta}}\leq
N\|D^{n}v\|_{H^{\gamma-n}_{p,\theta}}$ (3.8)

for any $v\in C^{\infty}_{0}(\mathbb{R}^{d};\mathbb{R}^{d_{1}})$. Indeed,
since $\theta+mp\neq d-1,d-1+p$ for any integer $m$

$\displaystyle\|M^{-n}v\|_{H^{\gamma}_{p,\theta}}$ $\displaystyle\leq$
$\displaystyle N\|M^{-1}v\|_{H^{\gamma}_{p,\theta-(n-1)p}}\leq
N\|v_{x}\|_{H^{\gamma-1}_{p,\theta-(n-1)p}}$ $\displaystyle\leq$
$\displaystyle N\|M^{-1}v_{x}\|_{H^{\gamma-1}_{p,\theta-(n-2)p}}\leq
N\|D^{2}v\|_{H^{\gamma-2}_{p,\theta-(n-2)p}}\ldots.$

For $-\infty\leq S<T\leq\infty$, we define the Banach spaces:

$\mathbb{H}^{\gamma}_{p,\theta}(S,T):=L_{p}((S,T),H^{\gamma}_{p,\theta}),\;\mathbb{H}^{\gamma}_{p,\theta}(T):=\mathbb{H}^{\gamma}_{p,\theta}(0,T),\;\mathbb{L}_{p,\theta}(S,T):=H^{0}_{p,\theta}(S,T),\;\mathbb{L}^{\gamma}_{p,\theta}(T):=\mathbb{L}^{\gamma}_{p,\theta}(0,T)$

with norms given by

$\|u\|^{p}_{\mathbb{H}^{\gamma}_{p,\theta}(S,T)}=\int^{T}_{S}\|u(t)\|^{p}_{H^{\gamma}_{p,\theta}}dt.$

######  Lemma 3.3.

For $\phi,\psi\in C^{\infty}_{0}((S,T)\times\mathbb{R}^{d}_{+})$, define
$(\phi,\psi)=\int_{S}^{T}\int_{\mathbb{R}^{d}_{+}}\phi(s,x)\psi(t,x)dtdx$. For
$p\in(1,\infty)$ and $\gamma,\theta\in\mathbb{R}$, define
$\gamma^{\prime},p^{\prime},\theta^{\prime}$ so that

$\gamma^{\prime}=-\gamma,\quad\frac{1}{p}+\frac{1}{p^{\prime}}=1,\quad\frac{\theta}{p}+\frac{\theta^{\prime}}{p^{\prime}}=d.$

Then for any $\phi\in C^{\infty}_{0}((S,T)\times\mathbb{R}^{d}_{+})$

$\displaystyle\|\phi\|_{\mathbb{H}^{\gamma}_{p,\theta}(S,T)}\leq
N\;\sup_{\psi\in
C^{\infty}_{0}((S,T)\times\mathbb{R}^{d}_{+})}\frac{(\phi,\psi)}{\|\psi\|_{\mathbb{H}^{\gamma^{\prime}}_{p^{\prime},\theta^{\prime}}(S,T)}}\leq
N\|\phi\|_{\mathbb{H}^{\gamma}_{p,\theta}(S,T)},$

where $N$ is independent of $\phi$. Moreover the relation $(\phi,\psi)$ can be
extended by continuity on all $\phi\in\mathbb{H}^{\gamma}_{p,\theta}(S,T)$ and
$\psi\in\mathbb{H}^{\gamma^{\prime}}_{p^{\prime},\theta^{\prime}}(S,T)$, and
then it identifies the dual to $\mathbb{H}^{\gamma}_{p,\theta}(S,T)$ with
$\mathbb{H}^{\gamma^{\prime}}_{p^{\prime},\theta^{\prime}}(S,T)$.

###### Proof.

See Theorem 2.5 of [11]; this actually proves the duality between
$H^{\gamma}_{p,\theta}$ and
$H^{\gamma^{\prime}}_{p^{\prime},\theta^{\prime}}$, but the proof of our claim
is essentially the same. The only difference is that one has to consider
integrations on the time variable, too. ∎

Finally, we set $U^{\gamma}_{p,\theta}:=M^{1-2/p}H^{\gamma-2/p}_{p,\theta}$,
meaning that any $u\in U^{\gamma}_{p,\theta}$ has the form $u=M^{1-2/p}\cdot
v$ with $v\in H^{\gamma-2/p}_{p,\theta}$ and
$\|u\|_{U^{\gamma}_{p,\theta}}:=\|M^{-1+2/p}u\|_{H^{\gamma-2/p}_{p,\theta}}=\|v\|_{H^{\gamma-2/p}_{p,\theta}}$.
Using these spaces, we define our solution spaces.

######  Definition 3.4.

We write $u\in\mathfrak{H}^{\gamma+2}_{p,\theta}(S,T)$ if $u\in
M\mathbb{H}^{\gamma+2}_{p,\theta}(S,T)$, $u(S,\cdot)\in
U^{\gamma+2}_{p,\theta}$ ($u(-\infty,\cdot):=0$ if $S=-\infty$), and for some
$\tilde{f}\in M^{-1}\mathbb{H}^{\gamma}_{p,\theta}(T)$ it holds
$u_{t}=\tilde{f}$ in the sense of distributions, that is for any $\phi\in
C^{\infty}_{0}(\mathbb{R}^{d};\mathbb{R}^{d_{1}})$ the equality

$(u(t,\cdot),\phi)=(u(S,\cdot),\phi)+\int^{t}_{S}(\tilde{f}(s,\cdot),\phi)ds$
(3.9)

holds for all $t\in(S,T)$. In this case we write $u_{t}=\tilde{f}$. The norm
in $\mathfrak{H}^{\gamma+2}_{p,\theta}(S,T)$ is defined by

$\|u\|_{\mathfrak{H}^{\gamma+2}_{p,\theta}(S,T)}=\|M^{-1}u\|_{\mathbb{H}^{\gamma+2}_{p,\theta}(S,T)}+\|Mu_{t}\|_{\mathbb{H}^{\gamma}_{p,\theta}(S,T)}+\|u(S,\cdot)\|_{U^{\gamma+2}_{p,\theta}}.$

Define
$\mathfrak{H}^{\gamma+2}_{p,\theta}(T):=\mathfrak{H}^{\gamma+2}_{p,\theta}(0,T)$
and
$\mathfrak{H}^{\gamma+2}_{p,\theta}:=\mathfrak{H}^{\gamma+2}_{p,\theta}(0,\infty)$.

######  Theorem 3.5.

(i) The space $\mathfrak{H}^{\gamma+2}_{p,\theta}(S,T)$ is a Banach space.

(ii) Let $0<T<\infty$. Then for any
$u\in\mathfrak{H}^{\gamma+2}_{p,\theta}(T)$,

$\sup_{t\leq T}\|u(t)\|_{\mathbb{H}^{\gamma+1}_{p,\theta}}\leq
N(d,p,\theta,T)\|u\|_{\mathfrak{H}^{\gamma+2}_{p,\theta}(T)}.$

(iii) Let $0<T<\infty$. For any nonnegative integer $n\geq\gamma+2$, the set

$\mathfrak{H}^{\gamma+2}_{p,\theta}(T)\bigcap\bigcup_{k=1}^{\infty}C([0,T],C^{n}_{0}(G_{k}))$

where $G_{k}=(1/k,k)\times\\{|x^{\prime}|<k\\}$ is dense in
$\mathfrak{H}^{\gamma+2}_{p,\theta}(T)$.

###### Proof.

See Theorem 2.9 and Theorem 2.11 of [14]. ∎

Here are some interior Hölder estimates of functions in the space
$\mathfrak{H}^{\gamma+2}_{p,\theta}(T)$.

######  Theorem 3.6.

Let $p>2$ and assume

$2/p<\alpha<\beta\leq 1,\quad\gamma+2-\beta-d/p=k+\varepsilon,$

where $k\in\\{0,1,2,\cdots\\}$ and $\varepsilon\in(0,1]$. Denote
$\sigma=\beta-1+\theta/p$. Then for any
$u\in\mathfrak{H}^{\gamma+2}_{p,\theta}(T)$ and multi-indices $i,j$ such that
$|i|\leq j$ and $|j|=k$,

(i) the functions $D^{i}u(t,x)$ are continuous in
$[0,T]\times\mathbb{R}^{d}_{+}$ and

$M^{\sigma+|i|}D^{i}u(t,\cdot)-M^{\sigma+|i|}D^{i}u(0,\cdot)\in
C^{\alpha/2-1/p}([0,T],C(\mathbb{R}^{d}_{+}));$

(ii) there exists a constant $N=N(p,d,\alpha,\beta)$ so that

$\displaystyle\sup_{t,s\leq
T}\left(\frac{\big{|}M^{\sigma+|i|}D^{i}(u(t)-u(s))\big{|}_{C(\mathbb{R}^{d}_{+})}}{|t-s|^{\alpha/2-1/p}}+\frac{\big{[}M^{\sigma+|j|+\varepsilon}D^{j}(u(t)-u(s))\big{]}_{C^{\varepsilon}}}{|t-s|^{\alpha/2-1/p}}\right)$
(3.10) $\displaystyle\leq$ $\displaystyle
NT^{(\beta-\alpha)/2}\|u\|_{\mathfrak{H}^{\gamma+2}_{p,\theta}(T)}.$

###### Proof.

See Theorem 4.7 of [8]. ∎

######  Remark 3.7.

(see Remark 4.8 of [8] for details) For instance, if $\theta=d$,
$\gamma\geq-1$ and $\kappa_{0}=1-\frac{2}{p}-\frac{d}{p}>0$, then for any
$\kappa\in(0,\kappa_{0})$ and $u\in\mathfrak{H}^{1}_{p,\theta}(T)$ with
$u(0)=0$,

$\sup_{t\leq
T}\sup_{x,y\in\mathbb{R}^{d}_{+}}\frac{|u(t,x)-u(t,y)|}{|x-y|^{\kappa}}<\infty.$
(3.11) $\sup_{x\in\mathbb{R}^{d}_{+}}\sup_{s,t\leq
T}\frac{|u(t,x)-u(s,x)|}{|t-s|^{\kappa/2}}<\infty.$ (3.12)

Indeed, for (3.11) take $j=0,\beta=\kappa_{0}-\kappa+2/p$ and
$\varepsilon=1-\beta-d/p=\kappa=-\sigma$, then $\sigma+|j|+\varepsilon=0$ and
(3.10) yields (3.11). Also for (3.12), take
$i=0,\alpha=\kappa+2/p,\beta=1-d/p$ then $\sigma+|i|=0$, $2/p<\alpha<\beta<1$
and $\alpha/2-1/p=\kappa/2$.

For any $d_{1}\times d_{1}$ matrix $C=(c_{kr})$ we let

$|C|:=\sqrt{\sum_{k,r}(c_{kr})^{2}}.$

We set $A^{ij}=(a^{ij}_{kr})_{k,r=1,\ldots,d_{1}}$ for each $i,j=1,\ldots,d$.
Throughout the article we assume the followings.

######  Assumption 3.8.

For each $i$ and $j$, $A^{ij}$ depends only on $t$ and there exist finite
constants $\delta,K>0$ so that

$\delta|\xi|^{2}\leq\sum_{i,j=1}^{d}(\xi^{i})^{*}A^{ij}\;\xi^{i}$ (3.13)

for all (real valued) $d_{1}\times d$-matrix $\xi$, where $\xi^{i}$ denotes
the $i$-th column of $\xi$. Also, there exists a constant $K<\infty$ such that

$\left|A^{ij}\right|\leq K,\quad\forall\;\;i,j=1,\ldots,d,$ (3.14)

where $*$ means matrix transposition.

We recall (1.2) and write it as

$u^{k}_{t}=a^{ij}_{kr}(t)u^{r}_{x^{i}x^{j}}+f^{k},\quad
u^{k}(S)=u^{k}_{0},\quad\quad k=1,2,\cdots,d_{1},$ (3.15)

assuming the summation convention on indices $i,j,r$; such convention will be
used throughout the article. In short, we will write (3.15) as

$u_{t}=A^{ij}(t)u_{x^{i}x^{j}}+f,\quad u(S)=u_{0},$ (3.16)

where we regard $u,u_{0},f$ as $d_{1}\times 1$ matrix-valued functions.

######  Definition 3.9.

A $d$-dimensional distribution-valued function $u$ defined on $(S,T)$ is a
solution of (3.16) in $\mathfrak{H}^{\gamma+2}_{p,\theta}(S,T)$ if $u\in
M\mathbb{H}^{\gamma+2}_{p,\theta}(S,T)$, $u(S)\in U^{\gamma+2}_{p,\theta}$
($u(-\infty,\cdot):=0$ if $S=-\infty$) and (3.16) holds in the sense of
distributions, that is (3.9) holds with $\tilde{f}=A^{ij}u_{x^{i}x^{j}}+f$.

The following is our $L_{p}$-theory for the parabolic system (3.16). The proof
is given in section 6.

######  Theorem 3.10.

Let $p\in(1,\infty)$ and $\gamma\geq 0$. Assume $\theta\in(d+1-p,d+p-1)$ if
$p\in(1,2]$ and $\theta\in(d-1,d+1)$ if $p\in(2,\infty)$. Then for any $f\in
M^{-1}\mathbb{H}^{\gamma}_{p,\theta}(T)$ and $u_{0}\in
U^{\gamma+2}_{p,\theta}$ system (3.16) admits a unique solution
$u\in\mathfrak{H}^{\gamma+2}_{p,\theta}(T)$, and for this solution we have

$\|u\|_{\mathfrak{H}^{\gamma+2}_{p,\theta}(T)}\leq
N\left(\|Mf\|_{\mathbb{H}^{\gamma}_{p,\theta}(T)}+\|u_{0}\|_{U^{\gamma+2}_{p,\theta}}\right),$
(3.17)

where $N=N(\gamma,p,\theta,\delta,K)$.

######  Remark 3.11.

Various interior Hölder estimates of the solution in Theorem 3.10 can be
obtained according to Theorem 3.6. Also see Lemma 4.11 and Lemma 4.14.

######  Remark 3.12.

(i) The proof of Theorem 3.10 is based on a sharp function estimate (Lemma
5.7). If $d_{1}=1$, then Lemma 5.7 can be proved for any
$\theta\in(d-1,d-1+p)$ as long as $p>1$; we will prove this in a subsequent
article for parabolic equations with (weighted) BMO-coefficients.

(ii) It is known (see Remark 3.6 of [14]) that if $\theta\not\in(d-1,d-1+p)$,
then Theorem 3.10 is not true even for the heat equation $u_{t}=\Delta u+f$.

Now we present our $L_{p}$-theory for the elliptic system (1.1). The proof is
given in section 6.

######  Theorem 3.13.

Let $p\in(1,\infty)$, $\gamma\geq 0$ and $A^{ij}$ be independent of $t$.
Assume $\theta\in(d+1-p,d+p-1)$ if $p\in(1,2]$ and $\theta\in(d-1,d+1)$ if
$p\in(2,\infty)$. Then for any $f=(f^{1},f^{2},\cdots,f^{d_{1}})\in
M^{-1}H^{\gamma}_{p,\theta}$ the system (1.1) admits a unique solution $u\in
MH^{\gamma+2}_{p,\theta}$, and for this solution we have

$\|M^{-1}u\|_{H^{\gamma+2}_{p,\theta}}\leq N\|Mf\|_{H^{\gamma}_{p,\theta}},$

where $N=N(\gamma,p,\theta,\delta,K)$.

######  Remark 3.14.

Theorem 3.10 and Theorem 3.13 hold not only for $\gamma\geq 0$ but also for
any $\gamma<0$. This can be easily proved by using the results for $\gamma\geq
0$ and repeating the arguments used for single equations (see the proof of
Theorem 5.6 of [10]).

##  4 Preliminary estimates : Some local estimates of solutions

In this section we prove a version of Theorem 3.10 for $\theta=d$. This result
is used to derive some local estimates of $D^{\alpha}u$ for any multi-index
$\alpha$, where $u$ is a solution of (3.16).

First, we introduce some results for systems defined on the entire space. For
$-\infty\leq S<T\leq\infty$ we denote
$\mathbb{H}^{\gamma}_{p}(S,T):=L_{p}((S,T),H^{\gamma}_{p})$ and
$\mathbb{H}^{\gamma}_{p}(T):=\mathbb{H}^{\gamma}_{p}(0,T)$.

######  Theorem 4.1.

Let $\gamma\in\mathbb{R}$ and $-\infty\leq S<T\leq\infty$. Let
$f\in\mathbb{H}^{\gamma}_{p}(S,T)$ and $u\in\mathbb{H}^{\gamma+2}_{p}(S,T)$
satisfy

$u_{t}=A^{ij}(t)u_{x^{i}x^{j}}+f,\quad t>S,x\in\mathbb{R}^{d}.$

Additionally assume $u(S,\cdot)=0$ if $S>-\infty$. Then

$\|u_{xx}\|^{p}_{\mathbb{H}^{\gamma}_{p}(S,T)}\leq
N(d,p,\delta,K)\|f\|^{p}_{\mathbb{H}^{\gamma}_{p}(S,T)}.$ (4.1)

Also if $-\infty<S<T<\infty$, then

$\|u\|^{p}_{\mathbb{H}^{\gamma+2}_{p}(S,T)}\leq
N(d,p,\delta,K,S,T)\|f\|^{p}_{\mathbb{H}^{\gamma}_{p}(S,T)}.$

###### Proof.

This is a classical result. See, for instance, Theorem 1.1 of [15]. Actually
in [15] the theorem is proved only when $\gamma=0$, but the general case
follows by the fact the operator
$(1-\Delta)^{\mu/2}:\mathbb{H}^{\gamma}_{p}(S,T)\to\mathbb{H}^{\gamma-\mu}_{p}(S,T)$
is an isometry. ∎

Theorem 4.1 yields the following result.

######  Corollary 4.2.

Let $u\in C^{\infty}_{0}(\mathbb{R}^{d+1};\mathbb{R}^{d_{1}})$. Then

$\|u_{xx}\|^{p}_{\mathbb{H}^{\gamma}_{p}(-\infty,\infty)}\leq
N(d,p,\delta,K)\;\|u_{t}-A^{ij}u_{x^{i}x^{j}}\|^{p}_{\mathbb{H}^{\gamma}_{p}(-\infty,\infty)}.$
(4.2)

######  Corollary 4.3.

Let $0<T<\infty$, $f^{i}\in\mathbb{L}_{p}(T)$, and $u\in\mathbb{H}^{1}_{p}(T)$
satisfies

$u_{t}=A^{ij}(t)u_{x^{i}x^{j}}+f^{i}_{x^{i}},\quad
t\in(0,T),x\in\mathbb{R}^{d}$

with zero initial condition $u(0)=0$. Then

$\|u_{x}\|^{p}_{\mathbb{L}_{p}(T)}\leq
N(d,p,\delta,K)\|f^{i}\|^{p}_{\mathbb{L}_{p}(T)}.$ (4.3)
$\|u\|^{p}_{\mathbb{H}^{1}_{p}(T)}\leq
N(d,p,\delta,K,T)\|f^{i}\|^{p}_{\mathbb{L}_{p}(T)}.$

###### Proof.

Remember

$\|f^{i}_{x}\|_{H^{-1}_{p}}\leq N\|f^{i}\|_{L_{p}},\quad\|u_{x}\|_{L_{p}}\leq
N(\|u_{xx}\|_{H^{-1}_{p}}+\|u\|_{L_{p}(T)}).$

By (4.1) with $\gamma=-1$,

$\|u_{x}\|_{\mathbb{L}(T)}\leq N(\|f^{i}\|_{\mathbb{L}(T)}+\|u\|_{L_{p}(T)}).$
(4.4)

Notice that, for any constant $c>0$, the function $u^{c}(t,x):=u(c^{2}t,cx)$
satisfies

$u^{c}_{t}=A^{ij}(c^{2}t)u^{c}_{x^{i}x^{j}}+(cf^{i}(c^{2}t,cx))_{x^{i}}.$

Thus for this function (4.4) with $c^{-2}T$ in place of $T$ becomes

$\|u_{x}\|_{\mathbb{L}(T)}\leq
N(\|f^{i}\|_{\mathbb{L}(T)}+c^{-1}\|u\|_{L_{p}}).$

Now we get (4.3) by taking $c\to\infty$. ∎

######  Corollary 4.4.

Let $u\in C^{\infty}_{0}(\mathbb{R}^{d};\mathbb{R}^{d_{1}})$ and $A^{ij}$ be
independent of $t$. Then

$\|u_{xx}\|^{p}_{H^{\gamma}_{p}}\leq
N(d,p,\delta,K)\;\|A^{ij}u_{x^{i}x^{j}}\|^{p}_{H^{\gamma}_{p}}.$ (4.5)

###### Proof.

Take a nonnegative smooth function $\eta(t)\in C^{\infty}_{0}(-1,1)$ so that
$\int_{\mathbb{R}}\eta^{p}(t)dt=1$. For each $n=1,2,\cdots$, define
$\eta_{n}(t)=n^{-1/p}\eta(t/n)$. Then applying (4.2) for
$v_{n}(t,x):=\eta_{n}(t)u(x)$,

$\|u_{xx}\|^{p}_{H^{\gamma}_{p}}\leq
N\|A^{ij}u_{x^{i}x^{j}}\|^{p}_{H^{\gamma}_{p}}+N\|u\|^{p}_{H^{\gamma}_{p}}\int_{\mathbb{R}}|\eta^{\prime}_{n}|^{p}dt$

Now it is enough to let $n\to\infty$. The corollary is proved. ∎

Remember that for any $t\in\mathbb{R}$, $(x^{1},x^{\prime})\in\mathbb{R}^{d}$,
we defined

$B_{r}(x)=(x^{1}-r,x^{1}+r)\times B^{\prime}_{r}(x^{\prime}),\quad
Q_{r}(t,x)=(t,t+r^{2})\times B_{r}(x),$

where $B^{\prime}_{r}(x^{\prime})$ is the open ball in $\mathbb{R}^{d-1}$ of
radius $r$ with center $x^{\prime}$. By
$C^{\infty}_{loc}(\mathbb{R}^{d+1};\mathbb{R}^{d_{1}})$ we denote the set of
$\mathbb{R}^{d_{1}}$-valued functions $u$ defined on $\mathbb{R}^{d+1}$ and
such that $\zeta u\in C^{\infty}_{0}(\mathbb{R}^{d+1};\mathbb{R}^{d_{1}})$ for
any $\zeta\in C^{\infty}_{0}(\mathbb{R}^{d+1};\mathbb{R})$.

######  Theorem 4.5.

Let $q\in(1,\infty)$ and $(t,x)\in\mathbb{R}^{d+1}$. Then there exists a
constant $N$, depending only on $q,d,d_{1},\delta$ and $K$ so that for any
$\lambda\geq 4,r>0$ and $u\in
C^{\infty}_{loc}(\mathbb{R}^{d+1};\mathbb{R}^{d_{1}})$, we have

$\displaystyle-\int_{Q_{r}(t,x)}-\int_{Q_{r}(t,x)}|u_{xx}(s,y)-u_{xx}(r,z)|^{q}\,dsdydrdz$
$\displaystyle\leq$ $\displaystyle N\lambda^{-q}-\int_{Q_{\lambda
r}(t,x)}|u_{xx}|^{q}\,dsdy+N\lambda^{d+2}-\int_{Q_{\lambda
r}(t,x)}|u_{t}+A^{ij}u_{x^{i}x^{j}}|^{q}\,dsdy.$

###### Proof.

See Theorem 6.1.2 of [13]. Actually this theorem is proved when $d_{1}=1$, and
the proof is based on Theorem 4.1. Since Theorem 4.1 holds for any
$d_{1}=1,2,\cdots$, the theorem can be proved by repeating the proof of
Theorem 6.1.2 of [13] word for word. ∎

######  Corollary 4.6.

Let $u=u(x)\in C^{\infty}_{loc}(\mathbb{R}^{d};\mathbb{R}^{d_{1}})$ and
$A^{ij}$ be independent of $t$. Then for any $x\in\mathbb{R}^{d}$,
$\lambda\geq 4$ and $r>0$,

$\displaystyle-\int_{B_{r}(x)}-\int_{B_{r}(x)}|u_{xx}(y)-u_{xx}(z)|^{q}dydz$
$\displaystyle\leq$ $\displaystyle N\lambda^{-q}-\int_{B_{\lambda
r}(x)}|u_{xx}|^{q}dy+N\lambda^{d+2}-\int_{B_{\lambda
r}(x)}|A^{ij}u_{x^{i}x^{j}}|^{q}dy.$

From now on we consider systems defined on a half space. Remember

$\mathbb{H}^{\gamma}_{p,\theta}(S,T):=L_{p}((S,T),H^{\gamma}_{p,\theta}),\quad\quad\|u\|^{p}_{\mathbb{H}^{\gamma}_{p,\theta}(S,T)}:=\int^{T}_{S}\|u(t,\cdot)\|^{p}_{H^{\gamma}_{p,\theta}}\,dt.$

######  Lemma 4.7.

Let $\gamma,\theta\in\mathbb{R}$ and $p\in(1,\infty)$.

(i) Let $-\infty\leq S<T\leq\infty$ and suppose $u(t,x)\in
C^{\infty}_{0}(\mathbb{R}\times\mathbb{R}^{d}_{+};\mathbb{R}^{d_{1}})$
satisfies

$u_{t}+A^{ij}(t)u_{x^{i}x^{j}}=f,\quad(t,x)\in(S,T)\times\mathbb{R}^{d}_{+}$

and assume $u(T,\cdot)=0$ if $T<\infty$.

$\|M^{-1}u\|^{p}_{\mathbb{H}^{\gamma+2}_{p,\theta}(S,T)}\leq
N(p,d,\theta,\delta,K)\left(\|M^{-1}u\|^{p}_{\mathbb{H}^{\gamma+1}_{p,\theta}(S,T)}+\|Mf\|^{p}_{\mathbb{H}^{\gamma}_{p,\theta}(S,T)}\right).$
(4.6)

(ii) If $u(x)\in C^{\infty}_{0}(\mathbb{R}^{d}_{+};\mathbb{R}^{d_{1}})$ and
$A^{ij}$ is independent of $t$, then

$\|M^{-1}u\|^{p}_{H^{\gamma+2}_{p,\theta}}\leq
N(p,d,\theta,\delta,K)\left(\|M^{-1}u\|^{p}_{H^{\gamma+1}_{p,\theta}}+\|MA^{ij}u_{x^{i}x^{j}}\|^{p}_{H^{\gamma}_{p,\theta}}\right).$
(4.7)

###### Proof.

(i). We proceed as in the proof of Lemma 5.8 of [10]. Denote $S_{n}=e^{-2n}S$
and $T_{n}=e^{-2n}T$. By Lemma 3.1(iii) and (3.3),

$\displaystyle\|M^{-1}u\|^{p}_{\mathbb{H}^{\gamma+2}_{p,\theta}(S,T)}$
$\displaystyle\leq$ $\displaystyle
N\sum_{n=-\infty}^{\infty}e^{n(\theta-p)}\|\zeta(x)u(t,e^{n}x)\|^{p}_{\mathbb{H}^{\gamma+2}_{p}(S,T)}$
(4.8) $\displaystyle=$ $\displaystyle
N\sum_{n=-\infty}^{\infty}e^{n(2+\theta-p)}\|\zeta(x)u(e^{2n}t,e^{n}x)\|^{p}_{\mathbb{H}^{\gamma+2}_{p}(S_{n},T_{n})}$
$\displaystyle\leq$ $\displaystyle
N\sum_{n=-\infty}^{\infty}e^{n(2+\theta-p)}\|(\zeta(x)u(e^{2n}t,e^{n}x))_{xx}\|^{p}_{\mathbb{H}^{\gamma}_{p}(S_{n},T_{n})},$

where the last inequality is due to (3.1). Denote
$v^{n}(t,x)=\zeta(x)u(e^{2n}t,e^{n}x)$, then it satisfies

$v^{n}_{t}+A^{ij}(e^{2n}t)v^{n}_{x^{i}x^{j}}=e^{2n}\zeta(x)f(e^{2n}t,e^{n}x)+2e^{n}A^{1j}(e^{2n}t)\zeta_{x}u_{x^{j}}(e^{2n}t,e^{n}x)+A^{11}(e^{2n}t)\zeta_{xx}u(e^{2n}t,e^{n}x)$

for $(t,x)\in(S_{n},T_{n})\times\mathbb{R}^{d}_{+}$. By (4.1),

$\displaystyle\|v^{n}_{xx}\|^{p}_{\mathbb{H}^{\gamma}_{p}(S_{n},T_{n})}$
$\displaystyle\leq$ $\displaystyle
Ne^{2np}\|\zeta(x)f(e^{2n}t,e^{n}x)\|^{p}_{\mathbb{H}^{\gamma}_{p}(S_{n},T_{n})}$
$\displaystyle+$ $\displaystyle
Ne^{np}\|\zeta_{x^{i}}u_{x^{j}}(e^{2n}t,e^{n}x)\|^{p}_{\mathbb{H}^{\gamma}_{p}(S_{n},T_{n})}+N\|\zeta_{xx}u(e^{2n}t,e^{n}x)\|^{p}_{\mathbb{H}^{\gamma}_{p}(S_{n},T_{n})},$

where $N$ is independent of $n$. Plugging this into (4.8) one gets

$\displaystyle\|M^{-1}u\|^{p}_{\mathbb{H}^{\gamma+2}_{p,\theta}(S,T)}$
$\displaystyle\leq$ $\displaystyle
N\sum_{n=-\infty}^{\infty}e^{n(\theta+p)}\|\zeta(x)f(t,e^{n}x)\|^{p}_{\mathbb{H}^{\gamma}_{p}(S,T)}$
$\displaystyle+$ $\displaystyle
N\sum_{n=-\infty}^{\infty}e^{n\theta}\|\zeta_{x}u_{x}(t,e^{n}x)\|^{p}_{\mathbb{H}^{\gamma}_{p}(S,T)}+N\sum_{n=-\infty}^{\infty}e^{n(\theta-p)}\|\zeta_{xx}u(t,e^{n}x)\|^{p}_{\mathbb{H}^{\gamma}_{p}(S,T)}.$

This, (3.4) and Lemma 3.1 easily lead us to (4.6). Indeed, for instance, by
(3.4)

$\sum_{n=-\infty}^{\infty}e^{n\theta}\|\zeta_{x}u_{x}(t,e^{n}x)\|^{p}_{\mathbb{H}^{\gamma}_{p}(S,T)}\leq
N\|u_{x}\|^{p}_{\mathbb{H}^{\gamma}_{p,\theta}(S,T)}$

and by Lemma 3.1(iv) applied to $M^{-1}u$ in place of $u$,

$\|u_{x}\|_{\mathbb{H}^{\gamma}_{p,\theta}(S,T)}=\|DM(M^{-1}u)\|_{\mathbb{H}^{\gamma}_{p,\theta}(S,T)}\leq
N\|M^{-1}u\|_{\mathbb{H}^{\gamma+1}_{p,\theta}(S,T)}.$

(ii) This is proved similarly based on (4.5). The lemma is proved. ∎

######  Remark 4.8.

Let $\gamma\geq 0$. By iterating (4.6), one gets

$\displaystyle\|M^{-1}u\|^{p}_{\mathbb{H}^{\gamma+2}_{p,\theta}(S,T)}$
$\displaystyle\leq$ $\displaystyle
N\|M^{-1}u\|^{p}_{\mathbb{L}_{p,\theta}(S,T)}+N\|Mf\|^{p}_{\mathbb{H}^{\gamma}_{p,\theta}(S,T)}$
$\displaystyle\leq$ $\displaystyle
N\|Mu_{xx}\|^{p}_{\mathbb{L}_{p,\theta}(S,T)}+N\|Mf\|^{p}_{\mathbb{H}^{\gamma}_{p,\theta}(S,T)},$

where for the second inequality we use (3.7) twice. We use both inequalities
later to estimate $\|M^{-1}u\|^{p}_{\mathbb{H}^{\gamma+2}_{p,\theta}(S,T)}$.

Let $(w^{1}_{t},w^{2}_{t},\cdots,w^{d}_{t})$ be a $d$-dimensional Wiener
process defined on a probability space $(\Omega^{\prime},\mathcal{F},P)$.
Denote

$\xi_{t}=w^{1}_{t}\sqrt{2}+2t,\quad\eta_{t}=(\sqrt{2}\int^{t}_{0}e^{\xi_{s}}dw^{2}_{s},\cdots,\sqrt{2}\int^{t}_{0}e^{\xi_{s}}dw^{d}_{s})$

and define $d\times d$ matrix-valued process $\sigma_{t}$ so that
$(\sigma_{t}x)^{1}=e^{\xi_{t}}x^{1}$ and
$(\sigma_{t}x)^{\prime}=x^{\prime}+x^{1}\eta_{t}$. It is easy to check (see
[10], p.1628) that $x_{t}(x):=\sigma_{t}x$ is the unique solution of the
stochastic differential equation

$dx_{t}=\sqrt{2}x^{1}_{t}dw_{t}+3x^{1}_{t}e_{1}dt,\quad x_{0}(x)=x,$

where $e_{1}=(1,0,\cdots,0)$. For any $f\in
C^{\infty}_{0}(\mathbb{R}^{d}_{+})$ and $x\in\mathbb{R}^{d}$, define

$\mathcal{E}f(x)=\mathbb{E}\int^{\infty}_{0}f(\sigma_{t}x)\,dt:=\int_{\Omega^{\prime}}\int^{\infty}_{0}f(\sigma_{t}x)\,dtdP.$

(See below for the convergence of this integral). Note that if $x^{1}\leq 0$
then $(\sigma_{t}x)^{1}\leq 0$ and thus $\mathcal{E}f(x)=0$. Denote

$\mathcal{L}u:=M^{2}\Delta u+3MD_{1}u=\sum_{i=1}^{d}(MD_{i})^{2}+2MD_{1}.$

######  Lemma 4.9.

Let $f\in C^{\infty}_{0}(\mathbb{R}^{d}_{+})$.

(i) $\mathcal{E}f\in L_{p}(\mathbb{R}^{d})$ and $f=\mathcal{L}(\mathcal{E}f)$
in the sense of distributions on $\mathbb{R}^{d}$.

(ii) There exist $f^{1},f^{2},\cdots,f^{d}\in L_{p}(\mathbb{R}^{d})$ so that
$f=MD_{i}f^{i}$ in the sense of distributions on $\mathbb{R}^{d}$, and

$\sum_{i=1}^{d}\|f^{i}\|_{L_{p}(\mathbb{R}^{d})}\leq
N\|f\|_{L_{p}(\mathbb{R}^{d}_{+})}.$

###### Proof.

By Theorem 2.11 of [10] (with $\theta=d$ and $b=3$ there), the map
$\mathcal{L}$ is a bounded one-to-one operator from $H^{2}_{p,d}$ onto
$L_{p,d}$, and its inverse ($:=\mathcal{L}^{-1}$) is also bounded. Denote
$u:=\mathcal{L}^{-1}f\in H^{2}_{p,d}$. By Lemma 3.1(i), there exists a
sequence $u_{n}\in C^{\infty}_{0}(\mathbb{R}^{d}_{+})$ so that $u_{n}\to u$ in
$H^{2}_{p,d}$. Denote $f_{n}(x):=\mathcal{L}u_{n}(x)$ for each
$x\in\mathbb{R}^{d}$. Then

$\mathcal{L}u_{n}\to\mathcal{L}u\,\,(=f)\quad\text{in}\quad
L_{p,d}\quad\quad\text{and}\quad\quad\|u_{n}-u_{m}\|_{H^{2}_{p,d}}\leq
N\|f_{n}-f_{m}\|_{L_{p,d}}.$ (4.9)

Obviously $u_{n}(x)=f_{n}(\sigma_{t}x)=0$ if $x^{1}\leq 0$. By Itô’s formula
(see (2.10) in [10] for details), we get

$u_{n}(x)=\mathbb{E}\int^{\infty}_{0}f_{n}(\sigma_{t}x)\,dt,\quad\quad\forall
x\in\mathbb{R}^{d}.$

The convergence of this improper integral is discussed in the proof of Theorem
2.11 of [10]. Actually there it is shown that for any $h\in
C^{\infty}_{0}(\mathbb{R}^{d}_{+})$ (here, $\theta=d$ and $b=3$ in our case),

$\mathbb{E}\int^{\infty}_{0}\|h(\sigma_{t}x)\|_{L_{p,d}}dt\leq
N\|h\|_{L_{p,d}}\int^{\infty}_{0}e^{-(\theta-d+1)(b-1)t+(\theta-d+1)^{2}t}dt=N\|h\|_{L_{p,d}},$
(4.10)

which also implies

$\|u_{n}-\mathcal{E}f\|_{L_{p,d}}=\|\mathbb{E}\int^{\infty}_{0}f_{n}(\sigma_{t}x)\,dt-\mathbb{E}\int^{\infty}_{0}f(\sigma_{t}x)\,dt\|_{L_{p,d}}\leq
N\|f_{n}-f\|_{L_{p,d}}\to 0\quad\text{as}\quad n\to\infty.$

Note $L_{p,d}=L_{p}(\mathbb{R}^{d}_{+})$. Since $u_{n}(x),f_{n}(x),f(x)$ and
$\mathcal{E}f$ vanish if $x^{1}\leq 0$, it follows that

$\|u_{n}-\mathcal{E}f\|_{L_{p}(\mathbb{R}^{d})}\to
0,\quad\quad\|f_{n}-f\|_{L_{p}(\mathbb{R}^{d})}\to 0$ (4.11)

as $n\to\infty$. Also (4.9) and fact
$\|u_{n}\|_{H^{2}_{p,d}}=\|\mathcal{L}^{-1}f_{n}\|_{H^{2}_{p,d}}\leq
N\|f_{n}\|_{L_{p,d}}$ show that $\\{MDu_{n}:n=1,2,\cdots\\}$ is a Cauchy
sequence in $L_{p}(\mathbb{R}^{d})$. Indeed, since each $u_{n}$ has compact
support in $\mathbb{R}^{d}_{+}$,

$\|MDu_{n}-MDu_{m}\|_{L_{p}(\mathbb{R}^{d})}=\|MDu_{n}-MDu_{m}\|_{L_{p,d}}\leq
N\|u_{n}-u_{m}\|_{H^{1}_{p,d}}\leq N\|f_{n}-f_{m}\|_{L_{p,d}}.$

Let $\mathcal{L}^{*}$ denote the adjoint operator of $\mathcal{L}$. For any
$\phi\in C^{\infty}_{0}(\mathbb{R}^{d})$, by (4.11),

$(f,\phi)=\lim_{n\to\infty}(f_{n},\phi)=\lim_{n\to\infty}(\mathcal{L}u_{n},\phi)=\lim_{n\to\infty}(u_{n},\mathcal{L}^{*}\phi)=(\mathcal{E}f,\mathcal{L}^{*}\phi)=(\mathcal{L}(\mathcal{E}f),\phi).$

Thus $f=\mathcal{L}(\mathcal{E}f)$ in the sense of distributions on
$\mathbb{R}^{d}$. Also since $u_{n}\to\mathcal{E}f$ in $L_{p}(\mathbb{R}^{d})$
and $\\{MDu_{n}\\}$ is a Cauchy sequence in $L_{p}(\mathbb{R}^{d})$, we have
$MD\mathcal{E}f\in L_{p}(\mathbb{R}^{d})$. Consequently,

$f=\mathcal{L}(\mathcal{E}f)=MD_{1}(MD_{1}\mathcal{E}f+2\mathcal{E}f)+\sum_{j=2}^{d}MD_{j}\mathcal{E}f=:\sum_{i=1}^{d}MD_{i}f^{i},$

and by (4.11),

$\sum_{i}\|f^{i}\|_{L_{p}(\mathbb{R}^{d})}=\lim_{n\to\infty}(\|u_{n}\|_{L_{p}}+\|MDu_{n}\|_{L_{p}})\leq\lim_{n\to\infty}\|u_{n}\|_{H^{2}_{p,d}}\leq
N\|f_{n}\|_{L_{p,d}}=N\|f\|_{L_{p,d}}.$

The lemma is proved. ∎

Now we prove a version of Theorem 3.10 for $\theta=d$.

######  Lemma 4.10.

Let $-\infty<S<T<\infty$, $p\in(1,\infty)$ and $n=0,1,2\cdots$. For any $f\in
M^{-1}\mathbb{H}^{n}_{p,d}(S,T)$, the equation

$u_{t}+A^{ij}(t)u_{x^{i}x^{j}}=f,\quad(t,x)\in(S,T)\times\mathbb{R}^{d}_{+}$

with the condition $u(T)=0$ has a unique solution
$u\in\mathfrak{H}^{n+2}_{p,d}(S,T)$, and for this solution

$\|M^{-1}u\|_{\mathbb{H}^{n+2}_{p,d}(S,T)}\leq
N(p,d,\delta,K)\|Mf\|_{\mathbb{H}^{n}_{p,d}(S,T)}.$ (4.12)

###### Proof.

As usual we only need to prove that the estimate (4.12) holds given that a
solution $u$ already exists. Furthermore we may assume $u(t,x)\in
C^{\infty}_{0}(\mathbb{R}\times\mathbb{R}^{d}_{+};\mathbb{R}^{d_{1}})$. Due to
Remark 4.8 and the inequality $\|M^{-1}u\|_{L_{p,d}}\leq
N(p,d)\|u_{x}\|_{L_{p,d}}$ (see Lemma 3.1(iv)), we only need to prove

$\|u_{x}\|_{\mathbb{L}_{p,d}(S,T)}\leq N\|Mf\|_{\mathbb{L}_{p,d}(S,T)}.$
(4.13)

By Lemma 4.9, we can write $Mf=MD_{i}f^{i}$ on $\mathbb{R}^{d}$ (thus
$f=D_{i}f^{i}$), where $f^{i}=(f^{i1},\cdots,f^{id_{1}})$, so that
$f^{i}\in\mathbb{L}_{p}(S,T)$ (not only in $\mathbb{L}_{p,d}(S,T)$) and

$\sum_{i=1}^{d}\|f^{i}\|_{\mathbb{L}_{p}(S,T)}\leq
N\|Mf\|_{\mathbb{L}_{p,d}(S,T)}.$

Thus by Corollary 4.3,

$\|u_{x}\|_{\mathbb{L}_{p,d}(S,T)}=\|u_{x}\|_{\mathbb{L}_{p}(S,T)}\leq
N\|f^{i}\|_{\mathbb{L}_{p}(S,T)}\leq N\|Mf\|_{\mathbb{L}_{p,d}(S,T)}.$

The lemma is proved. ∎

For $r,a>0$, denote

$Q_{r}(a)=Q_{r}(0,a,0)=(0,r^{2})\times(a-r,a+r)\times B^{\prime}_{r}(0),\quad
U_{r}=(-r^{2},r^{2})\times(-2r,2r)\times B^{\prime}_{r}(0).$

######  Lemma 4.11.

Let $0<s<r<\infty$, $u(t,x)\in
C^{\infty}_{0}(\mathbb{R}\times\mathbb{R}^{d}_{+};\mathbb{R}^{d_{1}})$ and

$u_{t}+A^{ij}(t)u_{x^{i}x^{j}}=0\quad\quad\text{for}\quad(t,x)\in Q_{r}(r).$

Then for any multi-index $\beta=(\beta^{1},\cdots,\beta^{d})$ there exists a
constant $N=N(p,|\beta|)$ so that the inequality

$\displaystyle\int_{Q_{s}(s)}\left(|M^{-1}D^{\beta}u|^{p}+|D^{\beta}u_{x}|^{p}+|MD^{\beta}u_{xx}|^{p}\right)(x^{1})^{\theta-d}dxdt$
(4.14) $\displaystyle\leq$ $\displaystyle
N(1+r)^{|\beta|p}\cdot(1+(r-s)^{-2})^{(|\beta|+1)p}\int_{Q_{r}(r)}|Mu(t,x)|^{p}(x^{1})^{\theta-d}dxdt$

holds for $\theta=d$.

###### Proof.

To prove (4.14) we use induction on $|\beta|$. Firstly, consider the case
$|\beta|=0$. We modify the proof of Lemma 2.4.4 of [13]. Denote $r_{0}=s$ and
$r_{m}=s+(r-s)\sum_{j=1}^{m}2^{-j}$ for $m=1,2,\cdots$. choose a smooth
function $\zeta_{m}$ so that $0\leq\zeta_{m}\leq 1$,

$\zeta_{m}=1\quad\text{on}\quad
U_{r_{m}},\quad\quad\zeta_{m}=0\quad\text{on}\quad\Omega\setminus
U_{r_{m+1}},$ $|\zeta_{mx}|\leq N(r-s)^{-1}2^{m},\quad|\zeta_{mxx}|\leq
N(r-s)^{-2}2^{2m},\quad|\zeta_{mt}|\leq N(r-s)^{-2}2^{2m}.$

Note that $(u\zeta_{m})(r^{2},x)=0$ on $\mathbb{R}^{d}_{+}$, and it satisfies

$(u\zeta_{m})_{t}+A^{ij}(u\zeta_{m})_{x^{i}x^{j}}=\zeta_{mt}u+2A^{ij}(u\zeta_{m+1})_{x^{i}}\zeta_{mx^{j}}+A^{ij}u\zeta_{mx^{i}x^{j}}=:f_{m},\quad\quad(t,x)\in(0,r^{2})\times\mathbb{R}^{d}_{+}.$

By Lemma 4.10 for $\gamma=0$,

$A_{m}:=\|M^{-1}u\zeta_{m}\|_{\mathbb{H}^{2}_{p,d}(r^{2})}\leq
N\|Mf_{m}\|_{\mathbb{L}_{p,d}(r^{2})}.$

Denote $B:=\left(\int_{Q_{r}(r)}|Mu|^{p}dxdt\right)^{1/p}$. Then

$\|\zeta_{mt}Mu+A^{ij}Mu\zeta_{mx^{i}x^{j}}\|_{\mathbb{L}_{p,d}(r^{2})}\leq
N(r-s)^{-2}2^{2m}(\int_{Q_{r}(r)}|Mu|^{p}dxdt)^{1/p}=N(r-s)^{-2}2^{2m}B,$
$\|A\zeta_{mx}M(u\zeta_{m+1})_{x}\|_{\L_{p,d}(r^{2})}\leq
N(r-s)^{-1}2^{m}\|M(u\zeta_{m+1})_{x}\|_{\mathbb{L}_{p,d}(r^{2})}\leq
N(r-s)^{-1}2^{m}\|u\zeta_{m+1}\|_{\mathbb{H}^{1}_{p,d}(r^{2})},$

and by Lemma 3.1 (v) (take
$p_{0}=p_{1}=p,\gamma=1,\gamma_{0}=0,\gamma_{1}=2,\theta=d,\theta_{0}=d+p,\theta_{1}=d-p$
and $\kappa=1/2$) for any $\varepsilon>0$

$(r-s)^{-1}2^{m}\|u\zeta_{m+1}\|_{\mathbb{H}^{1}_{p,d}(r^{2})}\leq\varepsilon
A_{m+1}+\varepsilon^{-1}(r-s)^{-2}2^{2m}B.$

It follows (with $\varepsilon$ different from the one above),

$A_{m}\leq\varepsilon A_{m+1}+N(1+\varepsilon^{-1})(r-s)^{-2}2^{2m}B.$

We take $\varepsilon=\frac{1}{16}$ and get

$\varepsilon^{m}A_{m}\leq\varepsilon^{m+1}A_{m+1}+N\varepsilon^{m}(1+\varepsilon^{-1})2^{2m}(r-s)^{-2}B,$
$A_{0}+\sum_{m=1}^{\infty}\varepsilon^{m}A_{m}\leq\sum_{m=1}^{\infty}\varepsilon^{m}A_{m}+N(r-s)^{-2}B.$

Note that the series $\sum_{m=1}\varepsilon^{m}A_{m}$ converges because
$A_{m}\leq N2^{2m}\|M^{-1}u\|_{\mathbb{H}^{2}_{p,d}(r^{2})}$. By Lemma
3.1(iii), for any $M^{-1}w\in H^{2}_{p,\theta}$,

$\|M^{-1}w\|_{H^{2}_{p,\theta}}\sim(\|M^{-1}w\|_{L_{p,\theta}}+\|w_{x}\|_{L_{p,\theta}}+\|Mw_{xx}\|_{L_{p,\theta}}).$
(4.15)

Therefore,

$\int_{Q_{s}(s)}\left(|M^{-1}u|^{p}+|u_{x}|^{p}+|Mu_{xx}|^{p}\right)dxdt\leq
NA^{p}_{0}\leq N(r-s)^{-2p}\int_{Q_{r}(r)}|u(t,x)|^{p}(x^{1})^{p}dxdt.$

Next assume that (4.14) holds whenever $s<r$ and $|\beta^{\prime}|=k$, that is

$\displaystyle\int_{Q_{s}(s)}\left(|M^{-1}D^{\beta^{\prime}}u|^{p}+|D^{\beta^{\prime}}u_{x}|^{p}+|MD^{\beta^{\prime}}u_{xx}|^{p}\right)(x^{1})^{\theta-d}dxdt$
$\displaystyle\leq$ $\displaystyle
N(1+r)^{kp}\cdot(1+(r-s)^{-2})^{(k+1)p}\int_{Q_{r}(r)}|Mu(t,x)|^{p}(x^{1})^{\theta-d}dxdt$

Let $|\beta|=k+1$ and $D^{\beta}=D_{i}D^{\beta^{\prime}}$ for some $i$ and
$\beta^{\prime}$ with $|\beta^{\prime}|=k$. Fix a smooth function $\eta$ so
that $\eta=1$ on $U_{s}$, $\eta=0$ on $\Omega\setminus U_{(r+s)/2}$,
$|\eta_{x}|\leq N(r-s)^{-1},|\eta_{xx}|\leq N(r-s)^{-2}$ and $|\eta_{t}|\leq
N(r-s)^{-2}$. Note that $v:=\eta D^{\beta}u$ satisfies $v(r^{2},\cdot)=0$ and

$v_{t}+A^{ij}v_{x^{i}x^{j}}=f:=\eta_{t}D^{\beta}u+2A^{ij}\eta_{x^{i}}D^{\beta}u_{x^{j}}+A^{ij}\eta_{x^{i}x^{j}}D^{\beta}u,\quad\quad(t,x)\in(0,r^{2})\times\mathbb{R}^{d}_{+}.$

By Lemma 4.10 for $\gamma=0$ (also note that $x^{1}\leq r$ on the support of
$\eta$ and $(r-s)^{-1}\leq 1+(r-s)^{-2}$),

$\displaystyle\|M^{-1}v\|^{p}_{\mathbb{H}^{2}_{p,d}(r^{2})}$
$\displaystyle\leq$ $\displaystyle
N\|M\eta_{t}D^{\beta}u+2A\eta_{x}MD^{\beta}u_{x}+MA\eta_{xx}D^{\beta}u\|^{p}_{\mathbb{L}_{p,d}(r^{2})}$
$\displaystyle\leq$ $\displaystyle
N(1+r)^{p}(1+(r-s)^{-2})^{p}\int_{Q_{(s+r)/2}((s+r)/2)}\left(|D^{\beta}u|^{p}+|MD^{\beta}u_{x}|^{p}\right)dxdt$
$\displaystyle\leq$ $\displaystyle
N(1+r)^{p}(1+(r-s)^{-2})^{p}\int_{Q_{(s+r)/2}((s+r)/2)}\left(|D^{\beta^{\prime}}u_{x}|^{p}+|MD^{\beta^{\prime}}u_{xx}|^{p}\right)dxdt.$

This and (4.15) show that the induction goes through, and hence the lemma is
proved. ∎

######  Remark 4.12.

The proof of Lemma 4.11 mainly depends on Lemma 4.10 and it can be easily
checked that the assertion of Lemma 4.11 holds for $\theta=\theta_{0}$
whenever Lemma 4.10 is true for $\theta=\theta_{0}$. Thus due to Theorem 3.10
(which will be proved in section 6), Lemma 4.11 holds for
$\theta\in(d+1-p,d+p-1)$ if $p\in(1,2]$ and $\theta\in(d-1,d+1)$ if
$p\in(2,\infty)$.

######  Lemma 4.13.

Let $u(t,x)\in
C^{\infty}_{0}(\mathbb{R}\times\mathbb{R}^{d}_{+};\mathbb{R}^{d_{1}})$. Then
for any $T>0$, $p>1$ and $n=0,1,2,\cdots$,

$\sup_{t\in[0,T]}\|u(t,\cdot)\|_{H^{n}_{p,\theta}}\leq
N(\|u\|_{\mathbb{H}^{n}_{p,\theta}(T)}+\|u_{t}\|_{\mathbb{H}^{n}_{p,\theta}(T)}).$

###### Proof.

See p. $66$ of [13]; actually in this book, weights are not used and hence we
give an outline of the proof. First of all, it is easy to check that for any
$\phi=\phi(t)\in W^{1}_{p}((0,T))$ (cf. p.32 of [13])

$\sup_{t\leq T}|\phi(t)|^{p}\leq
N\int^{T}_{0}(|\phi|^{p}+|\phi^{\prime}(t)|^{p})dt.$

Thus it suffices to prove

$\phi(t):=\|u(t,\cdot)\|_{H^{n}_{p,\theta}}\in
W^{1}_{p}((0,T)),\quad|\phi^{\prime}(t)|\leq\|u_{t}(t,\cdot)\|_{H^{n}_{p,\theta}}.$
(4.16)

One can prove (4.16) by repeating the proof of Exercise 2.4.8 on p.71 of [13].
It is enough to replace $H^{n}_{p}$ there by $H^{n}_{p,\theta}$. ∎

######  Lemma 4.14.

Let $\theta\leq d$, $p>1$, $s\in(0,r)$ and $u\in
C^{\infty}_{loc}(\Omega;\mathbb{R}^{d_{1}})$ satisfies
$u_{t}+A^{ij}(t)u_{x^{i}x^{j}}=0$ for $(t,x)\in Q_{r}(r)$. Then for any multi-
index $\beta=(\beta^{1},\beta^{2},\cdots,\beta^{d})$,

$\displaystyle\max_{(t,x)\in
Q_{s}(s)}(|D^{\beta}u_{xx}|^{p}+|D^{\beta}u_{t}|^{p})\leq
N\int_{Q_{r}(r)}|u|^{p}(x^{1})^{\theta-d+p}dxdt,$

where $N=N(s,r,\beta,p,\delta,K)$.

###### Proof.

Choose the smallest integer $n$ so that $np>d$. Let $v\in
C^{\infty}_{0}(\mathbb{R}^{d}_{+})$ satisfy $v(x)=0$ for $x^{1}\geq 2r$. The
by Lemma 3.1 $(ii)$ with $\gamma=n$, $i=0$, $\theta=d$ and $u=M^{-n}v$,

$\sup_{x}|v(x)|\leq N(r)\sup_{x}|M^{d/p}M^{-n}v(x)|\leq
N\|M^{-n}v\|_{H^{n}_{p,d}}\leq N(r,p,n)\|D^{n}v\|_{L_{p,d}},$ (4.17)

where for the last inequality we use Remark 3.2.

Fix $\kappa\in(s,r)$. Let $\psi$ be a smooth function so that $\psi(x)=1$ for
$(t,x)\in Q_{s}(s)$ and $\psi=0$ for $(t,x)\not\in U_{\kappa}$. It follows
from (4.17) and Lemma 4.13 that

$\displaystyle\max_{Q_{s}(s)}\left(|D^{\beta}u_{xx}|+|D^{\beta}u_{t}|\right)$
$\displaystyle\leq$ $\displaystyle N\max_{(t,x)\in Q_{s}(s)}|(D^{\beta}\psi
u)_{xx}|$ $\displaystyle\leq$ $\displaystyle
N\max_{t\in[0,s^{2}]}\|D^{n}(D^{\beta}\psi u)_{xx}\|_{L_{p,d}}$
$\displaystyle\leq$ $\displaystyle N\left(\|D^{n}(D^{\beta}\psi
u)_{xx}\|_{\mathbb{L}_{p,d}(s^{2})}+\|D^{n}(D^{\beta}\psi
u_{t})_{xx}\|_{\mathbb{L}_{p,d}(s^{2})}\right)$ $\displaystyle\leq$
$\displaystyle N\sum_{|\alpha|\leq
n+|\beta|+4}\int_{Q_{\kappa}(\kappa)}|D^{\alpha}u|^{p}\,dxdt$
$\displaystyle\leq$ $\displaystyle N\int_{Q_{r}(r)}|u|^{p}(x^{1})^{p}dxdt\leq
N\int_{Q_{r}(r)}|u|^{p}(x^{1})^{\theta-d+p}dxdt,$

where the last inequality is due to the fact that $1\leq
N(r)(x^{1})^{\theta-d}$ for $x^{1}\leq 2r$. The lemma is proved. ∎

######  Remark 4.15.

Actually by inspecting the proof of Lemma 4.14 it can be easily shown that if
Lemma 4.11 holds for some $\theta_{0}\in(d-1,d-1+p)$ then Lemma 4.14 holds for
any $\theta\in(d-1,\theta_{0}]$.

##  5 Main estimates : Sharp function estimations

Remember that we denote

$\nu_{\alpha}(dx)=\nu^{1}_{\alpha}(dx^{1})dx^{\prime}:=(x^{1})^{\alpha}dx^{1}dx^{\prime}.$

The following is a weighted version of Poincaré’s inequality.

######  Lemma 5.1.

Let $\alpha\geq 0$, $p\in[1,\infty)$, $D_{r}(a):=(a-r,a+r)\times
B^{\prime}_{r}(0)\subset\mathbb{R}^{d}_{+}$ , and $u\in
C^{\infty}_{loc}(\mathbb{R}^{d}_{+};\mathbb{R}^{d_{1}})$. Then

$\displaystyle\int_{D_{r}(a)}\int_{D_{r}(a)}|u(x)-u(y)|^{p}\nu_{\alpha}(dx)\;\nu_{\alpha}(dy)\leq
2^{\alpha+2}(2r)^{p}|D_{r}(a)|\int_{D_{r}(a)}|u_{x}(x)|^{p}\nu_{\alpha}(dx),$
(5.1)

where $|D_{r}(a)|:=\nu_{\alpha}(D_{r}(a))$ and we define

$\displaystyle\int_{A}|f(x)|^{p}\nu_{\alpha}(dx)=\sum_{k=1}^{d_{1}}\int_{A}|f^{k}(x)|^{p}\nu_{\alpha}(dx)$

for $\mathbb{R}^{d_{1}}$-valued function $f$ and $A\subset\Omega$.

###### Proof.

We use the outline of the proof of Theorem 10.2.5 of [13]. Without loss of
generality we may assume $d_{1}=1$. For $x,y\in D_{r}(a)$ we have

$\displaystyle|u(x)-u(y)|^{p}\leq(2r)^{p}\int^{1}_{0}|u_{x}(tx+(1-t)y)|^{p}dt$

and the left-hand side of $(\ref{2010.03.17.1})$ is less than

$\displaystyle(2r)^{p}\int^{1}_{0}I(t)dt=2(2r)^{p}\int^{1}_{1/2}I(t)dt,$

where

$\displaystyle
I(t):=\int_{D_{r}(a)}\int_{D_{r}(a)}|u_{x}(tx+(1-t)y)|^{p}\nu_{\alpha}(dx)\;\nu_{\alpha}(dy)$

and $I$ satisfies $I(t)=I(1-t)$. For each $t\in[1/2,1]$ and $y$, substituting
$w=tx+(1-t)y$ and noticing $x^{1}=(w^{1}-(1-t)y^{1})/t\leq w^{1}/t$ since
$y^{1}\geq 0$, we get

$\displaystyle I(t)$ $\displaystyle\leq$ $\displaystyle
t^{-\alpha-1}\int_{D_{r}(a)}\left(\int_{tD_{r}(a)+(1-t)y}|u_{x}(w)|^{p}\nu_{\alpha}(dw)\right)\nu_{\alpha}(dy)$
$\displaystyle\leq$ $\displaystyle
2^{\alpha+1}\int_{D_{r}(a)}\left(\int_{D_{r}(a)}|u_{x}(x)|^{p}\nu_{\alpha}(dx)\right)\nu_{\alpha}(dy)$
$\displaystyle=$ $\displaystyle
2^{\alpha+1}|D_{r}(a)|\int_{D_{r}(a)}|u_{x}(x)|^{p}\nu_{\alpha}(dx)$

with the observation $tD_{r}(a)+(1-t)y:=\\{tz+(1-t)y:z\in D_{r}(a)\\}\subset
D_{r}(a)$. Now, (5.1) follows. ∎

######  Lemma 5.2.

Let $\alpha>-1$. Recall $\nu_{\alpha}^{1}(dx^{1})=(x^{1})^{\alpha}dx^{1}$. For
any $B^{1}_{r}(a)\subset\mathbb{R}_{+}$ we have a non-negative function
$\zeta\in C^{\infty}_{0}(\mathbb{R}_{+};\mathbb{R})$ such that

$\displaystyle supp(\zeta)\in
B^{1}_{r/2}(a),\quad\int_{B^{1}_{r}(a)}\zeta(x^{1})\nu^{1}_{\alpha}(dx^{1})=1,\quad\sup\zeta\cdot|B^{1}_{r}(a)|\leq
N,\quad\sup|\zeta_{x^{1}}|\cdot|B^{1}_{r}(a)|\leq\frac{N}{r},$ (5.2)

where $N=N(\alpha)$ and $|B^{1}_{r}(a)|=\nu_{\alpha}^{1}(B^{1}_{r}(a))$.

###### Proof.

Choose a nonnegative smooth function $\psi=\psi(x^{1})\in
C^{\infty}_{0}(B^{1}_{1/2}(0))$ so that
$\int_{\mathbb{R}}\psi(x^{1})dx^{1}=1$. Define

$\zeta(x^{1})=\frac{(x^{1})^{-\alpha}}{r}\psi(\frac{x^{1}-a}{r}).$

Then the first and the second of (5.2) are obvious.

Case 1: Let $\alpha\geq 0$. Since $r\leq a$ and
$(a+r)^{\alpha+1}-(a-r)^{\alpha+1}\leq 2r(\alpha+1)(2a)^{\alpha}$, the third
follows:

$\displaystyle\sup|\zeta|\cdot|B^{1}_{r}(a)|\leq$ $\displaystyle N$
$\displaystyle\sup_{|x^{1}-a|\leq
r/2}\frac{(x^{1})^{-\alpha}}{r}\cdot((a+r)^{\alpha+1}-(a-r)^{\alpha+1})$
$\displaystyle\leq$ $\displaystyle
N\frac{(a/2)^{-\alpha}}{r}\cdot((a+r)^{\alpha+1}-(a-r)^{\alpha+1})\leq N.$

Similarly, the last inequality holds by

$\displaystyle\sup|\zeta_{x^{1}}|\cdot|B^{1}_{r}(a)|$ $\displaystyle\leq$
$\displaystyle N\sup_{|x^{1}-a|\leq
r/2}\left(\frac{(x^{1})^{-\alpha}}{r^{2}}+\frac{(x^{1})^{-\alpha-1}}{r}\right)\cdot((a+r)^{\alpha+1}-(a-r)^{\alpha+1})$
$\displaystyle\leq$
$\displaystyle\frac{N}{r}(1+\frac{(2a)^{\alpha+1}}{(a/2)^{\alpha+1}})\leq\frac{N}{r}.$

Case 2: Let $\alpha\in(-1,0)$. First assume $r\leq a/2$. Then by mean value
theorem $(a+r)^{\alpha+1}-(a-r)^{\alpha+1}\leq 2r(\alpha+1)(a/2)^{\alpha}$ and
thus the right term of (5) is bounded by a constant $N$. If $r\in[a/2,a]$,
then

$\sup_{|x^{1}-a|\leq
r/2}\frac{(x^{1})^{-\alpha}}{r}\cdot((a+r)^{\alpha+1}-(a-r)^{\alpha+1})\leq\frac{(2a)^{-\alpha}}{a/2}(2a)^{\alpha+1}\leq
N.$

One can handle $\sup|\zeta_{x^{1}}|\cdot|B^{1}_{r}(a)|$ similarly. The lemma
is proved. ∎

Now we consider the system

$\displaystyle
u_{t}+A^{ij}u_{x^{i}x^{j}}=f^{i}_{x^{i}}+g,\quad(t,x)\in\Omega=\mathbb{R}\times\mathbb{R}^{d}_{+};\quad
f^{i}=(f^{1i},\ldots,f^{d_{1}i}),$ (5.4)

i.e.,

$\displaystyle
u^{k}_{t}+a^{ij}_{kr}u^{r}_{x^{i}x^{j}}=f^{ki}_{x^{i}}+g^{k},\quad
k=1,2,\ldots,d_{1}.$

Recall that for $t\in\mathbb{R}$, $a\in\mathbb{R}_{+}$ and
$x^{\prime}\in\mathbb{R}^{d-1}$

$Q_{r}(t,a,x^{\prime}):=(t,t+r^{2})\times(a-r,a+r)\times
B^{\prime}_{r}(x^{\prime}),\quad Q_{r}(a):=Q_{r}(0,a,0).$

By $C^{\infty}_{loc}(\Omega;\mathbb{R}^{d_{1}})$ we denote the set of
$\mathbb{R}^{d_{1}}$-valued functions $u$ defined on $\Omega$ and such that
$\zeta u\in C^{\infty}_{0}(\Omega;\mathbb{R}^{d_{1}})$ for any $\zeta\in
C^{\infty}_{0}(\Omega;\mathbb{R})$.

######  Lemma 5.3.

Let $\alpha\geq 0$, $p\in[1,\infty)$, $f^{i},g\in
C^{\infty}_{loc}(\Omega;\mathbb{R}^{d_{1}})$. Assume that $u\in
C^{\infty}_{loc}(\Omega;\mathbb{R}^{d_{1}})$ satisfies (5.4) on
$Q_{r}(a)\subset\Omega$. Then

$\displaystyle\int_{Q_{r}(a)}\left|u(t,x)-u_{Q_{r}(a)}\right|^{p}\mu_{\alpha}(dtdx)\leq
Nr^{p}\int_{Q_{r}(a)}(|u_{x}(t,x)|^{p}+|f(t,x)|^{p}+r^{p}|g(t,x)|^{p})\mu_{\alpha}(dtdx),$
(5.5)

where $N=N(\theta,\alpha,p,d,d_{1},K)$.

###### Proof.

We follow the outline of the proof of Theorem 4.2.1 in [13]. We take the
scalar function $\zeta$ corresponding to $B^{1}_{r}(a)$ and $\alpha$ from
Lemma 5.2 and take a nonnegative function $\phi=\phi(x^{\prime})\in
C^{\infty}_{0}(B^{\prime}_{1}(0))$ with unit integral. Denote
$\eta(x^{\prime})=r^{-d+1}\phi(\frac{x^{\prime}}{r})$,
$D_{r}(a):=(a-r,a+r)\times B^{\prime}_{r}(0)$ as before, and for
$t\in(0,r^{2})$ set

$\displaystyle\bar{u}(t):=\int_{D_{r}(a)}\zeta(y^{1})\eta(y^{\prime})u(t,y)\nu_{\alpha}(dy).$

Then by Jensen’s inequality and the weighted version of Poincaré’s inequality
(Lemma 5.1),

$\displaystyle\int_{D_{r}(a)}|u(t,x)-\bar{u}(t)|^{p}\nu_{\alpha}(dx)$ (5.6)
$\displaystyle=$
$\displaystyle\int_{D_{r}(a)}\Big{|}\int_{D_{r}(a)}(u(t,x)-u(t,y))\zeta(y^{1})\eta(y^{\prime})\nu_{\alpha}(dy)\Big{|}^{p}\nu_{\alpha}(dx)$
$\displaystyle\leq$
$\displaystyle\int_{D_{r}(a)}\left(\int_{D_{r}(a)}|u(t,x)-u(t,y)|^{p}\zeta(y^{1})\eta(y^{\prime})\nu_{\alpha}(dy)\right)\nu_{\alpha}(dx)$
$\displaystyle\leq$
$\displaystyle|\sup\;\zeta|\cdot|\sup\,\eta|\,\int_{D_{r}(a)}\int_{D_{r}(a)}|u(t,x)-u(t,y)|^{p}\nu_{\alpha}(dx)\nu_{\alpha}(dy)$
$\displaystyle\leq$ $\displaystyle
Nr^{-d+1}|\sup\;\zeta|\cdot\nu_{\alpha}(D_{r}(a))\;r^{p}\int_{D_{r}(a)}|u_{x}(t,x)|^{p}\nu_{\alpha}(dx)$
$\displaystyle\leq$ $\displaystyle
Nr^{-d+1}|\sup\;\zeta|\cdot\nu_{\alpha}^{1}(B^{1}_{r}(a))\;r^{d-1}r^{p}\int_{D_{r}(a)}|u_{x}(t,x)|^{p}\nu_{\alpha}(dx)$
$\displaystyle\leq$ $\displaystyle
N\;r^{p}\int_{D_{r}(a)}|u_{x}(t,x)|^{p}\nu_{\alpha}(dx).$

We observe that for any constant vector $c\in\mathbb{R}^{d}$ the left-hand
side of (5.5) is less than $2\cdot 2^{p}$ times

$\displaystyle\int_{Q_{r}(a)}|u(t,x)-c|^{p}\mu_{\alpha}(dtdx)\leq
2^{p}\int_{Q_{r}(a)}|u(t,x)-\bar{u}(t)|^{p}\mu_{\alpha}(dtdx)+2^{p}\;\nu_{\alpha}(D_{r}(a))\int^{r^{2}}_{0}|\bar{u}(t)-c|^{p}dt.$

By (5.6) the first term is less than (5.5). To estimate the second term, we
take

$\displaystyle c=\frac{1}{r^{2}}\int^{r^{2}}_{0}\bar{u}(t)dt.$

Then by Poincaré’s inequality without a weight in variable $t$ we have

$\displaystyle\nu_{\alpha}(D_{r}(a))\int^{r^{2}}_{0}|\bar{u}(t)-c|^{p}dt$
(5.7) $\displaystyle\leq$ $\displaystyle
N\;\nu_{\alpha}(D_{r}(a))\;(r^{2})^{p}\int^{r^{2}}_{0}\Big{|}\int_{D_{r}(a)}\zeta(x^{1})\eta(x^{\prime})u_{t}(t,x)\nu_{\alpha}(dx)\Big{|}^{p}dt.$

Remember $u_{t}=-A^{ij}(t)u_{x^{i}x^{j}}+f^{i}_{x^{i}}+g$. We show that (5.7)
is less than (5.5). In fact, for handling the integral with $g$, using
Jensen’s inequality and taking the supremum out of the integral, we have

$\displaystyle\nu_{\alpha}(D_{r}(a))\;r^{2p}\int^{r^{2}}_{0}\Big{|}\int_{D_{r}(a)}\zeta(x^{1})\eta(x^{\prime})g(t,x)\nu_{\alpha}(dx)\Big{|}^{p}dt$
$\displaystyle\leq$
$\displaystyle\nu_{\alpha}(D_{r}(a))\;r^{2p}\;|\sup\;\zeta|\,|\sup\eta|\,\int^{r^{2}}_{0}\int_{D_{r}(a)}|g(t,x)|^{p}\nu_{\alpha}(dx)dt$
$\displaystyle\leq$ $\displaystyle
N\nu_{\alpha}^{1}(B^{1}_{r}(a))r^{d-1}\;r^{2p}\;|\sup\;\zeta|\,r^{-d+1}\int^{r^{2}}_{0}\int_{D_{r}(a)}|g(t,x)|^{p}\nu_{\alpha}(dx)dt$
$\displaystyle\leq$ $\displaystyle
N(\theta,p,d)\;r^{2p}\int_{Q_{r}(a)}|g(t,x)|^{p}\mu_{\alpha}(dtdx),$

where we used $|\sup\zeta|\;\nu_{\alpha}^{1}(B^{1}_{r}(a))\leq N$ (Lemma 5.2).

Next, we handle the integral with $-A^{ij}u_{x^{i}x^{j}}$. Fix $i,j$. Firstly,
assume either $i$ or $j$ is $1$; say $j=1$. We use integration by parts and
observe

$\displaystyle\nu_{\alpha}(D_{r}(a))\;(r^{2})^{p}\int^{r^{2}}_{0}\Big{|}\int_{D_{r}(a)}\zeta(x^{1})\eta(x^{\prime})A^{ij}(t)u_{x^{i}x^{j}}(t,x)\nu_{\alpha}(dx)\Big{|}^{p}dt$
$\displaystyle\leq$
$\displaystyle\nu_{\alpha}(D_{r}(a))\;r^{2p}\int^{r^{2}}_{0}\Big{|}\int_{D_{r}(a)}\zeta_{x^{1}}(x^{1})\eta(x^{\prime})A^{ij}(t)u_{x^{i}}(t,x)\nu_{\alpha}(dx)\Big{|}^{p}dt$
$\displaystyle\quad+\nu_{\alpha}(D_{r}(a))\;r^{2p}|\alpha|^{p}\int^{r^{2}}_{0}\Big{|}\int_{D_{r}(a)}\frac{1}{x}\zeta(x^{1})\eta(x^{\prime})A(t)u_{x^{i}}(t,x)\nu_{\alpha}(dx)\Big{|}^{p}dt$
$\displaystyle=:$ $\displaystyle\quad I_{1}+I_{2}.$

For $I_{2}$ we use the fact $|A^{ij}u_{x^{i}}|\leq|A^{ij}||u_{x^{i}}|\leq
K|u_{x}|$ and $1/x\leq 2/r$ on the support of $\zeta$. The argument handling
the case of $g$ easily shows

$\displaystyle I_{2}\leq
N(K,\theta,p,d)\;r^{p}\int_{Q_{r}(a)}|u_{x}(t,x)|^{p}\mu_{\alpha}(dtdx).$

For $I_{1}$ we use Hölder’s inequality and get

$\displaystyle\nu_{\alpha}(D_{r}(a))\cdot|\int_{D_{r}(a)}\zeta_{x^{1}}\eta
A^{ij}u_{x^{i}}\;\nu_{\alpha}(dx)|^{p}$ $\displaystyle\leq$
$\displaystyle\nu_{\alpha}(D_{r}(a))^{p}\int_{D_{r}(a)}|\zeta_{x^{1}}\eta
A^{ij}u_{x^{i}}|^{p}\;\nu_{\alpha}(dx)$ $\displaystyle\leq$ $\displaystyle
N(\nu_{\alpha}^{1}(B^{1}_{r}(a))^{p}r^{(d-1)p}\cdot|\sup\zeta_{x^{1}}|^{p}r^{(-d+1)p}\int_{D_{r}(a)}|u_{x}|^{p}\nu_{\alpha}(dx).$

Since $\nu_{\alpha}^{1}(B^{1}_{r}(a))\cdot|\sup\zeta_{x}|\leq N/r$, it easily
follows that

$\displaystyle I_{1}\leq
N(K,\theta,p,d)\;r^{p}\int_{Q_{r}(a)}|u_{x}(t,x)|^{p}\mu_{\alpha}(dtdx).$

Secondly, if $i,j\neq 1$, by integration by parts, Hölder’s inequality and the
inequality $\sup|\eta_{x^{\prime}}|\leq Nr^{-d}$,

$\displaystyle\nu_{\alpha}(D_{r}(a))\;r^{2p}\int^{r^{2}}_{0}\Big{|}\int_{D_{r}(a)}\zeta(x^{1})\eta(x^{\prime})\left[-A^{ij}(t)u_{x^{i}x^{j}}(t,x)\right]\nu_{\alpha}(dx)\Big{|}^{p}dt$
$\displaystyle=$
$\displaystyle\nu_{\alpha}(D_{r}(a))\;r^{2p}\int^{r^{2}}_{0}\Big{|}\int_{D_{r}(a)}\zeta(x^{1})\eta_{x^{j}}(x^{\prime})A^{ij}(t)u_{x^{i}}(t,x)\nu_{\alpha}(dx)\Big{|}^{p}dt$
$\displaystyle\leq$
$\displaystyle\nu_{\alpha}(D_{r}(a))^{p}\;r^{2p}\int^{r^{2}}_{0}\int_{D_{r}(a)}\Big{|}\zeta(x^{1})\eta_{x^{j}}(x^{\prime})A^{ij}(t)u_{x^{i}}(t,x)\Big{|}^{p}\nu_{\alpha}(dx)\;dt$
$\displaystyle\leq$ $\displaystyle
N\nu_{\alpha}(D_{r}(a))^{p}\;r^{2p}\cdot\sup|\zeta|^{p}\cdot
r^{-dp}\int^{r^{2}}_{0}\int_{D_{r}(a)}|u_{x}|^{p}\nu_{\alpha}(dx)dt$
$\displaystyle\leq$ $\displaystyle
Nr^{p}\int_{Q_{r}(a)}|u_{x}|^{p}\mu_{\alpha}(dxdt).$

For the integral with $f^{i}_{x^{i}}$ we use a similar calculation to the one
of $-A^{ij}u_{x^{i}x^{j}}$ and get for each $i$

$\displaystyle\nu_{\alpha}(D_{r}(a))\,r^{2p}\int^{r^{2}}_{0}\Big{|}\int_{D_{r}(a)}\zeta(x^{1})\eta(x^{\prime})f_{x^{i}}(t,x)\nu_{\alpha}(dx)\Big{|}^{p}dt$
$\displaystyle\leq$ $\displaystyle
N(K,\theta,p,d)\;r^{p}\int_{Q_{r}(a)}|f(t,x)|^{p}\mu_{\alpha}(dtdx).$

The lemma is proved. ∎

######  Lemma 5.4.

Let $\alpha\geq 0,p\in[1,\infty)$, $0<r\leq a$ and $u\in
C^{\infty}_{loc}(\Omega;\mathbb{R}^{d_{1}})$.

(i) There is a constant $N=N(K,\theta,\alpha,p,d,d_{1})$ such that for any
$i=1,\cdots,d$ we have

$\displaystyle\int_{Q_{r}(a)}\left|u_{x^{i}}(t,x)-(u_{x^{i}})_{Q_{r}(a)}\right|^{p}\mu_{\alpha}(dtdx)\leq
Nr^{p}\int_{Q_{r}(a)}(|u_{xx}(t,x)|^{p}+|u_{t}(t,x)|^{p})\mu_{\alpha}(dtdx)$
(5.8)

(ii) Denote
$\kappa_{0}=\kappa_{0}(r,a):=(\nu^{1}_{\alpha}(B^{1}_{r}(a))^{-1}\cdot\int^{a+r}_{a-r}x^{1}\nu^{1}_{\alpha}(dx^{1})$.
Then

$\displaystyle\int_{Q_{r}(a)}\left|u(t,x)-u_{Q_{r}(a)}+\kappa_{0}(u_{x^{1}})_{Q_{r}(a)}-\sum_{i=1}^{d}x^{i}(u_{x^{i}})_{Q_{r}(a)}\right|^{p}\mu_{\alpha}(dtdx)$
(5.9) $\displaystyle\leq$ $\displaystyle
Nr^{p}\int_{Q_{r}(a)}(|u_{x}(t,x)-(u_{x})_{Q_{r}(a)}|^{p}+r^{p}|u_{t}(t,x)|^{p}+r^{p}|u_{xx}(t,x)|^{p})\mu_{\alpha}(dtdx)$
$\displaystyle\leq$ $\displaystyle
Nr^{2p}\int_{Q_{r}(a)}(|u_{xx}(t,x)|^{p}+|u_{t}(t,x)|^{p})\mu_{\alpha}(dtdx)$

###### Proof.

(i) For (5.8) we use the fact that for $v=u_{x^{i}}$,
$v_{t}-A^{jm}v_{x^{j}x^{m}}=(u_{t}-A^{jm}u_{x^{j}x^{m}})_{x^{i}}$ and apply
Lemma 5.3 with $f^{i}=u_{t}-A^{jm}u_{x^{j}x^{m}}$ for all $i$.

(ii) To prove (5.9), denote
$v(t,x):=u(t,x)-(u)_{Q_{r}(a)}+\kappa_{0}(u_{x^{1}})_{Q_{r}(a)}-\sum_{i}x^{i}(u_{x^{i}})_{Q_{r}(a)}$.
Then

$v_{Q_{r}(a)}=\kappa_{0}(u_{x^{1}})_{Q_{r}(a)}-\sum_{i}\frac{(u_{x^{i}})_{Q_{r}(a)}}{|Q_{r}(a)|}\int_{Q_{r}(a)}x^{i}\nu_{\alpha}(dx)dt=0,$
$v-v_{Q_{r}(a)}=v,\quad v_{x^{i}}=u_{x^{i}}-(u_{x^{i}})_{Q_{r}(a)},\quad
v_{t}-A^{ij}v_{x^{i}x^{j}}=g:=u_{t}-A^{ij}u_{x^{i}x^{j}}.$

Now it is enough to use Lemma 5.3 and (5.8). The lemma is proved. ∎

From this point on we fix $\alpha:=\theta-d+p$ (note $\alpha>0$) and denote

$\nu:=\nu_{\alpha},\quad\nu^{1}:=\nu_{\alpha}^{1},\quad\quad\mu(dxdt)=\nu(dx)dt=(x^{1})^{\theta-d+p}dxdt.$

######  Theorem 5.5.

Let $\theta\in(d-1,d]$, $0<r\leq a$ and $\lambda r/a\geq 2$.

(i) Assume that $u\in C^{\infty}_{loc}(\Omega;\mathbb{R}^{d_{1}})$ satisfies
$u_{t}+A^{ij}(t)u_{x^{i}x^{j}}=0$ in $Q_{\lambda
r}(t_{0},a,x^{\prime}_{0})\cap\Omega$. Then there is a constant
$N=N(K,\delta,\theta,p,d,d_{1})$ so that

$\displaystyle-\int_{Q_{r}(t_{0},a,x^{\prime}_{0})}|u_{xx}(t,x)-(u_{xx})_{Q_{r}(t_{0},a,x^{\prime}_{0})}|^{p}\mu(dtdx)$
(5.10) $\displaystyle\leq$ $\displaystyle\frac{N}{(1+\lambda
r/a)^{p}}\;-\int_{Q_{\lambda
r}(t_{0},a,x^{\prime}_{0})\cap\Omega}|u_{xx}(t,x)|^{p}\mu(dtdx).$

(ii) If $u\in C^{\infty}_{loc}(\mathbb{R}^{d}_{+};\mathbb{R}^{d_{1}})$,
$A^{ij}$ is independent of $t$ and $A^{ij}u_{x^{i}x^{j}}=0$ in $B_{\lambda
r}(a,x^{\prime}_{0})\cap\mathbb{R}^{d}_{+}$, then

$\displaystyle-\int_{B_{r}(a,x^{\prime}_{0})}|u_{xx}(x)-(u_{xx})_{B_{r}(a,x^{\prime}_{0})}|^{p}\nu(dx)$
(5.11) $\displaystyle\leq$ $\displaystyle\frac{N}{(1+\lambda
r/a)^{p}}\;-\int_{B_{\lambda
r}(a,x^{\prime}_{0})\cap\mathbb{R}^{d}_{+}}|u_{xx}(x)|^{p}\nu(dx).$

###### Proof.

(ii) is a consequence of (i). To prove (i), without loss of generality we may
assume $t_{0}=0$, $x^{\prime}_{0}=0$ and thus
$Q_{r}(t_{0},a,x^{\prime}_{0})=Q_{r}(a)$.

Step 1. First, we consider the case $a=1$. Note that

$r\leq 1,\quad 2\leq\lambda r,\quad\beta:=\frac{1+\lambda r}{2}\leq\lambda
r,\quad\frac{r}{\beta}\leq\frac{1}{\beta}\leq\frac{2}{3},\quad\quad
2\beta=1+\lambda r.$

Thus,

$Q_{\beta}(\beta)\subset Q_{\lambda r}(1)\cap\Omega,\quad
Q_{r/\beta}(\beta^{-1})\subset Q_{2/3}(2/3).$

Denote $w(t,x)=u(\beta^{2}t,\beta x)$, then obviously

$w_{t}+A^{ij}(\beta^{2}t)w_{x^{i}x^{j}}=0,\quad\quad\text{for}\quad(t,x)\in
Q_{1}(1)$

and

$\displaystyle-\int_{Q_{r}(1)}|u_{xx}(t,x)-(u_{xx})_{Q_{r}(1)}|^{p}(x^{1})^{\theta-d+p}dxdt$
$\displaystyle\leq$ $\displaystyle
N(d)\sup_{Q_{r}(1)}(|u_{xxx}|^{p}+|u_{xxt}|^{p})$ $\displaystyle\leq$
$\displaystyle
N(d)\beta^{-3p}\,\,\sup_{Q_{r/\beta}(\beta^{-1})}(|w_{xxx}|^{p}+|w_{xxt}|^{p})$
$\displaystyle\leq$ $\displaystyle
N(d)\beta^{-3p}\,\,\sup_{Q_{2/3}(2/3)}(|w_{xxx}|^{p}+|w_{xxt}|^{p}).$

Applying Lemma 4.14 to
$v(t,x)=w(t,x)-w_{Q_{1}(1)}+\kappa_{0}(w_{x^{1}})_{Q_{1}(1)}-\sum_{i=1}^{d}x^{i}(w_{x^{i}})_{Q_{1}(1)}$,
and then using Lemma 5.4

$\displaystyle\beta^{-3p}\,\,\sup_{Q_{2/3}(2/3)}(|w_{xxx}|^{p}+|w_{xxt}|^{p})$
$\displaystyle\leq$ $\displaystyle
N\beta^{-3p}\int_{Q_{1}(1)}|v|^{p}(x^{1})^{\theta-d+p}dxdt$
$\displaystyle\leq$ $\displaystyle
N\beta^{-3p}\int_{Q_{1}(1)}|w_{xx}|^{p}(x^{1})^{\theta-d+p}dxdt$
$\displaystyle=$ $\displaystyle
N\beta^{-2p-2-\theta}\int_{Q_{\beta}(\beta)}|u_{xx}|^{p}(x^{1})^{\theta-d+p}dxdt.$

This leads to (5.10) since $|Q_{\lambda
r}(1)\cap\Omega|\sim\beta^{p+\theta+2}$.

Step 2. Let $a\neq 1$. Define $v(t,x):=u(a^{2}t,ax)$. Then
$v_{t}+A^{ij}(a^{2}t)v_{x^{i}x^{j}}=0$ in $Q_{\lambda r/a}(1)\cap\Omega$. As
easy to check,

$|Q_{r/a}(1)|=a^{-\theta-p-2}|Q_{r}(a)|,\quad(v_{xx})_{Q_{r/a}(1)}=a^{2}(u_{xx})_{Q_{r}(a)},\quad|Q_{\lambda
r/a}(1)\cap\Omega|=a^{-\theta-p-2}|Q_{\lambda r}(a)\cap\Omega|,$

and consequently

$-\int_{Q_{r/a}(1)}|v_{xx}(t,x)-(v_{xx})_{Q_{r/a}(1)}|^{p}(x^{1})^{\theta-d+p}dxdt=a^{2p}-\int_{Q_{r}(a)}|u_{xx}(t,x)-(u_{xx})_{Q_{r}(a)}|^{p}(x^{1})^{\theta-d+p}dxdt,$
$-\int_{Q_{\lambda
r/a}(1)\cap\Omega}|v_{xx}(t,x)|^{p}(x^{1})^{\theta-d+p}dxdt=a^{2p}-\int_{Q_{\lambda
r}(a)\cap\Omega}|u_{xx}(t,x)|^{p}(x^{1})^{\theta-d+p}dxdt.$

It follows

$\displaystyle-\int_{Q_{r}(a)}|u_{xx}(t,x)-(u_{xx})_{Q_{r}(a)}|^{p}(x^{1})^{\theta-d+p}dxdt$
$\displaystyle=$ $\displaystyle a^{-2p}-\int_{Q_{\lambda
r/a}(1)\cap\Omega}|v_{xx}(t,x)|^{p}(x^{1})^{\theta-d+p}dxdt$
$\displaystyle\leq$ $\displaystyle a^{-2p}\cdot\frac{N}{(1+\lambda
r/a)^{p}}\;-\int_{Q_{\lambda
r/a}(1)\cap\Omega}|v_{xx}(t,x)|^{p}(x^{1})^{\theta-d+p}dxdt$ $\displaystyle=$
$\displaystyle\frac{N}{(1+\lambda r/a)^{p}}\;-\int_{Q_{\lambda
r}(a)\cap\Omega}|u_{xx}(t,x)|^{p}(x^{1})^{\theta-d+p}dxdt.$

The theorem is proved. ∎

######  Remark 5.6.

Note that Theorem 5.5 is based on Lemma 4.14. It follows from Remark 4.12 and
Remark 4.15 that if $p\geq 2$ then Theorem 5.5 holds for any
$\theta\in(d-1,d+1)$ (not only for $\theta\in(d-1,d]$). Obviously we cannot
use this result yet since Remark 4.12 is valid only after we prove Theorem
3.10.

######  Lemma 5.7.

Assume $\theta\in(d-1,d]$ if $p\in(2,\infty)$ and $\theta\in(d-p+1,d]$ if
$p\in(1,2]$. Denote $q:=\theta-d+p$ which is in $(1,p]$.

(i) Let $u\in C^{\infty}_{0}(\mathbb{R}\times\mathbb{R}^{d}_{+})$ and
$f:=u_{t}+A^{ij}(t)u_{x^{i}x^{j}}$. Suppose that $A^{ij}(t)$ is infinitely
differentiable and has bounded derivatives. Then for any $\varepsilon>0$,
$Q_{r}(t_{0},a,x^{\prime}_{0})\subset\Omega$ and $(t,x)\in
Q_{r}(t_{0},a,x^{\prime}_{0})$

$-\int_{Q_{r}(t_{0},a,x^{\prime}_{0})}|u_{xx}-(u_{xx})_{Q_{r}(t_{0},a,x^{\prime}_{0})}|^{q}\mu(dyds)\leq\varepsilon\mathbb{M}(|u_{xx}|^{q})(t,x)+N\mathbb{M}(|f|^{q})(t,x),$
(5.12)

where $N=N(\varepsilon,\theta,q,d,d_{1},\delta,K)$.

(ii) Furthermore, if $u\in C^{\infty}_{0}(\mathbb{R}^{d}_{+})$ and $A^{ij}$ is
independent of $t$, then for any $\varepsilon>0$,
$B_{r}(a,x^{\prime}_{0})\subset\mathbb{R}^{d}_{+}$ and $x\in
B_{r}(a,x^{\prime}_{0})$

$-\int_{B_{r}(a,x^{\prime}_{0})}|u_{xx}-(u_{xx})_{B_{r}(a,x^{\prime}_{0})}|^{q}\nu(dy)\leq\varepsilon\mathbb{M}(|u_{xx}|^{q})(x)+N\mathbb{M}(|A^{ij}u_{x^{i}x^{j}}|^{q})(x),$
(5.13)

where $N=N(\varepsilon,\theta,q,d,d_{1},\delta,K)$.

###### Proof.

(i) Without loss of generality we may take $t_{0}=0$ and $x^{\prime}_{0}=0$;
$Q_{r}(t_{0},a,x^{\prime}_{0})=Q_{r}(a)$. In fact, for other cases it is
enough to consider the function
$v(t,x):=u(t_{0}+t,x^{1},x^{\prime}_{0}+x^{\prime})$ in place of
$u(t,x^{1},x^{\prime})$.

Step 1. We prove that there exists $\kappa=\kappa(\varepsilon)\in(0,1)$ so
that (5.12) holds if $(r/a)\leq\kappa$.

Let $m$ denote the Lebesque measure on $\mathbb{R}^{d+1}$. Assume $\lambda\geq
4$ and $\lambda r\leq a/4$. Then $(3a/4)\leq x^{1}\leq(5a/4)$ if $x^{1}\in
B^{1}_{\lambda r}(a)$, and therefore

$(3/5)^{p+\theta-d}\frac{dtdx}{m(Q_{r}(a))}\leq\frac{\mu(dtdx)}{|Q_{r}(a)|}\leq(5/3)^{p+\theta-d}\frac{dtdx}{m(Q_{r}(a))}\quad\quad\text{on}\quad
Q_{r}(a),$ $(3/5)^{p+\theta-d}\frac{dtdx}{m(Q_{\lambda
r}(a))}\leq\frac{\mu(dtdx)}{|Q_{\lambda
r}(a)|}\leq(5/3)^{p+\theta-d}\frac{dtdx}{m(Q_{\lambda
r}(a))}\quad\quad\text{on}\quad Q_{\lambda r}(a).$

Denote $c_{0}:=(5/3)^{p+\theta-d}$. By Theorem 4.5,

$\displaystyle-\int_{Q_{r}(a)}|u_{xx}-(u_{xx})_{Q_{r}(a)}|^{q}\mu(dsdy)$
$\displaystyle\leq$
$\displaystyle\int_{Q_{r}(a)}\int_{Q_{r}(a)}|u_{xx}(s,y)-u_{xx}(\tau,\xi)|^{q}\frac{\mu(dsdy)}{|Q_{r}(a)|}\frac{\mu(d\tau
d\xi)}{|Q_{r}(a)|}$ $\displaystyle\leq$ $\displaystyle
c^{2}_{0}\int_{Q_{r}(a)}\int_{Q_{r}(a)}|u_{xx}(s,y)-u_{xx}(\tau,\xi)|^{q}\frac{dsdy}{m(Q_{r}(a))}\frac{d\tau
d\xi}{m(Q_{r}(a))}$ $\displaystyle\leq$ $\displaystyle
Nc^{2}_{0}\lambda^{d+2}\int_{Q_{\lambda r}(a)}|f|^{q}\frac{dyds}{m(Q_{\lambda
r}(a))}+Nc^{2}_{0}\lambda^{-q}\int_{Q_{\lambda
r}(a)}|u_{xx}|^{q}\frac{dyds}{m(Q_{\lambda r}(a))}$ $\displaystyle\leq$
$\displaystyle Nc^{3}_{0}\lambda^{d+2}-\int_{Q_{\lambda
r}(a)}|f|^{q}\mu(dyds)+Nc^{3}_{0}\lambda^{-q}-\int_{Q_{\lambda
r}(a)}|u_{xx}|^{q}\mu(dyds)$ $\displaystyle\leq$ $\displaystyle
N\lambda^{d+2}\mathbb{M}(|f|^{q})(t,x)+N\lambda^{-q}\mathbb{M}(|u_{xx}|^{q})(t,x),$

where $N$ depends only on $d,d_{1},p,\theta,\delta,K$. Note that the above
inequality holds as long as $r\lambda/a\leq 1/4$. Now we fix $\lambda$ so that
$N\lambda^{-q}=\varepsilon/2$, i.e. $\lambda=(2N/\varepsilon)^{1/q}$ and
define $\kappa=1/{(4\lambda)}=1/4\cdot(2N/\varepsilon)^{-1/q}$. Then whenever
$r/a\leq\kappa$ we have $(r/a)\lambda\leq 1/4$ and thus (5.12) follows.

Step 2. For given $\varepsilon$, take $\kappa=\kappa(\varepsilon)$ from Step
1. Assume $r/a\geq\kappa$. Choose $\lambda$, which will be specified later, so
that $r\lambda>4a$; this $\lambda$ is different from the one in step 1. Take a
$\zeta\in C^{\infty}_{0}(\mathbb{R}^{d+1})$ so that $\zeta(t,x)=1$ for
$(t,x)\in Q_{\lambda r/2}(a)\cap\Omega$ and $\zeta(t,x)=0$ if
$(t,x)\not\in(-\lambda^{2}r^{2},\lambda^{2}r^{2})\times(-a,a+\lambda r)\times
B^{\prime}_{\lambda r}$. Denote

$g=f\zeta,\quad h=f(1-\zeta).$

Take a large $T$ so that $u(t,x)=0$ if $t\geq T$. By Lemma 4.10 we can define
$v$ as the solution of

$v_{t}+A^{ij}v_{x^{i}x^{j}}=h,\quad t\in(S,T),\quad v(T,\cdot)=0$ (5.14)

so that $v\in\mathfrak{H}^{n}_{p,d}(S,T)$ for any $n$ and $S>-\infty$. Also
let $\bar{v}\in\mathfrak{H}^{n}_{p,d}(S,T+1)$ be the solution of

$\bar{v}_{t}+A^{ij}\bar{v}_{x^{i}x^{j}}=h,\quad
t\in(S,T+1),\quad\bar{v}(T+1,\cdot)=0.$

Then by considering the equation for $\bar{v}$ on $(T,T+1)$, since $h(t)=0$
for $t\geq T$, we conclude $\bar{v}(t)=0$ for $t\in[T,T+1]$. Thus $\bar{v}$
also satisfies (5.14) and $v=\bar{v}$. It follows from (3.6) that $v$ is
infinitely differentiable in $x$ (and hence in $t$) in $\Omega$. By applying
Theorem 5.5 with $\bar{p}=q,\bar{\theta}=d$ and $\lambda/2$ in places of
$p,\theta$ and $\lambda$ respectively,

$\displaystyle-\int_{Q_{r}(a)}|v_{xx}(t,x)-(v_{xx})_{Q_{r}(a)}|^{q}\bar{\mu}(dyds)$
$\displaystyle\leq$ $\displaystyle N\frac{1}{(1+\lambda
r/2a)^{q}}-\int_{Q_{\lambda
r/2}(a)\cap\Omega}|v_{xx}(t,x)|^{q}\bar{\mu}(dyds)$ (5.15) $\displaystyle\leq$
$\displaystyle N\frac{1}{(1+\lambda r/a)^{q}}-\int_{Q_{\lambda
r}(a)\cap\Omega}|v_{xx}(t,x)|^{q}\bar{\mu}(dyds),$

where
$\bar{\mu}(dsdy):=(y^{1})^{\bar{\theta}-d+\bar{p}}dyds=(y^{1})^{q}dyds=\mu(dyds)$.
On the other hand, $w:=u-v$ satisfies $w(T,\cdot)=0$ and

$w_{t}+A^{ij}w_{x^{i}x^{j}}=g,\quad t\in(0,T).$

By Lemma 4.10,

$\int_{Q_{r}(a)}|w_{yy}|^{q}(y^{1})^{q}dyds\leq\int_{Q_{\lambda
r}(a)\cap\Omega}|w_{yy}|^{q}(y^{1})^{q}dyds\leq N\int_{Q_{\lambda
r}(a)\cap\Omega}|f|^{q}(y^{1})^{q}\;dyds,$
$\displaystyle-\int_{Q_{r}(a)}|w_{yy}|^{q}\mu(dyds)$ $\displaystyle\leq$
$\displaystyle N\frac{\lambda^{d+1}(1+\lambda
r/a)^{p+\theta-d+1}}{(1+r/a)^{p+\theta-d+1}-(1-r/a)^{p+\theta-d+1}}-\int_{Q_{\lambda
r}(a)\cap\Omega}|f|^{q}\mu(dyds)$ (5.16) $\displaystyle\leq$ $\displaystyle
N(\kappa)\lambda^{d+1}(1+\lambda r/a)^{p+\theta-d+1}-\int_{Q_{\lambda
r}(a)\cap\Omega}|f|^{q}\mu(dyds),$

where for the second inequality we use
$(1+r/a)^{p+\theta-d+1}-(1-r/a)^{p+\theta-d+1}\geq(1+\kappa)^{p+\theta-d+1}-1$.
Observing that $u=v+w$,

$\displaystyle I:$ $\displaystyle=$
$\displaystyle-\int_{Q_{r}(a)}|u_{yy}(t,x)-(u_{yy})_{Q_{r}(a)}|^{q}\mu(dyds)$
$\displaystyle\leq$ $\displaystyle
N(q)-\int_{Q_{r}(a)}|w_{yy}(t,x)-(w_{yy})_{Q_{r}(a)}|^{q}\mu(dyds)+N(q)-\int_{Q_{r}(a)}|v_{yy}(t,x)-(v_{yy})_{Q_{r}(a)}|^{q}\mu(dyds)$
$\displaystyle\leq$ $\displaystyle
N(q)-\int_{Q_{r}(a)}|w_{yy}(t,x)|^{q}\mu(dyds)+N(q)-\int_{Q_{r}(a)}|v_{yy}(t,x)-(v_{yy})_{Q_{r}(a)}|^{q}\mu(dyds)$

and thus by (5.15) and (5.16),

$\displaystyle I$ $\displaystyle\leq$ $\displaystyle N\lambda^{d+1}(1+\lambda
r/a)^{p+\theta-d+1}-\int_{Q_{\lambda r}(a)\cap\Omega}|f|^{q}\mu(dyds)$
$\displaystyle+N\frac{1}{(1+\lambda r/a)^{q}}-\int_{Q_{\lambda
r}(a)\cap\Omega}|v_{yy}(t,x)|^{q}\mu(dyds)$ $\displaystyle\leq$ $\displaystyle
N\lambda^{d+1}(1+\lambda
r/a)^{p+\theta-d+1}-\int_{(0,\lambda^{2}r^{2})\times(0,a+\lambda
r)}|f|^{q}\mu(dyds)$ $\displaystyle+N\frac{1}{(1+\lambda
r/a)^{q}}-\int_{Q_{\lambda
r}(a)\cap\Omega}\left(|u_{yy}(t,x)|^{q}+|w_{yy}(t,x)|^{q}\right)\mu(dyds)$
$\displaystyle\leq$ $\displaystyle N\lambda^{d+1}(1+\lambda
r/a)^{p+\theta-d+1}-\int_{Q_{\lambda r}(a)\cap\Omega}|f|^{q}\mu(dyds)$
$\displaystyle+N\frac{1}{(1+\lambda r/a)^{q}}-\int_{Q_{\lambda
r}(a)\cap\Omega}|u_{yy}(t,x)|^{q}\mu(dyds).$

Now to prove the first assertion it is enough to choose $\lambda$ so large
that $N\frac{1}{(1+\lambda r/a)^{q}}\leq\varepsilon$. Also note that since
$r/a\geq\kappa$, we have

$N\lambda^{d+1}(1+\lambda r/a)^{p+\theta-d+1}\leq N(\lambda,\kappa).$

(ii) The second assertion is proved similarly based on Corollary 4.6 and
(5.11) in place Theorem 4.5 and (5.10). The lemma is proved.

∎

##  6 Proof of Theorem 3.10 and Theorem 3.13

Firstly, we give an $L_{p}$-theory for the following backward system defined
on $\mathbb{R}\times\mathbb{R}^{d}_{+}$.

######  Theorem 6.1.

Let $p\in(1,\infty)$. Assume $\theta\in(d-1,d+1)$ if $p\in(2,\infty)$, and
$\theta\in(d+1-p,d+p-1)$ if $p\in(1,2]$. Then for any
$f\in\mathbb{L}_{p,\theta}(-\infty,\infty)$ the system

$u_{t}+A^{ij}(t)u_{x^{i}x^{j}}=f$

has a unique solution $u$ in $M\mathbb{H}^{2}_{p,\theta}(-\infty,\infty)$ and
for this solution we have

$\|Mu_{t}\|_{\mathbb{L}_{p,\theta}(-\infty,\infty)}+\|M^{-1}u\|_{\mathbb{H}^{2}_{p,\theta}(-\infty,\infty)}\leq
N\|Mf\|_{\mathbb{L}_{p,\theta}(-\infty,\infty)}.$ (6.1)

###### Proof.

If $A^{ij}u_{x^{i}x^{j}}=\Delta u=(\Delta u^{1},\ldots,\Delta u^{d_{1}})$,
then the theory of single equations is applied and the theorem is true for any
$\theta\in(d-1,d-1+p)$; see Theorem 5.6 in [10]. Actually the mentioned
theorem is proved for parabolic equations defined on
$(0,T)\times\mathbb{R}^{d}_{+}$, but one can easily check that the proofs in
[10] work for equations defined on $\mathbb{R}\times\mathbb{R}^{d}_{+}$.

For $\lambda\in[0,1]$ and $d_{1}\times d_{1}$ identity matrix $I$ we define

${A}^{ij}_{\lambda}=({a}^{ij}_{kr,\lambda}):=(1-\lambda)A^{ij}+\delta^{ij}\lambda\delta
I.$

Then for each $\lambda\in[0,1]$ the coefficient matrices
$\\{A^{ij}_{\lambda}:i,j=1,\ldots,d\\}$ satisfy Assumption 3.8 with the same
$\delta,K$. Thus due to the method of continuity, we only need to prove that a
priori estimate (6.1) holds given that a solution $u$ already exists.
Furthermore, since $C^{\infty}_{0}(\mathbb{R}\times\mathbb{R}^{d}_{+})$ is
dense in $M\mathbb{H}^{2}_{p,\theta}(-\infty,\infty)$, we may assume that
$u\in C^{\infty}_{0}(\mathbb{R}\times\mathbb{R}^{d}_{+})$. By Remark 4.8, we
only need to prove the following:

$\int^{\infty}_{-\infty}\int_{\mathbb{R}^{d}_{+}}|u_{xx}(t,x)|^{p}\mu(dtdx)\leq
N\int^{\infty}_{-\infty}\int_{\mathbb{R}^{d}_{+}}|f(t,x)|^{p}\mu(dtdx).$ (6.2)

To prove this we certainly may assume that $A^{ij}$ are infinitely
differentiable and have bounded derivatives (remember that the constant $N$ in
(5.12) do not depend on the regularity of $A^{ij}$).

Case 1. Assume that either (i) $p\in(2,\infty)$ and $\theta\in(d-1,d]$ or (ii)
$p\in(1,2]$ and $\theta\in(d-p+1,d]$.

Define $q:=\theta-d+p$. Recall that the range of $q\in(1,p]$. By Lemma 5.7, if
$u\in C^{\infty}_{0}(\mathbb{R}\times\mathbb{R}^{d}_{+})$, then for any
$\varepsilon>0$

$(u_{xx})^{\sharp}(t,x)\leq\varepsilon\mathbb{M}^{1/q}(|u_{xx}|^{q})(t,x)+N(\varepsilon)\mathbb{M}^{1/q}(|u_{t}+A^{ij}u_{x^{i}x^{j}}|^{q})(t,x).$

By Theorem 2.10 $($Fefferman-Stein$)$ and Theorem 2.12 $($Hardy-Littlewood$)$,

$\displaystyle\|Mu_{xx}\|_{\mathbb{L}_{p,\theta}(-\infty,\infty)}$
$\displaystyle=$ $\displaystyle\|u_{xx}\|_{L_{p}(\Omega,\mu)}$
$\displaystyle\leq$ $\displaystyle N\|(u_{xx})^{\sharp}\|_{L_{p}(\Omega,\mu)}$
$\displaystyle\leq$ $\displaystyle
N\varepsilon\|\mathbb{M}^{1/q}(|u_{xx}|^{q})\|_{L_{p}(\Omega,\mu)}+N\cdot
N(\varepsilon)\|\mathbb{M}^{1/q}(|u_{t}+A^{ij}u_{x^{i}x^{j}}|^{q})\|_{L_{p}(\Omega,\mu)}$
$\displaystyle=$ $\displaystyle
N\varepsilon\|\mathbb{M}(|u_{xx}|^{q})\|^{1/q}_{L_{p/q}(\Omega,\mu)}+N\cdot
N(\varepsilon)\|\mathbb{M}(|u_{t}+A^{ij}u_{x^{i}x^{j}}|^{q})\|^{1/q}_{L_{p/q}(\Omega,\mu)}$
$\displaystyle\leq$ $\displaystyle
N\varepsilon\||u_{xx}|^{q}\|^{1/q}_{L_{p/q}(\Omega,\mu)}+N\cdot
N(\varepsilon)\||u_{t}+A^{ij}u_{x^{i}x^{j}}|^{q}\|^{1/q}_{L_{p/q}(\Omega,\mu)}$
$\displaystyle=$ $\displaystyle
N\varepsilon\|u_{xx}\|_{L_{p}(\Omega,\mu)}+N\cdot
N(\varepsilon)\|u_{t}+A^{ij}u_{x^{i}x^{j}}\|_{L_{p}(\Omega,\mu)}.$

This obviously yields (6.2).

Case 2. Assume that either (i) $p\in(2,\infty)$ and $\theta\in[d,d+1)$ or (ii)
$p\in(1,2]$ and $\theta\in[d,d+p-1)$. By Remark 4.8 we only need to prove the
following:

$\int^{\infty}_{-\infty}\int_{\mathbb{R}^{d}_{+}}|M^{-1}u(t,x)|^{p}(x^{1})^{\theta-d}dxdt\leq
N\int^{\infty}_{-\infty}\int_{\mathbb{R}^{d}_{+}}|Mf(t,x)|^{p}(x^{1})^{\theta-d}dxdt.$
(6.3)

To prove this, we use a duality (Lemma 3.3). Denote $p^{\prime}=p/(p-1)$ and
choose $\bar{\theta}$ so that $\theta/p+\bar{\theta}/p^{\prime}=d$. Then
$\bar{\theta}\in(d-1,d]$ if $p^{\prime}\in(2,\infty)$ and
$\bar{\theta}\in(d-p^{\prime}+1,d]$ if $p^{\prime}\in(1,2]$.

Changing the variable $t\to-t$ shows that the result of case 1 is applicable
to the operator $u_{t}-A^{ij}u_{x^{i}x^{j}}$ in place of
$u_{t}+A^{ij}u_{x^{i}x^{j}}$. Therefore for any $v\in
M\mathbb{H}^{2}_{p^{\prime},\bar{\theta}}(-\infty,\infty)$, by integration by
parts,

$\displaystyle\int_{\mathbb{R}^{d+1}_{+}}M^{-1}uM(v_{t}-A^{ij}v_{x^{i}x^{j}})dxdt$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d+1}_{+}}u(v_{t}-A^{ij}v_{x^{i}x^{j}})dxdt$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d+1}_{+}}M(-u_{t}-A^{ij}u_{x^{i}x^{j}})M^{-1}vdxdt$
$\displaystyle\leq$
$\displaystyle\|M(u_{t}+A^{ij}u_{x^{i}x^{j}})\|_{\mathbb{L}_{p,\theta}(-\infty,\infty)}\|M^{-1}v\|_{\mathbb{L}_{p^{\prime},\bar{\theta}}(-\infty,\infty)}$
$\displaystyle\leq$ $\displaystyle
N\|M(u_{t}+A^{ij}u_{x^{i}x^{j}})\|_{\mathbb{L}_{p,\theta}(-\infty,\infty)}\|M(v_{t}-A^{ij}v_{x^{i}x^{j}})\|_{\mathbb{L}_{p^{\prime},\bar{\theta}}(-\infty,\infty)}.$

Since, by Case 1, $\\{v_{t}-A^{ij}v_{x^{i}x^{j}}:v\in
M\mathbb{H}^{2}_{p^{\prime},\bar{\theta}}(-\infty,\infty)\\}$ is dense in
$M^{-1}\mathbb{L}_{p^{\prime},\bar{\theta}}(-\infty,\infty)$, it follows that

$\|M^{-1}u\|_{\mathbb{L}_{p,\theta}(-\infty,\infty)}\leq
N\|M(u_{t}+A^{ij}u_{x^{i}x^{j}})\|_{\mathbb{L}_{p,\theta}(-\infty,\infty)}.$

The theorem is proved. ∎

Proof of Theorem 3.10 As usual, we assume $u_{0}=0$. For details see the proof
of Theorem 5.1 in [9].

Case 1. Let $T=\infty$. As before we only prove the a priori estimate. Suppose
$u\in\mathfrak{H}^{\gamma+2}_{p,\theta}(\infty)$ satisfies

$u_{t}=A^{ij}u_{x^{i}x^{j}}+f,\quad t\in(0,T)\,;\quad u(0,\cdot)=0.$ (6.4)

Define $v(t,x)=u(t,x)I_{t>0}$ and $\bar{f}=fI_{t>0}$, then $v\in
M^{-1}\mathbb{H}^{2}_{p,\theta}(-\infty,\infty)$ and $v$ satisfies (see
Definition 3.9)

$v_{t}=A^{ij}u_{x^{i}x^{j}}+\bar{f},\quad(t,x)\in\mathbb{R}^{d+1}_{+}.$

By Theorem 6.1,

$\|Mu_{xx}\|_{\mathbb{L}_{p,\theta}(\infty)}\leq
N\|Mf\|_{\mathbb{L}_{p,\theta}(\infty)}.$

By Remark 4.8, this certainly proves (3.17).

Case 2. Let $T<\infty$. The existence of the solution in
$\mathfrak{H}^{\gamma+2}_{p,\theta}(T)$ is obvious. Now suppose that
$u\in\mathfrak{H}^{\gamma+2}_{p,\theta}(T)$ is a solution of (6.4). By the
result of Case 1, the system

$v_{t}=\Delta v+(A^{ij}u_{x^{i}x^{j}}+f-\Delta u)I_{t\leq T},\quad t>0\,;\quad
v(0,\cdot)=0$ (6.5)

has a unique solution $v\in\mathfrak{H}^{\gamma+2}_{p,\theta}(0,\infty)$. Then
$v-u$ satisfies

$(v-u)_{t}=\Delta(v-u),\quad t\in(0,T)\,;\quad(v-u)(0,\cdot)=0.$

If follows from the theory of single equations (see, for instance, Theorem 5.6
in [10]), $u=v$ for $t\in[0,T]$. For $t\geq 0$, define

$A^{ij}_{T}=(a^{ij}_{T,kr}),\quad a^{ij}_{T,kr}=a^{ij}_{kr}I_{t\leq
T}+\delta^{ij}\delta^{kr}I_{t>T}.$

Then (6.5) and the fact $u=v$ for $t\in[0,T]$ show that $v$ satisfies (replace
$u$ by $v$ for $t\leq T$ in (6.5))

$v_{t}=A^{ij}_{T}v_{x^{i}x^{j}}+fI_{t<T},\quad t>0\,;\,\,v(0,\cdot)=0.$ (6.6)

By Case 1, $v\in\mathfrak{H}^{\gamma+2}_{p,\theta}(\infty)$ is the unique
solution of (6.6), and $u=v$ on $[0,T]$ whenever $u$ is a solution of (6.4) on
$[0,T]$. This obviously yields the uniqueness. The theorem is proved. $\Box$

Proof of Theorem 3.13 The proof is very similar to that of the proof of
Theorem 3.10 and is based on (5.13). We leave the details to the readers as an
exercise.

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