# Parabolic Littlewood-Paley inequality for a class of time-dependent
operators of arbitrary order, and applications to higher order stochastic PDE

Ildoo Kim  Department of Mathematics, Korea University, 1 Anam-dong, Sungbuk-
gu, Seoul, 136-701, Republic of Korea<EMAIL_ADDRESS>,  Kyeong-Hun Kim
Department of Mathematics, Korea University, 1 Anam-dong, Sungbuk-gu, Seoul,
136-701, Republic of Korea<EMAIL_ADDRESS>and  Sungbin Lim  Department
of Mathematics, Korea University, 1 Anam-dong, Sungbuk-gu, Seoul, 136-701,
Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

In this paper we prove a parabolic version of the Littlewood-Paley inequality
for a class of time-dependent local and non-local operators of arbitrary
order, and as an application we show this inequality gives a fundamental
estimate for the $L_{p}$-theory of the stochastic partial differential
equations.

###### Key words and phrases:

Parabolic Littlewood-Paley inequality, Stochastic partial differential
equations, Time-dependent high order operators, Non-local operators of
arbitrary order

###### 2010 Mathematics Subject Classification:

42B25, 26D10, 60H15, 35G05, 47G30

The research of the second author was supported by Basic Science Research
Program through the National Research Foundation of Korea(NRF) funded by the
Ministry of Education, Science and Technology (20110015961)

##  1\. Introduction

The classical Littlewood-Paley inequality says (see [12]) that for any
$p\in(1,\infty)$ and $f\in L_{p}(\mathbf{R}^{d})$,

$\int_{\mathbf{R}^{d}}\left(\int^{\infty}_{0}|\sqrt{-\Delta}\,e^{t\Delta}f|^{2}dt\right)^{p/2}dx\leq
N(p)\|f\|^{p}_{p},$ (1.1)

where $e^{t\Delta}f(x):=S_{t}f=p(t,\cdot)*f(\cdot)=\frac{1}{(4\pi
t)^{d/2}}\int_{\mathbf{R}^{d}}f(x-y)e^{\frac{-|y|^{2}}{4t}}dy$. In [5, 7]
Krylov proved the following parabolic version, in which $H$ is a Hilbert space
: for any $p\in[2,\infty),-\infty\leq a<b\leq\infty$, $f\in
L_{p}((a,b)\times\mathbf{R}^{d},H)$,

$\|(\int^{t}_{a}|(\sqrt{-\Delta}e^{(t-s)\Delta}f)(s,x)|^{2}_{H}\,ds)^{1/2}\|^{p}_{L_{p}((a,b)\times\mathbf{R}^{d})}\leq
N(p)\||f|_{H}\|^{p}_{L_{p}((a,b)\times\mathbf{R}^{d})}.$ (1.2)

Some related works and the significance of the parabolic Littlewood-Paley
inequality in the $L_{p}$-theory of stochastic PDEs will be discussed later.

If $f=f(x)$ and $H=\mathbf{R}$ then by (1.2) with $a=0$ and $b=2$,

$\displaystyle\int_{\mathbf{R}^{d}}[\int^{1}_{0}|\sqrt{-\Delta}e^{s\Delta}f|^{2}ds]^{p/2}dx$
$\displaystyle\leq\int_{\mathbf{R}^{d}}\int^{2}_{1}[\int^{t}_{0}|\sqrt{-\Delta}e^{(t-s)\Delta}f|^{2}ds]^{p/2}dtdx\leq
2N(p)\|f\|^{p}_{L_{p}(\mathbb{R}^{d})}.$

This and the scaling
$(\sqrt{-\Delta}S_{t}f(c\cdot))(x)=\sqrt{-\Delta}(cS_{c^{2}t}f)(cx)$ yield
(1.1). Hence (1.2) is a generalization of (1.1). Note that by putting
$K_{0}(t,x)=\sqrt{-\Delta}p(t,x)$, we get
$\sqrt{-\Delta}e^{(t-s)\Delta}f=K_{0}(t-s,\cdot)*f(s,\cdot)$ and therefore
(1.2) becomes

$\|(\int^{t}_{a}|K_{0}(t-s,\cdot)*f(s,\cdot)(x)|^{2}_{H}ds)^{1/2}\|^{p}_{L_{p}((a,b)\times\mathbf{R}^{d})}\leq
N\||f|_{H}\|^{p}_{L_{p}((a,b)\times\mathbf{R}^{d})}.$ (1.3)

In this article we extend (1.3) to a class of time-dependent operators. For a
wide class of differential operators $A(t)$ with symbol $\psi(t,\xi)$, one can
define the kernel

$p(t,s,x)=p_{A}(t,s,x)=\mathcal{F}^{-1}(\exp(\int_{s}^{t}\psi(r,\xi))dr))(x)$

so that the solution of

$u_{t}=A(t)u+f,\quad u(0)=0$

is given by

$u=\int^{t}_{0}p(t,s,\cdot)*f(s,\cdot)ds.$

We provide a classification of operators $A(t)$ for which (1.3) holds with
formally

$K_{A}(t,s,x)=\sqrt{-A(t)}p(t,s,x).$

More generally, we provide sufficient conditions on measurable functions
$K(t,s,x)$ on $\mathbf{R}^{d+2}$ so that

$\|(\int^{t}_{a}|K(t,s,\cdot)*f(s,\cdot)(x)|^{2}_{H}ds)^{1/2}\|^{p}_{L_{p}((a,b)\times\mathbf{R}^{d})}\leq
N\||f|_{H}\|^{p}_{L_{p}((a,b)\times\mathbf{R}^{d})}$ (1.4)

holds for any $f\in C^{\infty}_{0}(\mathbf{R}^{d+1},H)$ with constant $N$
independent of $f,a$ and $b$. The functions $K(t,s,x)$ are assumed to satisfy
the conditions described in Assumptions 2.1 and 2.2.

For concrete examples we introduce the operators $A_{1}(t)$ of $2m$-order
($m=1,2,3,\cdots$) and $A_{2}(t)$ of order $\gamma\in(0,\infty)$

$A_{1}(t)u:=(-1)^{m-1}\sum_{|\alpha|=|\beta|=m}a^{\alpha\beta}(t)D^{\alpha+\beta}u,\quad\quad
A_{2}(t)u:=-a(t)(-\Delta)^{\gamma/2}$ (1.5)

where $-a(t)(-\Delta)^{\gamma/2}$ is the operator with symbol
$-a(t)|\xi|^{\gamma}$ and the coefficients $a(t)$ and $a^{\alpha\beta}(t)$ are
bounded measurable in $t$ and satisfy the ellipticity conditions

$0<\nu<\Re[a(t)]<\nu^{-1},$

and

$\displaystyle\nu|\xi|^{2m}\leq\sum_{|\alpha|=|\beta|=m}\xi^{\alpha}\xi^{\beta}\Re\left[a^{\alpha\beta}(t)\right]\leq\nu^{-1}|\xi|^{2m}.$

Here $\Re[z]$ is the real part of $z$. Let $p_{1}(t,s,x)$ and $p_{2}(t,s,x)$
be the kernels related to $A_{1}(t)$ and $A_{2}(t)$ respectively. We prove
that (1.4) holds with

$K_{1}(t,s,x):=D^{m}p_{1}(t,s,x),\quad
K_{2}(t,s,x):=(-\Delta)^{\gamma/4}p_{2}(t,s,x).$

Letting the function $f$ depend only on $x$, one can obtain elliptic versions
of these results. For instance, we have for any $\gamma\in(0,\infty)$ and
$f\in L_{p}(\mathbf{R}^{d})$,

$\int_{\mathbf{R}^{d}}\left(\int^{\infty}_{0}|(-\Delta)^{\gamma/2}e^{-t(-\Delta)^{\gamma}}f|^{2}dt\right)^{p/2}dx\leq
N(p,\gamma)\|f\|^{p}_{p},\quad\quad\forall\,\gamma\in(0,\infty),$

which is an extension of (1.1), the classical (elliptic) Littlewood-Paley
inequality.

Among many other examples of (1.4) are the product $A_{1}(t)A_{2}(t)$ and
$(-\Delta)^{k}\mathcal{L}_{0}(t)$ ($k=0,1,2,\cdots)$, where

$\mathcal{L}_{0}(t)u=\int_{\mathbf{R}^{d}}\Big{(}u(x+y)-u(x)-\chi(y)(\nabla
u(x),y)\Big{)}m(t,y)\frac{dy}{|y|^{d+\gamma}},$ (1.6)

$\gamma\in(0,2)$, $\chi(y)=I_{\gamma>1}+I_{\gamma=1}I_{|y|\leq 1}$, and
$m(t,y)\geq 0$ satisfies a certain condition described in Corollary 2.8. Note
that if $m(t,y)\equiv 1$ then $\mathcal{L}_{0}=-(-\Delta)^{\gamma/2}$.

One of important applications of the parabolic Littlewood-Paley inequality is
the theory of stochastic partial differential equations of the type

$du=A_{i}(\omega,t)u\,\,dt+\sum_{k=1}^{\infty}f^{k}\,\,dw^{k}_{t},\quad
u(0,x)=0.$ (1.7)

Here $f=(f^{1},f^{2},\cdots)$ is an $\ell_{2}$-valued random function
depending on $(t,x)$, and $w^{k}_{t}$ are independent one-dimensional Wiener
processes defined on a probability space $(\Omega,P)$. The operators
$A_{i}=A_{i}(\omega,t)$ are defined in (1.5), but this time we allow the
coefficients $a(\omega,t)$ and $a^{\alpha\beta}(\omega,t)$ to depend also on
$\omega\in\Omega$. It turns out that if $f=(f^{1},f^{2},\cdots)\in
L_{p}(\Omega\times(0,\infty)\times\mathbf{R}^{d},\ell_{2})$ satisfies a
certain measurability condition, the solutions of these problems are given by

$u_{i}(t,x)=\sum_{k=1}^{\infty}\int^{t}_{0}p_{i}(t,s,\cdot)*f^{k}(s,\cdot)(x)dw^{k}_{s},\quad
i=1,2$ (1.8)

where $p_{i}(t,s,x)$ are introduced above, but they are random due to the
randomness of the coefficients. The derivation of formula (1.8) can be found
in [8] when $A_{i}=\Delta$, and by repeating the arguments in [8] one can
derive (1.8) for such $A_{i}$. By Burkholder-Davis-Gundy inequality (see [6]),
we have

$\displaystyle\|D^{m}u_{1}$
$\displaystyle(t,\cdot)\|^{p}_{L_{p}(\Omega\times(0,\infty)\times\mathbf{R}^{d})}$
$\displaystyle\leq
N(p)\|[\int^{t}_{0}|D^{m}p_{1}(t,s,\cdot)*f(s,\cdot)(x)|^{2}_{\ell_{2}}ds]^{1/2}\|^{p}_{L_{p}(\Omega\times(0,\infty)\times\mathbf{R}^{d})}.$
(1.9)

The corresponding inequality for $u_{2}$ also holds with $p_{2}$ and
$(-\Delta)^{\gamma/4}$ in place of $p_{1}$ and $D^{m}$ respectively. Actually
if $f$ is not random, then $u_{1}$ and $u_{2}$ become Gaussian processes and
the reverse inequalities also hold. Thus to prove

$D^{m}u_{1},\,\,(-\Delta)^{\gamma/4}u_{2}\,\,\in
L_{p}(\Omega\times(0,\infty)\times\mathbf{R}^{d})$

and to get a legitimate start of the $L_{p}$-theory of stochastic PDEs of type
(1.7), one has to estimate the right-hand side of (1.9). Obviously (1.4) with
$K_{1}$ and (1.9) imply

$\|D^{m}u_{1}(t,\cdot)\|^{p}_{L_{p}(\Omega\times(0,\infty)\times\mathbf{R}^{d})}\leq
N(p,m)\||f|_{\ell_{2}}\|^{p}_{L_{p}(\Omega\times(0,\infty)\times\mathbf{R}^{d})}.$
(1.10)

Using (1.10) and following the ideas in [8], one can construct an
$L_{p}$-theory of the general $2m$-order stochastic PDEs. Similarly one can
construct an $L_{p}$-theory of stochastic PDEs with the operator
$A_{2}(\omega,t)$. We acknowledge that if the coefficients $a^{\alpha\beta}$
are independent of $t$ then inequality (1.10) for high order stochastic PDEs
is also introduced in [11] on the basis of $H^{\infty}$-functional calculus
which is far different from our approach. One of advantages of our approach is
that no regularity condition of the coefficients with respect to time variable
is required.

Below is a short description on the related works. As mentioned above
parabolic Littlewood-Paley inequality related to the Laplacian $\Delta$ was
first proved by Krylov in [5, 7]. This result is considered as a foundation of
the $L_{p}$-theory of the second-order stochastic partial differential
equations. Recently the parabolic Littlewood-Paley inequality was proved for
the fractional Laplacian $(-\Delta)^{\gamma/2}$, $\gamma\in(0,2)$, in [1, 2],
and a slight extension of the result of [1, 2] was made to the operator
$\mathcal{L}_{0}(t)$ in [10] and to the operator with symbol
$-\phi(|\xi|^{2})$ in [4], where $\mathcal{L}_{0}(t)$ is from (1.6) and $\phi$
is a Bernstein function satisfying

$c^{-1}\lambda^{\delta_{1}}\phi(t)\leq\phi(\lambda t)\leq
c\lambda^{\delta_{2}}\phi(t),\quad\quad\forall\,\,\lambda,t\geq 1,$
$\phi(\lambda t)\leq
c\lambda^{\delta_{3}}\phi(t),\quad\quad\forall\,\,\lambda,t\leq 1$

with some constants $c>1$, $0<\delta_{1}\leq\delta_{2}<1$ and
$\delta_{3}\in(0,1]$. The operators considered in [1, 2, 4, 10] are of order
less than $2$, they (except $\mathcal{L}_{0}(t)$) do not depend on $t$. The
novelty of this article is that it extends existing results which have been
proved for lower order operators independent of $t$ to the time-dependent
local and non-local operators of arbitrary order.

Next we briefly describe our approach to prove (1.4). We estimate the sharp
function of $(\int^{t}_{a}|K(t,s,\cdot)*f(s,\cdot)(x)|^{2}_{H}ds)^{1/2}$ in
terms of the maximal function of $|f|_{H}$, then apply Fefferman-Stein theorem
and Hardy-Littlewood maximal theorem. The operators considered in [1, 2, 4]
are the infinitesimal generators of certain Lévy processes, and the related
kernels $p(t,x)$ are transition densities of these processes. Thus to estimate
the sharp function of
$(\int^{t}_{a}|K(t,s,\cdot)*f(s,\cdot)(x)|^{2}_{H}ds)^{1/2}$, appropriate
bounds of the transition densities can be used as in [1, 2, 4]. But for high
order operators there is no such related Lévy process and this method can not
be applied. Instead, we modify the idea in [10] and make a good use of
Parseval’s identity which enables us to avoid using estimates of the kernels
related to the operators.

Finally we introduce some notation used in the article. As usual
$\mathbf{R}^{d}$ stands for the Euclidean space of points
$x=(x^{1},...,x^{d})$, $B_{r}(x):=\\{y\in\mathbf{R}^{d}:|x-y|<r\\}$ and
$B_{r}:=B_{r}(0)$. For multi-indices $\alpha=(\alpha_{1},...,\alpha_{d})$,
$\alpha_{i}\in\\{0,1,2,...\\}$, $x\in\mathbf{R}^{d}$, and functions $u(x)$ we
set

$u_{x^{i}}=\frac{\partial u}{\partial x^{i}}=D_{i}u,\quad\quad
D^{\alpha}u=D_{1}^{\alpha_{1}}\cdot...\cdot D^{\alpha_{d}}_{d}u,$
$x^{\alpha}=(x^{1})^{\alpha_{1}}(x^{2})^{\alpha_{2}}\cdots(x^{d})^{\alpha_{d}},\quad\quad|\alpha|=\alpha_{1}+\cdots+\alpha_{d}.$

We also use $D^{m}_{x}$ to denote a partial derivative of order $m$ with
respect to $x$. For an open set $U\subset\mathbf{R}^{d}$ and a nonnegative
integer $n$, we write $u\in C^{n}(U)$ if $u$ is $n$-times continuously
differentiable in $U$. By $C^{n}_{0}(U)$ (resp. $C^{\infty}_{0}(U)$) we denote
the set of all functions in $C^{n}(U)$ (resp. $C^{\infty}(U)$) with compact
supports. The standard $L_{p}$-space on $U$ with Lebesgue measure is denoted
by $L_{p}(U)$. Similarly, by $C_{0}^{\infty}(\mathbf{R}^{d},H)$ we denote the
set of $H$-valued infinitely differentiable functions with compact support. We
use “$:=$” to denote a definition. $a\wedge b=\min\\{a,b\\}$, $a\vee
b=\max\\{a,b\\}$ and $\lfloor a\rfloor$ is the biggest integer which is less
than or equal to $a$. By $\mathcal{F}$ and $\mathcal{F}^{-1}$ we denote the
Fourier transform and the inverse Fourier transform, respectively. That is,
$\mathcal{F}(f)(\xi):=\int_{\mathbf{R}^{d}}e^{-ix\cdot\xi}f(x)dx$ and
$\mathcal{F}^{-1}(f)(x):=\frac{1}{(2\pi)^{d}}\int_{\mathbf{R}^{d}}e^{i\xi\cdot
x}f(\xi)d\xi$. For a Borel set $X\subset\mathbf{R}^{d}$, we use $|X|$ to
denote its Lebesgue measure and by $I_{X}(x)$ we denote the indicator of $A$.
For a sequence $a=(a_{1},a_{2},a_{3},\ldots)$, we define
$|a|_{\ell_{2}}=\big{(}\sum_{k=1}^{\infty}a_{k}^{2}\big{)}^{1/2}$. If we write
$N=N(a,\ldots,z)$, this means that the constant $N$ depends only on
$a,\ldots,z$.

##  2\. Main results

In this section we prove (1.4), a generalized version of the parabolic
Littlewood-Paley inequality, under the following conditions on the kernel
$K(t,s,x)$ and provide a classification of operators $A(t)$ for which (1.4)
holds with $K=K_{A}$ (see (1)). Three interesting examples related to the
operators $A_{1}(t),A_{2}(t)$ and $(-\Delta)^{k}\mathcal{L}_{0}(t)$ are also
presented.

Assumption 2.1 below is needed to prove (1.4) for $p=2$.

######  Assumption 2.1.

The kernel $K(t,s,x)$ is a measurable function defined on $\mathbf{R}^{d+2}$
satisfying

$\displaystyle\int_{s}^{\infty}|\mathcal{F}\\{K(t,s,\cdot)\\}(\xi)|^{2}dt\leq
C_{0}$ (2.1)

with constant $C_{0}$ independent of $(s,\xi)$.

Take a constant $c_{2}>\frac{1}{2}$ and denote

$\displaystyle c_{3}:=\frac{2(d+1)(c_{2}+1)+3}{2(d+2)}.$ (2.2)

######  Assumption 2.2.

(i) For almost all $t$ and each $s<t$, $K(t,s,\cdot)$ $D_{x}K(t,s,\cdot)$ and
$\frac{\partial}{\partial t}K(t,s,x)$ are locally integrable functions of $x$.

(ii) There exist functions $F_{i}(t,s,x)$ and positive constants
$\sigma_{i},\kappa_{i}$ ($i=1,2,3)$ and $C$ such that for almost all $t$ and
each $s<t$ and $x\in\mathbf{R}^{d}\setminus\\{0\\}$,

$\displaystyle\big{|}D_{x}K(t,s,x)\big{|}\leq
C\big{|}(t-s)^{-\sigma_{1}}F_{1}\big{(}t,s,(t-s)^{-\kappa_{1}}x\big{)}\big{|},$
(2.3)

$\displaystyle\big{|}D^{2}_{x}K(t,s,x)\big{|}\leq
C\Big{(}(t-s)^{-\sigma_{2}}\big{|}F_{2}(t,s,(t-s)^{{-\kappa_{2}}}x)\big{|}\wedge(t-s)^{{-c_{2}}}\Big{)},$
(2.4)

$\displaystyle\big{|}\frac{\partial}{\partial t}D_{x}K(t,s,x)\big{|}\leq
C\Big{(}(t-s)^{{-\sigma_{3}}}\big{|}F_{3}(t,s,(t-s)^{{-\kappa_{3}}}x)\big{|}\wedge(t-s)^{{-c_{3}}}\Big{)}.$
(2.5)

(iii) For these $F_{i}$ ($i=1,2,3$), we have

$\displaystyle\sup_{s<t}\int_{\mathbf{R}^{d}}|x|^{\mu_{1}}|F_{1}(t,s,x)|^{2}dx<\infty,$
(2.6)

$\displaystyle\sup_{s<t}\int_{|x|\geq(t-s)^{c_{2}-c_{3}+1-\kappa_{2}}}|x|^{\mu_{2}}|F_{2}(t,s,x)|^{2}dx<\infty,$
(2.7)

$\displaystyle\sup_{s<t}\int_{|x|\geq(t-s)^{c_{2}-c_{3}+1-\kappa_{3}}}|x|^{\mu_{3}}|F_{3}(t,s,x)|^{2}dx<\infty,$
(2.8)

where $\mu_{i}>d+2$ ($i=1,2,3$) satisfy the following system

$\displaystyle\left\\{\begin{array}[]{c}(c_{2}-c_{3}+1-\kappa_{1})\mu_{1}=d(\kappa_{1}+c_{2}-c_{3}+1)+2(c_{2}-c_{3}-\sigma_{1})+3\\\
\\\
(c_{2}-c_{3}+1-\kappa_{2})\mu_{2}=d(\kappa_{2}-c_{2}+c_{3}-1)+2(c_{2}-\sigma_{2})\\\
\\\
(c_{2}-c_{3}+1-\kappa_{3})\mu_{3}=d(\kappa_{3}-c_{2}+c_{3}-1)+2(c_{3}-\sigma_{3})\\\
\end{array}\right\\}.$ (2.14)

######  Remark 2.3.

(i) Suppose

$\sup_{s<t}\int_{\mathbf{R}^{d}}|F_{i}(t,s,x)|^{2}dx<\infty,\quad(i=1,2,3)$

and

$\sup_{s<t}\int_{\mathbf{R}^{d}}|x|^{\hat{\mu}_{i}}|F_{i}(t,s,x)|^{2}dx<\infty\quad(i=1,2,3)$

with some $(\hat{\mu}_{1},\hat{\mu}_{2},\hat{\mu}_{3})\in\mathbf{R}^{3}$. Then
obviously (2.6)-(2.8) hold for any $\mu_{i}\leq\hat{\mu}_{i}$ ($i=1,2,3$).

(ii) Suppose, for example, $c_{2}-c_{3}+1-\kappa_{2}=0$. Then in (2.14) we are
assuming

$d(\kappa_{2}-c_{2}+c_{3}-1)+2(c_{2}-\sigma_{2})=0.$

In this case, we have a freedom of choosing $\mu_{2}$, that is we can choose
arbitrary $\mu_{2}>d+2$ satisfying (2.6).

(iii) Put

$\displaystyle\delta_{0}:=c_{2}-c_{3}+1,\quad\Theta(\theta,\vartheta):=\theta
d-2\vartheta.$ (2.15)

One can easily check

$\Theta(\theta_{1}+\theta_{2},\vartheta_{1}+\vartheta_{2})=\Theta(\theta_{1},\vartheta_{1})+\Theta(\theta_{2},\vartheta_{2}),$

and (2.14) is equivalent to

$\displaystyle\left[\begin{array}[]{ccc}\delta_{0}-\kappa_{1}&&\\\
&\delta_{0}-\kappa_{2}&\\\ &&\delta_{0}-\kappa_{3}\\\
\end{array}\right]\left[\begin{array}[]{c}\mu_{1}\\\ \mu_{2}\\\ \mu_{3}\\\
\end{array}\right]=\left[\begin{array}[]{c}\Theta(\kappa_{1}+\delta_{0},\sigma_{1}-\delta_{0})+1\\\
\Theta(\kappa_{2}-\delta_{0},\sigma_{2}-c_{2})\\\
\Theta(\kappa_{3}-\delta_{0},\sigma_{3}-c_{3})\\\ \end{array}\right].$ (2.25)

Note that to prove (1.4) we may assume $a=-\infty$ and $b=\infty$. Recall $H$
denote a Hilbert space. Here are the main results of this article. The proofs
of Theorems 2.4 and 2.5 are given in Sections 4 and 5 respectively.

######  Theorem 2.4.

Let $p\geq 2$. Suppose that Assumptions 2.1 and 2.2 hold. Then for any $f\in
C^{\infty}_{0}(\mathbf{R}^{d+1},H)$,

$\displaystyle\left\|\Big{(}\int_{-\infty}^{t}\big{|}K(t,s,\cdot)\ast
f(s,\cdot)(x)\big{|}_{H}^{2}ds\Big{)}^{1/2}\right\|_{L_{p}(\mathbf{R}^{d+1})}\leq
N\||f|_{H}\|_{L_{p}(\mathbf{R}^{d+1})},$

where $N$ is independent of $f$.

Let $A(t)$ be a non-positive operator with the symbol $\psi(t,\xi)$, that is

$\mathcal{F}(A(t)u)(\xi)=\psi(t,\xi)\mathcal{F}(u)(\xi),\quad\forall\,u\in
C^{\infty}_{0}(\mathbf{R}^{d}).$

Define the kernel $p(t,s,x)$ by the formula

$\displaystyle p(t,s,x)=I_{0\leq
s<t}\mathcal{F}^{-1}\Big{(}\exp\big{(}\int_{s}^{t}\psi(r,\xi))dr\big{)}\Big{)}(x).$
(2.26)

######  Theorem 2.5.

Fix $p\geq 2$ and $\gamma>0$. Assume there exist constants $\nu>0$ such that
for any multi-index $|\alpha|\leq\lfloor\frac{d}{2}\rfloor+2$,

$\Re[\psi(t,\xi)]\leq-\nu|\xi|^{\gamma},$ (2.27)
$|D^{\alpha}\psi(t,\xi)|\leq\nu^{-1}|\xi|^{\gamma-|\alpha|}$ (2.28)

hold for almost every $t>0$ and $\xi\neq 0$. Then for any $f\in
C^{\infty}_{0}(\mathbf{R}^{d+1},H)$

$\displaystyle\left\|\Big{(}\int_{0}^{t}|\Delta^{\gamma/4}p(t,s,\cdot)\ast
f(s,\cdot)(x)|_{H}^{2}~{}ds\Big{)}^{1/2}\right\|_{L_{p}(\mathbf{R}^{d+1})}\leq
N\||f|_{H}\|_{L_{p}(\mathbf{R}^{d+1})},$

where $N$ depends only on $p,\nu,\gamma$ and $d$.

For applications of Theorem 2.5 we recall the operators $A_{i}(t)$ from (1.5),
that is,

$A_{1}(t)u=(-1)^{m-1}\sum_{|\alpha|=|\beta|=m}a^{\alpha\beta}(t)D^{\alpha+\beta}u,\quad\quad
A_{2}(t)=-a(t)(-\Delta)^{\gamma/2}$

where the coefficients $a^{\alpha\beta}$ and $a(t)$ are bounded complex-valued
measurable functions satisfying $\nu<\Re[a(t)]<\nu^{-1}$ and

$\displaystyle\nu|\xi|^{2m}\leq\sum_{|\alpha|=|\beta|=m}\xi^{\alpha}\xi^{\beta}\Re\left[a^{\alpha\beta}(\omega,t)\right]\leq\nu^{-1}|\xi|^{2m},\quad\forall\xi\in\mathbf{R}^{d}.$

Denote

$p_{1}(t,s,z)=p_{1,m}(t,s,x)=I_{0\leq
s<t}\mathcal{F}^{-1}\Big{(}\exp\big{\\{}-\int_{s}^{t}a^{\alpha\beta}(r)\xi^{\alpha}\xi^{\beta}dr\big{\\}}\Big{)}(x),$
$p_{2}(t,s,x)=p_{2,\gamma}(t,s,z)=I_{0\leq
s<t}\mathcal{F}^{-1}\Big{(}\exp\big{\\{}-|\xi|^{\gamma}\int_{s}^{t}a(r)dr\big{\\}}\Big{)}(x).$

######  Corollary 2.6.

Let $p\geq 2$. Then for any $f\in C^{\infty}_{0}(\mathbf{R}^{d+1},H)$,

$\displaystyle\left\|\Big{(}\int_{0}^{t}|\Delta^{m/2}p_{1}(t,s,\cdot)\ast
f(s,\cdot)(x)|_{H}^{2}~{}ds\Big{)}^{1/2}\right\|_{L_{p}(\mathbf{R}^{d+1})}\leq
N\||f|_{H}\|_{L_{p}(\mathbf{R}^{d+1})},$

where $N$ depends only on $p,\nu,m$ and $d$.

###### Proof.

It is obvious that the symbol
$\psi(t,\xi)=-a^{\alpha\beta}(t)\xi^{\alpha}\xi^{\beta}$ satisfies (2.27) and
(2.28) with $\gamma=2m$ and any multi-index $\alpha$. Thus the corollary
follows from Theorem 2.5. $\Box$

######  Corollary 2.7.

Let $p\geq 2$. Then for any $f\in C^{\infty}_{0}(\mathbf{R}^{d+1},H)$,

$\displaystyle\left\|\Big{(}\int_{0}^{t}|\Delta^{\gamma/4}p_{2}(t,s,\cdot)\ast
f(s,\cdot)(x)|_{H}^{2}~{}ds\Big{)}^{1/2}\right\|_{L_{p}(\mathbf{R}^{d+1})}\leq
N\||f|_{H}\|_{L_{p}(\mathbf{R}^{d+1})},$

where $N$ depends only on $p,\nu,\gamma$ and $d$.

###### Proof.

The symbol related to the operator $A_{2}(t)$ is $-a(t)|\xi|^{\gamma}$, and
therefore the corollary follows from Theorem 2.5. $\Box$

Recall we defined $(-\Delta)^{\gamma/2}$ as the operator with symbol
$|\xi|^{\gamma}$ for any $\gamma\in(0,\infty)$. For further applications of
Theorem 2.5, we consider a product of $(-\Delta)^{k}$ and an integro-
differential operator $\mathcal{L}_{0}=\mathcal{L}_{0,\gamma}$. We remark that
in place of $(-\Delta)^{k}$ one can consider many other pseudo-differential or
high order differential operators.

Fix $\gamma\in(0,2)$, and for $k=0,1,2,\cdots$ denote

$\displaystyle\mathcal{L}_{k}(t)u=(-\Delta)^{k}\mathcal{L}_{0,\gamma}u:=$
$\displaystyle\int_{\mathbf{R}^{d}\setminus\\{0\\}}\Big{(}(-\Delta)^{k}u(t,x+y)-(-\Delta)^{k}u(t,x)-\chi(y)(\nabla(-\Delta)^{k}u(t,x),y)\Big{)}m(t,y)\frac{dy}{|y|^{d+\gamma}}$

where $\chi(y)=I_{\gamma>1}+I_{|y|\leq 1}I_{\gamma=1}$ and $m(t,y)\geq 0$ is
measurable function satisfying the following conditions :

(i) If $\gamma=1$ then

$\displaystyle\int_{\partial B_{1}}wm(t,w)~{}S_{1}(dw)=0,\quad\forall t>0,$
(2.29)

where $\partial B_{1}$ is the unit sphere in $\mathbf{R}^{d}$ and $S_{1}(dw)$
is the surface measure on it.

(ii) The function $m=m(t,y)$ is zero-order homogeneous and differentiable in
$y$ up to $d_{0}=\lfloor\frac{d}{2}\rfloor+2$.

(iii) There is a constant $K$ such that for each $t\in\mathbf{R}$

$\sup_{|\alpha|\leq d_{0},|y|=1}|D^{\alpha}_{y}m^{(\alpha)}(t,y)|\leq K.$

It turns out that the operator $\mathcal{L}_{k}$ is a pseudo differential
operator with symbol

$\displaystyle\psi(t,\xi)=-c_{1}|\xi|^{2k}\int_{\partial
B_{1}}|(w,\xi)|^{\gamma}[1-i\varphi^{(\gamma)}(w,\xi)]m(t,w)~{}S_{1}(dw),$
$\displaystyle\varphi^{(\gamma)}(w,\xi)=c_{2}\frac{(w,\xi)}{|(w,\xi)|}I_{\gamma\neq
1}-\frac{2}{\pi}\frac{(w,\xi)}{|(w,\xi)|}\ln|(w,\xi)|I_{\gamma=1},$

and $c_{1}(\gamma,d)$, $c_{2}(\gamma,d)$ are certain positive constants.

(iv) There is a constant $N_{0}>0$ such that the symbol $\psi(t,\xi)$ of
$\mathcal{L}_{k}$ satisfies

$\sup_{t,|\xi|=1}\Re[\psi(t,\xi)]\leq-N_{0}.$ (2.30)

One can check that (2.30) holds if there exists a constant $c>0$ so that
$m(t,y)>c$ on a set $E\subset\partial B_{1}$ of positive $S_{1}(dw)$-measure.

######  Corollary 2.8.

Let $p\geq 2$ and $p(t,s,x)$ be the kernel related to $\mathcal{L}_{k}(t)$.
Then under above conditions (i)-(iv) on $m(t,y)$ it holds that for any $f\in
C^{\infty}_{0}(\mathbf{R}^{d+1},H)$

$\displaystyle\left\|\Big{(}\int_{0}^{t}|\Delta^{k/2+\gamma/4}p(t,s,\cdot)\ast
f(s,\cdot)(x)|_{H}^{2}~{}ds\Big{)}^{1/2}\right\|_{L_{p}(\mathbf{R}^{d+1})}\leq
N\||f|_{H}\|_{L_{p}(\mathbf{R}^{d+1})},$

where $N$ depends only on $p,\gamma,k,d,N_{0}$ and $K$.

###### Proof.

Note that for $\xi\neq 0$

$\displaystyle\psi(t,\xi)$
$\displaystyle=|\xi|^{2k+\gamma}\psi\Big{(}t,\frac{\xi}{|\xi|}\Big{)}=:|\xi|^{2k+\gamma}\tilde{\psi}(t,\xi).$

The above equality is obvious if $\gamma\neq 1$, and if $\gamma=1$ then by
(2.29)

$\displaystyle\psi(t,\xi)$
$\displaystyle=|\xi|^{2k+1}\psi\Big{(}t,\frac{\xi}{|\xi|}\Big{)}+|\xi|^{2k}\ln|\xi|\int_{\partial
B_{1}}(w,\xi)m(t,w)~{}S_{1}(dw)$
$\displaystyle=|\xi|^{2k+1}\psi\Big{(}t,\frac{\xi}{|\xi|}\Big{)}.$

By using condition (iii) one can check (see e.g. [9, Remark 2.6]) that for any
multi-index $\alpha$, $|\alpha|\leq d_{0}$, there exists a constant
$N=N(\alpha)$ such that

$|D^{\alpha}\tilde{\psi}(t,\xi)|\leq N|\xi|^{-|\alpha|}.$

Thus it is obvious that the given symbol $\psi$ satisfies (2.27) and (2.28).
The corollary is proved.

$\Box$

##  3\. Some preliminary estimates

For $f\in C^{\infty}_{0}(\mathbf{R}^{d+1},H)$, we define

$\displaystyle\mathcal{G}f(t,x):=\left(\int_{-\infty}^{t}\big{|}K(t,s,\cdot)\ast
f(s,\cdot)(x)\big{|}_{H}^{2}ds\right)^{1/2}.$

######  Lemma 3.1.

Let Assumption 2.1 hold and $f\in C^{\infty}_{0}(\mathbf{R}^{d+1},H)$. Then
for any $-\infty\leq a\leq b\leq\infty$,

$\displaystyle\|\mathcal{G}f\|^{2}_{L_{2}((a,b)\times\mathbf{R}^{d})}\leq
N\||f|_{H}\|^{2}_{L_{2}((-\infty,b)\times\mathbf{R}^{d})},$ (3.1)

where $N=N(d,C_{0})$.

###### Proof.

By the continuity of $f$, the range of $f$ belongs to a separable subspace of
$H$. Thus by using a countable orthonormal basis of this subspace and the
Fourier transform one easily finds

$\displaystyle\|\mathcal{G}f\|^{2}_{L_{2}((a,b)\times\mathbf{R}^{d})}$
$\displaystyle=(2\pi)^{d}\int_{\mathbf{R}^{d}}\int_{a}^{b}\int_{-\infty}^{t}|\mathcal{F}\\{K(t,s,\cdot)\\}(\xi)|^{2}\,|\mathcal{F}(f)(s,\xi)|^{2}_{H}dsdtd\xi$
$\displaystyle\leq(2\pi)^{d}\int_{\mathbf{R}^{d}}\int_{-\infty}^{b}\int_{a}^{b}I_{0\leq
t-s}|\mathcal{F}\\{K(t,s,\cdot)\\}(\xi)|^{2}dt|\mathcal{F}(f)(s,\xi)|^{2}_{H}dsd\xi$
$\displaystyle\leq(2\pi)^{d}\int_{\mathbf{R}^{d}}\int_{-\infty}^{b}\left(\int_{s}^{\infty}|\mathcal{F}\\{K(t,s,\cdot)\\}(\xi)|^{2}dt\right)|\mathcal{F}(f)(s,\xi)|^{2}_{H}dsd\xi.$

From (2.1), we have

$\displaystyle\|\mathcal{G}f\|^{2}_{L_{2}((a,b)\times\mathbf{R}^{d})}\leq
N\int_{-\infty}^{b}\int_{\mathbf{R}^{d}}|\mathcal{F}(f)(s,\xi)|^{2}_{H}~{}d\xi
ds.$

The last expression is equal to the right-hand side of (3.1), and therefore
the lemma is proved. $\Box$

######  Corollary 3.2.

Let $r_{1},r_{2}>0$. Suppose that Assumption 2.1 holds, $f\in
C^{\infty}_{0}(\mathbf{R}^{d+1},H)$, and $f(t,x)=0$ for $x\not\in B_{3r_{1}}$.
Then

$\int_{-2r_{2}}^{0}\int_{B_{r_{1}}}|\mathcal{G}f(s,y)|^{2}dyds\leq
N(d,C_{0})\int_{-\infty}^{0}\int_{B_{3r_{1}}}|f(s,y)|_{H}^{2}dyds.$

###### Proof.

Applying Lemma 3.1 with $a=-2r_{2}$ and $b=0$ and using the condition on $f$,
we gt

$\displaystyle\int_{-2r_{2}}^{0}\int_{B_{r_{1}}}|\mathcal{G}f(s,y)|^{2}dyds$
$\displaystyle\leq\int_{-\infty}^{0}\int_{\mathbf{R}^{d}}|\mathcal{G}f(s,y)|^{2}dyds$
$\displaystyle\leq
N\int_{-\infty}^{0}\int_{\mathbf{R}^{d}}|f(s,y)|_{H}^{2}dyds$
$\displaystyle=N\int_{-\infty}^{0}\int_{B_{3r_{1}}}|f(s,y)|_{H}^{2}dyds.$

Hence the corollary is proved. $\Box$

For $R\geq 0$ and real-valued locally integrable functions $h(x)$ on
$\mathbf{R}^{d}$, define the maximal functions

$\mathbb{M}^{R}_{x}h(x):=\sup_{r>R}\frac{1}{|B_{r}(x)|}\int_{B_{r}(x)}|h(y)|dy,\quad\quad\mathbb{M}_{x}h(x):=\mathbb{M}^{0}_{x}h(x).$

Similarly, for real-valued locally integrable functions $h=h(t)$ on
$\mathbf{R}$ we introduce

$\mathbb{M}^{R}_{t}h(t):=\sup_{r>R}\frac{1}{2r}\int_{-r}^{r}|h(t+s)|\,ds,\quad\quad\mathbb{M}_{t}h(t):=\mathbb{M}^{0}_{t}h(t).$

For functions $h=h(t,x)$, set

$\mathbb{M}^{R}_{x}h(t,x):=\mathbb{M}^{R}_{x}(h(t,\cdot))(x),\quad\mathbb{M}^{R}_{t}h(t,x)=\mathbb{M}^{R}_{t}(h(\cdot,x))(t).$

Obviously if $R_{1}\geq R_{2}$, then

$\mathbb{M}^{R_{1}}_{x}h(x)\leq\mathbb{M}^{R_{2}}_{x}h(x)$

and if $R_{n}\downarrow R$, then

$\mathbb{M}^{R_{n}}_{x}h(x)\uparrow\mathbb{M}^{R}_{x}h(x).$

The same properties hold for $\mathbb{M}^{R}_{t}$.

Let $S_{1}(dw)$ denote the counting measure on $\\{-1,1\\}$ if $d=1$ and the
surface measure on the unit sphere if $d\geq 2$. The following lemma is a
slight modification of [10, Lemma 8].

######  Lemma 3.3.

Let $f\in C_{0}(\mathbf{R}^{d})$, and $v(x)$ be a locally integrable and
continuously differentiable function on $\mathbf{R}^{d}$. Let
$x,y\in\mathbf{R}^{d}$, $|x-y|\leq R_{1}$ and $f(y-z)=0$ if $|z|\leq R_{2}$
with some constants $R_{1},R_{2}\geq 0$ Then it holds that

$\displaystyle\big{|}(f\ast v)(y)|\leq
N\big{(}\mathbb{M}^{R_{1}+R_{2}}_{x}f^{2}(x)\big{)}^{1/2}\int_{R_{2}}^{\infty}(R_{1}+\rho)^{d}\Big{(}\int_{\partial
B_{1}}\big{(}\nabla v(\rho w),w\big{)}^{2}S_{1}(dw)\Big{)}^{1/2}d\rho,$

where $N=N(d)$.

###### Proof.

Since the case $d=1$ is easier, we assume $d\geq 2$. Using the polar
coordinates and Fubini’s theorem we get

$\displaystyle\int_{|z|>R_{2}}f(y-z)v(z)~{}dz$
$\displaystyle=\int_{R_{2}}^{\infty}\int_{\partial B_{1}}f(y-\rho w)v(\rho
w)\rho^{d-1}~{}S_{1}(dw)d\rho$ $\displaystyle=\int_{\partial
B_{1}}\left[\int_{R_{2}}^{\infty}v(\rho
w)\left(\frac{d}{d\rho}\int_{R_{2}}^{\rho}f(y-\gamma
w)\gamma^{d-1}~{}d\gamma\right)S_{1}(dw)\right]d\rho.$

By integration by parts and the assumption on $v$, for almost all $w$,

$\displaystyle\int_{R_{2}}^{\infty}v(\rho
w)\left(\frac{d}{d\rho}\int_{R_{2}}^{\rho}f(y-\gamma
w)\gamma^{d-1}~{}d\gamma\right)d\rho$
$\displaystyle=-\int_{R_{2}}^{\infty}(\nabla v(\rho
w),w)\int_{R_{2}}^{\rho}f(y-\gamma w)\gamma^{d-1}~{}d\gamma d\rho.$

In the above we use the fact that there exists a sequence $\rho_{n}\to\infty$,
which might be dependent on $w$, so that $v(\rho_{n}w)\to 0$ as $n\to\infty$
and that $\int^{\rho}_{R_{2}}f(y-\gamma w)\gamma^{d-1}d\gamma$ is a bounded
function of $\rho$. Also note that the limits of two improper integrals exist
since the first one is actually an integral over finite interval.

By the assumption $|x-y|\leq R_{1}$, for any $\rho>R_{2}$

$\displaystyle\int_{B_{\rho}}f^{2}(y-z)~{}dz=\int_{B_{\rho}(y)}f^{2}(z)~{}dz$
$\displaystyle\leq\int_{B_{R_{1}+\rho}(x)}f^{2}(z)~{}dz$ $\displaystyle\leq
N(d)(R_{1}+\rho)^{d}\mathbb{M}^{R_{1}+R_{2}}_{x}f^{2}(x).$

Finally using Fubini’s theorem, Hölder’s inequality, and the assumption that
$f(y-z)=0$ if $|z|\leq R_{2}$, we get

$\displaystyle|(f\ast v)(y)|$
$\displaystyle\leq\Big{|}\int_{R_{2}}^{\infty}\int_{\partial B_{1}}(\nabla
v(\rho w),w)\int_{R_{2}}^{\rho}f(y-\gamma w)\gamma^{d-1}~{}d\gamma
S_{1}(dw)d\rho\Big{|}$
$\displaystyle\leq\int_{R_{2}}^{\infty}\Big{(}\int_{\partial
B_{1}}\int_{R_{2}}^{\rho}\Big{|}(\nabla v(\rho
w),w)\Big{|}^{2}\gamma^{d-1}d\gamma S_{1}(dw)\Big{)}^{1/2}$
$\displaystyle\hskip 113.81102pt\times\Big{(}\int_{\partial
B_{1}}\int_{R_{2}}^{\rho}f^{2}(y-\gamma w)\gamma^{d-1}~{}d\gamma
S_{1}(dw)\Big{)}^{1/2}d\rho$
$\displaystyle\leq\int_{R_{2}}^{\infty}\rho^{d/2}\Big{(}\int_{\partial
B_{1}}\Big{|}(\nabla v(\rho
w),w)\Big{|}^{2}S_{1}(dw)\Big{)}^{1/2}\Big{(}\int_{|z|\leq\rho}f^{2}(y-z)~{}dz\Big{)}^{1/2}d\rho$
$\displaystyle\leq
N\big{(}\mathbb{M}^{R_{1}+R_{2}}_{x}f^{2}(x)\big{)}^{1/2}\int_{R_{2}}^{\infty}(R_{1}+\rho)^{d}\Big{(}\int_{\partial
B_{1}}\Big{|}(\nabla v(\rho w),w)\big{|}^{2}S_{1}(dw)\Big{)}^{1/2}d\rho.$

The lemma is proved. $\Box$

For $r_{1},r_{2}>0$ denote

$Q_{r_{2},r_{1}}:=(-2r_{2},0)\times B_{r_{1}}.$

######  Lemma 3.4.

Suppose there exist constants $\sigma,\kappa>0$ and $\mu>d+2$ so that

$\big{|}D_{x}K(t,s,x)\big{|}\leq
C\big{|}(t-s)^{-\sigma}F_{1}\big{(}t,s,(t-s)^{-\kappa}x\big{)}\big{|},$ (3.2)
$-2{\sigma}+{\kappa}(\mu+d)>-1,$ (3.3)

and

$H_{1,K}(\mu):=\sup_{s<t}\int_{|x|>r_{1}(s-r)^{-\kappa}}|x|^{\mu}|F_{1}(t,s,x)|^{2}dx<\infty.$
(3.4)

Let $f\in C^{\infty}_{0}(\mathbf{R}^{d+1},H)$ with support in
$(-10r_{2},10r_{2})\times\mathbf{R}^{d}\setminus B_{2r_{1}}$. Then for any
$x\in B_{r_{1}}$ we have

$\int_{Q_{r_{2},r_{1}}}|\mathcal{G}f(s,y)|^{2}~{}dsdy\leq
NH_{1,K}(\mu)r_{1}^{\mu}r_{2}^{-2{\sigma}+{\kappa}(\mu+d)+1}\int_{-10r_{2}}^{0}\mathbb{M}^{3r_{1}}_{x}|f|_{H}^{2}(s,x)ds,$

where $N=N(d,\mu,{\sigma},{\kappa},C_{0},C)$.

###### Proof.

Let $x\in B_{r_{1}}$, $(s,y)\in Q_{r_{2},r_{1}}$ and $r\leq s$. Then
$|x-y|\leq 2r_{1}$, and $|z|\leq r_{1}$ implies $|y-z|\leq 2r_{1}$ and
$f(r,y-z)=0$ due to the assumption on $f$. Therefore,

$\displaystyle|K(s,r,\cdot)\ast f(r,\cdot)(y)|_{H}\leq\int_{|z|\geq
r_{1}}|K(s,r,z)||f|_{H}(r,y-z)~{}dz.$

Applying Lemma 3.3 with $R_{1}=2r_{1}$ and $R_{2}=r_{1}$, we get

$\displaystyle|K(s,r,\cdot)\ast f(r,\cdot)(y)|_{H}^{2}$ $\displaystyle\leq
N\mathbb{M}^{3r_{1}}_{x}|f|_{H}^{2}(r,x)\left(\int_{r_{1}}^{\infty}(2r_{1}+\rho)^{d}\big{[}\int_{\partial
B_{1}}\Big{|}\nabla K(s,r,\rho
w)\Big{|}^{2}S_{1}(dw)\big{]}^{1/2}d\rho\right)^{2}$ $\displaystyle\leq
N\mathbb{M}^{3r_{1}}_{x}|f|_{H}^{2}(r,x)\left(\int_{r_{1}}^{\infty}\rho^{d}\big{[}\int_{\partial
B_{1}}\Big{|}\nabla K(s,r,\rho
w)\Big{|}^{2}S_{1}(dw)\big{]}^{1/2}d\rho\right)^{2}.$ (3.5)

By (3.2) and the change of variable $(s-r)^{-\kappa}\rho\to\rho$, the last
term is less than or equal to constant times of

$(s-r)^{-2{\sigma}+2\kappa(d+1)}\mathbb{M}^{3r_{1}}_{x}|f|_{H}^{2}(r,x)\left(\int_{r_{1}(s-r)^{-\kappa}}^{\infty}\rho^{d}\big{[}\int_{\partial
B_{1}}\Big{|}F_{1}(s,r,\rho
w)\Big{|}^{2}S_{1}(dw)\big{]}^{1/2}d\rho\right)^{2}.$

By Hölder inequality and the definition of $H_{1,K}(\mu)$,

$\displaystyle\left(\int_{r_{1}(s-r)^{-\kappa}}^{\infty}\rho^{d}\big{[}\int_{\partial
B_{1}}\Big{|}F_{1}(s,r,\rho
w)\Big{|}^{2}S_{1}(dw)\big{]}^{1/2}d\rho\right)^{2}$
$\displaystyle\leq\left(\int^{\infty}_{r_{1}(s-r)^{-\kappa}}\rho^{d+1-\mu}~{}d\rho\right)\cdot\left(\int^{\infty}_{r_{1}(s-r)^{-\kappa}}\int_{\partial
B_{1}}\rho^{\mu+d-1}\Big{|}F_{1}(s,r,\rho w)\Big{|}^{2}S_{1}(dw)d\rho\right)$
$\displaystyle\leq Nr_{1}^{d+2-\mu}(s-r)^{\kappa(\mu-d-2)}H_{1,K}(\mu).$

Coming back to (3.5) and remembering the definition of $\mathcal{G}f$, we get

$\displaystyle\int_{Q_{r_{2},r_{1}}}|\mathcal{G}f(s,y)|^{2}~{}dsdy$
$\displaystyle\leq
NH_{1,K}(\mu)r_{1}^{2d+2-\mu}\int_{-2r_{2}}^{0}\int_{-10r_{2}}^{s}(s-r)^{-2{\sigma}+{\kappa}(\mu+d)}\mathbb{M}^{3r_{1}}_{x}|f|_{H}^{2}(r,x)drds$
$\displaystyle\leq
NH_{1,K}(\mu)r_{1}^{2d+2-\mu}\int_{-10r_{2}}^{0}\left[\int_{r}^{0}(s-r)^{-2{\sigma}+{\kappa}(\mu+d)}ds\right]\mathbb{M}^{3r_{1}}_{x}|f|_{H}^{2}(r,x)dr$
$\displaystyle\leq
NH_{1,K}(\mu)r_{1}^{2d+2-\mu}\int_{-10r_{2}}^{0}(-r)^{-2{\sigma}+{\kappa}(\mu+d)+1}\mathbb{M}^{3r_{1}}_{x}|f|_{H}^{2}(r,x)dr$
$\displaystyle\leq
NH_{1,K}(\mu)r_{1}^{2d+2-\mu}r_{2}^{-2{\sigma}+{\kappa}(\mu+d)+1}\int_{-10r_{2}}^{0}\mathbb{M}^{3r_{1}}_{x}|f|_{H}^{2}(r,x)dr.$

The lemma is proved. $\Box$

Recall that $\Theta(\theta,\vartheta):=\theta d-2\vartheta$.

######  Lemma 3.5.

Suppose that

$\displaystyle\big{|}D^{2}_{x}K(t,s,x)\big{|}\leq
C\Big{(}(t-s)^{-\sigma}\big{|}F_{2}(t,s,(t-s)^{{-\kappa}}x)\big{|}\wedge(t-s)^{{-c}}\Big{)}$
(3.6)

holds with some constants $\sigma,\kappa,c>0$ and there exists $\delta>0$ such
that

$\displaystyle r_{2}^{\delta}=r_{1},\quad\Theta(2\delta,c-\delta)<-1.$

Moreover assume that there exists $\mu>d+2$ so that

$\Theta(\kappa+\delta,\sigma-\delta)-(\delta-\kappa)\mu<-1,$ (3.7)

and

$H_{2,K}(\mu):=\sup_{r\leq
s}\int_{|x|\geq(s-r)^{\delta-\kappa}}|x|^{\mu}|F_{2}(s,r,x)|^{2}dx<\infty.$

Let $f\in C^{\infty}_{0}(\mathbf{R}^{d+1},H)$, and $f(t,x)=0$ for
$t\geq-8r_{2}$. Then for any $(t,x)\in Q_{r_{2},r_{1}}$ we have

$\displaystyle\sup_{Q_{r_{2},r_{1}}}|\nabla\mathcal{G}f|^{2}$
$\displaystyle\leq
N\Big{(}H_{2,K}(\mu)r_{2}^{\Theta(\kappa+\delta,\sigma-\delta)-(\delta-\kappa)\mu+1}\wedge
r_{2}^{\Theta(2\delta,c-\delta)+1}\Big{)}\mathbb{M}^{6r_{2}}_{t}\mathbb{M}^{2r_{1}}_{x}|f|_{H}^{2}(t,x),$

where $N=N(d,\mu,\delta,c,\sigma,\kappa,C_{0},C)$.

###### Proof.

Let $(t,x),(s,y)\in Q_{r_{2},r_{1}}$ and $r\leq s$. By Minkowski’s inequality

$\left|\frac{\|f(s+h,\cdot)\|-\|f(s,\cdot)\|}{h}\right|\leq\frac{\|f(s+h,\cdot)-f(s,\cdot)\|}{|h|},$

the derivative of a norm is less than or equal to the norm of the derivative
if both exist. Thus,

$\displaystyle\Big{|}\frac{\partial}{\partial x_{i}}\mathcal{G}f(s,y)\Big{|}$
$\displaystyle=\Big{|}\frac{\partial}{\partial
x_{i}}\Big{(}\int_{-\infty}^{s}|K(s,r,\cdot)\ast
f(r,\cdot)(y)|_{H}^{2}dr\Big{)}^{1/2}\Big{|}$
$\displaystyle\leq\Big{(}\int_{-\infty}^{s}\Big{|}\frac{\partial}{\partial
x_{i}}K(s,r,\cdot)\ast f(r,\cdot)(y)\Big{|}_{H}^{2}dr\Big{)}^{1/2}.$

Applying Lemma 3.3 with $R_{1}=2r_{1}$ and $R_{2}=0$ we get

$\displaystyle\Big{|}\frac{\partial}{\partial x_{i}}K(s,r,\cdot)\ast
f(r,\cdot)(y)\Big{|}_{H}^{2}\leq
N\mathbb{M}_{x}^{2r_{1}}|f|_{H}^{2}(r,x)(\mathcal{I}_{1}^{2}+\mathcal{I}_{2}^{2})$

where

$\mathcal{I}_{1}=\int_{(s-r)^{\delta}}^{\infty}(2r_{1}+\rho)^{d}\big{(}\int_{\partial
B_{1}}\Big{|}D^{2}_{x}K(s,r,\rho w)\Big{|}^{2}S_{1}(dw)\big{)}^{1/2}d\rho,$
$\mathcal{I}_{2}=\int_{0}^{(s-r)^{\delta}}(2r_{1}+\rho)^{d}\big{(}\int_{\partial
B_{1}}\Big{|}D^{2}_{x}K(s,r,\rho w)\Big{|}^{2}S_{1}(dw)\big{)}^{1/2}d\rho$

Thus,

$\Big{|}\frac{\partial}{\partial x_{i}}\mathcal{G}(s,y)\Big{|}^{2}\leq
N\int_{-\infty}^{s}\mathbb{M}_{x}^{2r_{1}}|f|_{H}^{2}(r,x)(\mathcal{I}_{1}^{2}+\mathcal{I}_{2}^{2})~{}dr.$

Since $f(r,x)=0$ if $r\geq-8r_{2}$, we may assume $r<-8r_{2}$. So

$\displaystyle|s-r|^{\delta}\geq 6r^{\delta}_{2}=6r_{1}.$ (3.8)

First, we estimate $\mathcal{I}_{1}$. Due to (3.8) and (3.6),

$\displaystyle\mathcal{I}_{1}$
$\displaystyle=\int_{(s-r)^{\delta}}^{\infty}(2r_{1}+\rho)^{d}\big{(}\int_{\partial
B_{1}}\Big{|}D^{2}_{x}K(s,r,\rho w)\Big{|}^{2}S_{1}(dw)\big{)}^{1/2}d\rho$
$\displaystyle\leq
N\int_{(s-r)^{\delta}}^{\infty}\rho^{d}\big{(}\int_{\partial
B_{1}}\Big{|}D^{2}_{x}K(s,r,\rho w)\Big{|}^{2}S_{1}(dw)\big{)}^{1/2}d\rho$
$\displaystyle\leq
N(s-r)^{-\sigma}\int_{(s-r)^{\delta}}^{\infty}\rho^{d}\big{(}\int_{\partial
B_{1}}|F_{2}(s,r,(s-r)^{{-\kappa}}\rho w)|^{2}S_{1}(dw)\big{)}^{1/2}d\rho.$

By the change of variable $(s-r)^{-\kappa}\rho\to\rho$, the last therm is less
than or equal to

$\displaystyle
N(s-r)^{{-\sigma}+{\kappa}(d+1)}\int_{(s-r)^{\delta-\kappa}}^{\infty}\rho^{d}\big{(}\int_{\partial
B_{1}}|F_{2}(s,r,\rho w)|^{2}S_{1}(dw)\big{)}^{1/2}d\rho$ $\displaystyle\leq
N(s-r)^{{-\sigma}+{\kappa}(d+1)}\left[\int_{(s-r)^{\delta-\kappa}}\rho^{d-\mu+1}d\rho\right]^{1/2}\left[\int_{|z|\geq(s-r)^{\delta-\kappa}}|z|^{\mu}|F_{2}(s,r,z)|^{2}dz\right]^{1/2}$
$\displaystyle\leq
N(s-r)^{\frac{\Theta(\kappa+\delta,\sigma-\delta)-(\delta-\kappa)\mu}{2}}\left[\int_{|z|\geq(s-r)^{\delta-\kappa}}|z|^{\mu}|F_{2}(s,r,z)|^{2}dz\right]^{1/2}$
$\displaystyle\leq
NH_{2,K}^{1/2}(\mu)(s-r)^{\frac{\Theta(\kappa+\delta,\sigma-\delta)-(\delta-\kappa)\mu}{2}}.$

Note $|s-r|\geq r/2$ for $r\leq-8r_{2}$. Thus, by the integration by parts and
the assumption on $f$,

$\displaystyle\int_{-\infty}^{s}\mathbb{M}_{x}^{2r_{1}}|f|_{H}^{2}(r,x)\mathcal{I}_{1}^{2}~{}dr$
$\displaystyle\leq
NH_{2,K}(\mu)\int_{-\infty}^{-8r_{2}}(s-r)^{\Theta(\kappa+\delta,\sigma-\delta)-(\delta-\kappa)\mu}\mathbb{M}_{x}^{2r_{1}}|f|_{H}^{2}(r,x)~{}dr$
$\displaystyle\leq
NH_{2,K}(\mu)\int_{-\infty}^{-8r_{2}}|r|^{\Theta(\kappa+\delta,\sigma-\delta)-(\delta-\kappa)\mu-1}\left[\int_{r}^{0}\mathbb{M}_{x}^{2r_{1}}|f|_{H}^{2}(\bar{s},x)~{}d\bar{s}\right]dr$
$\displaystyle\leq
NH_{2,K}(\mu)\mathbb{M}^{6r_{2}}_{t}\mathbb{M}_{x}^{2r_{1}}|f|_{H}^{2}(t,x)\int_{-\infty}^{-8r_{2}}|r|^{\Theta(\kappa+\delta,\sigma-\delta)-(\delta-\kappa)\mu}~{}dr$
$\displaystyle\leq
NH_{2,K}(\mu)r_{2}^{\Theta(\kappa+\delta,\sigma-\delta)-(\delta-\kappa)\mu+1}\mathbb{M}^{6r_{2}}_{t}\mathbb{M}_{x}^{2r_{1}}|f|_{H}^{2}(t,x).$

Next we estimate $\mathcal{I}_{2}$. Using (3.6) and (3.8),

$\displaystyle\mathcal{I}_{2}$
$\displaystyle\leq\int_{0}^{(s-r)^{\delta}}\big{(}2r_{1}+\rho\big{)}^{d}\big{[}\int_{\partial
B_{1}}\Big{|}D^{2}_{x}K(s,r,\rho w)\Big{|}^{2}S_{1}(dw)\big{]}^{1/2}\,d\rho$
$\displaystyle\leq
N(s-r)^{{-c}}\int_{0}^{(s-r)^{\delta}}\big{(}2r_{1}+\rho\big{)}^{d}d\rho\leq
N(s-r)^{\frac{\Theta(2\delta,c-\delta)}{2}}.$

Applying the integration by parts again, we obtain

$\displaystyle\int_{-\infty}^{s}$
$\displaystyle\mathbb{M}_{x}^{2r_{1}}|f|_{H}^{2}(r,x)\mathcal{I}_{2}^{2}~{}dr$
$\displaystyle\leq
N\int_{-\infty}^{-8r_{2}}(s-r)^{\Theta(2\delta,c-\delta)}\mathbb{M}_{x}^{2r_{1}}|f|_{H}^{2}(r,x)~{}dr$
$\displaystyle\leq
N\int_{-\infty}^{-8r_{2}}|r|^{\Theta(2\delta,c-\delta)-1}\left[\int_{r}^{0}\mathbb{M}_{x}^{2r_{1}}|f|_{H}^{2}(\bar{s},x)~{}d\bar{s}\right]dr$
$\displaystyle\leq
N\mathbb{M}^{6r_{2}}_{t}\mathbb{M}_{x}^{2r_{1}}|f|_{H}^{2}(t,x)\int_{-\infty}^{-8r_{2}}|r|^{\Theta(2\delta,c-\delta)}~{}dr$
$\displaystyle\leq
Nr_{2}^{\Theta(2\delta,c-\delta)+1}\mathbb{M}^{6r_{2}}_{t}\mathbb{M}_{x}^{2r_{1}}|f|_{H}^{2}(t,x).$

Finally, we get

$\displaystyle\Big{|}\frac{\partial}{\partial
x_{i}}\mathcal{G}f(s,y)\Big{|}^{2}$ $\displaystyle\leq
N\Big{(}H_{2,K}(\mu)r_{2}^{\Theta(\kappa+\delta,\sigma-\delta)-(\delta-\kappa)\mu+1}\wedge
r_{2}^{\Theta(2\delta,c-\delta)+1}\Big{)}\mathbb{M}^{6r_{2}}_{t}\mathbb{M}_{x}^{2r_{1}}|f|_{H}^{2}(t,x).$

The lemma is proved. $\Box$

######  Lemma 3.6.

Suppose that

$\left|\frac{\partial^{2}}{\partial x\partial t}K(t,s,x)\right|\leq
C\Big{(}(t-s)^{{-\sigma}}\big{|}F_{3}(t,s,(t-s)^{{-\kappa}}x)\big{|}\wedge(t-s)^{{-c}}\Big{)}$

holds with some constants $\sigma,\kappa,c>0$ and there exists a constant
$\delta>0$ such that

$\displaystyle r_{2}^{\delta}=r_{1},\quad\Theta(2\delta,c-\delta)<-1.$

Moreover assume that there exists $\mu>d+2$ so that

$\displaystyle\Theta(\kappa+\delta,\sigma-\delta)-(\delta-\kappa)\mu<-1,$

and

$\displaystyle H_{3,K}(\mu):=\sup_{r\leq
s}\int_{|x|\geq(s-r)^{\delta-\kappa}}|x|^{\mu}|F_{3}(s,r,x)|^{2}dx<\infty.$

Let $f\in C^{\infty}_{0}(\mathbf{R}^{d+1},H)$ and $f(t,x)=0$ for
$t\geq-8r_{2}$. Then for any $(t,x)\in Q_{r_{2},r_{1}}$ we have

$\displaystyle\sup_{Q_{r_{2},r_{1}}}|D_{t}\mathcal{G}f|^{2}$
$\displaystyle\leq
N\Big{(}H_{3,K}(\mu)r_{2}^{\Theta(\kappa+\delta,\sigma-\delta)-(\delta-\kappa)\mu+1}\wedge
r_{2}^{\Theta(2\delta,c-\delta)+1}\Big{)}\mathbb{M}^{6r_{2}}_{t}\mathbb{M}_{x}^{2r_{1}}|f|_{H}^{2}(t,x),$

where $N=N(d,\mu,\delta,c,\sigma,\kappa,C_{0},C)$.

###### Proof.

The proof of this lemma is quite similar to the previous one. Note that by
Minkowski’s inequality

$\displaystyle|D_{s}\mathcal{G}f(s,y)|$
$\displaystyle=\big{|}D_{s}[\int_{-\infty}^{-8r_{2}}|K(s,r,\cdot)\ast
f(r,\cdot)(y)|_{H}^{2}dr]^{1/2}\big{|}$
$\displaystyle\leq[\int_{-\infty}^{-8r_{2}}|D_{s}K(s,r,\cdot)\ast
f(r,\cdot)(y)|_{H}^{2}dr]^{1/2}.$

The other parts are easily obtained by following the proof of the previous
lemma. $\Box$

##  4\. Proof of Theorem 2.4

First, observe that from (2.2) and (2.14) we have

$\displaystyle-2{\sigma_{1}}+\kappa_{1}(\mu_{1}+d)>-1.$ (4.1)

Indeed,

$\displaystyle-2{\sigma_{1}}+\kappa_{1}(\mu_{1}+d)$
$\displaystyle=\mu_{1}(c_{2}-c_{3}+1)-d(c_{2}-c_{3}+1)-2(c_{2}-c_{3})-3$
$\displaystyle=(c_{2}-c_{3}+1)(\mu_{1}-d-2)-1>-1,$

since $c_{2}-c_{3}+1=\frac{2c_{2}-1}{2(d+2)}>0$ and $\mu_{1}>d+2$.

Also, we can derive the following relation from (2.2) (note that
$c_{2}>\frac{1}{2}$)

$\displaystyle\Theta(2\delta_{0},c_{2}-\delta_{0})=-2\delta_{0}-1=\frac{1-2c_{2}}{d+2}-1<-1$
(4.2)

and

$\displaystyle\Theta(2\delta_{0},c_{3}-\delta_{0})=\Theta(2\delta_{0},c_{2}-\delta_{0})+2(c_{2}-c_{3})=-3.$
(4.3)

Take $\delta_{0}$ from (2.15). If Assumption 2.2 holds, then $\delta_{0}>0$
due to (4.2). For $R>0$ set

$Q_{R}=(-2R,0)\times B_{R^{\delta_{0}}}.$

By $-\int_{Q_{R}}f~{}dsdy$ we denote the mean average of $f$ on $Q_{R}$, i.e.

$-\int_{Q_{R}}f~{}dsdy:=\frac{1}{|Q_{R}|}\int_{Q_{R}}f(s,y)~{}dsdy.$

Recall

$\displaystyle\mathcal{G}f(t,x):=\Big{(}\int_{-\infty}^{t}\big{|}K(t,s,\cdot)\ast
f(s,\cdot)(x)\big{|}_{H}^{2}ds\Big{)}^{1/2}.$

To continue the proof we need the following lemma.

######  Lemma 4.1.

Suppose that Assumption 2.1 and 2.2 hold. Then for any $(t,x)\in Q_{R}$

$\displaystyle\frac{1}{|Q_{R}|^{2}}\int_{Q_{R}}\int_{Q_{R}}\big{|}\mathcal{G}f(s,y)-\mathcal{G}f(r,z)\big{|}^{2}~{}dsdydrdz\leq
N\mathbb{M}_{t}\mathbb{M}_{x}|f|_{H}^{2}(t,x),$

where the constant $N$ is independent of $f$, $R$, and $(t,x)$.

###### Proof.

Let $(t,x)\in Q_{R}$. We take a function $\zeta\in C_{0}^{\infty}(\mathbf{R})$
such that $0\leq\zeta\leq 1$, $\zeta=1$ on $[-8R,8R]$, and $\zeta=0$ outside
of $[-10R,10R]$. Define

$\mathcal{A}(s,y):=f(s,y)\zeta(s),\quad\mathcal{B}(s,y):=f(s,y)-\mathcal{A}(s,y)=f(s,y)(1-\zeta(s)).$

Then

$K(t,s,\cdot)\ast\mathcal{A}(s,\cdot)=\zeta(s)K(t,s,\cdot)\ast
f(s,\cdot),\quad\mathcal{G}f\leq\mathcal{G}\mathcal{A}+\mathcal{G}\mathcal{B}\quad\text{and}\quad\mathcal{G}\mathcal{B}\leq\mathcal{G}f.$

The first inequality comes from Minkowski’s inequality. The second inequality
comes from the fact
$|K(t,s,\cdot)\ast\mathcal{B}(s,\cdot)(y)|=(1-\zeta(s))|K(t,s,\cdot)\ast
f(s,\cdot)(y)|$ and $|1-\zeta(s)|\leq 1$. So for any constant $c$,

$\displaystyle|\mathcal{G}f-c|\leq|\mathcal{G}\mathcal{A}|+|\mathcal{G}\mathcal{B}-c|.$
(4.4)

This is because if $\mathcal{G}f\geq c$, then

$|\mathcal{G}f-c|=\mathcal{G}f-c\leq\mathcal{G}\mathcal{A}+\mathcal{G}\mathcal{B}-c\leq|\mathcal{G}\mathcal{A}|+|\mathcal{G}\mathcal{B}-c|$

and if $\mathcal{G}f<c$, then

$|\mathcal{G}f-c|=c-\mathcal{G}f\leq
c-\mathcal{G}\mathcal{B}\leq|\mathcal{G}\mathcal{A}|+|\mathcal{G}\mathcal{B}-c|.$

First we prove

$\displaystyle\int_{Q_{R}}|\mathcal{G}\mathcal{A}(s,y)|^{2}~{}dsdy\leq
N|Q_{R}|\mathbb{M}_{t}\mathbb{M}_{x}|f|^{2}_{H}(t,x).$ (4.5)

Take $\eta\in C_{0}^{\infty}(\mathbf{R}^{d})$ such that $0\leq\eta\leq 1$,
$\eta=1$ in $B_{2R^{\delta_{0}}}$, and $\eta=0$ outside of
$B_{3R^{\delta_{0}}}$. Set $\mathcal{A}_{1}=\eta\mathcal{A}$ and
$\mathcal{A}_{2}=(1-\eta)\mathcal{A}$. By Minkowski’s inequality,
$\mathcal{G}\mathcal{A}\leq\mathcal{G}\mathcal{A}_{1}+\mathcal{G}\mathcal{A}_{2}$.
$\mathcal{G}\mathcal{A}_{1}$ can be estimated by Corollary 3.2. Indeed,

$\displaystyle\int_{-2R}^{0}\int_{B_{R^{\delta_{0}}}}|\mathcal{G}\mathcal{A}_{1}(s,y)|^{2}dsdy$
$\displaystyle\leq
N\int_{-\infty}^{0}\int_{B_{3R^{\delta_{0}}}}|\mathcal{A}_{1}(s,y)|_{H}^{2}dsdy$
$\displaystyle\leq
N\int_{-10R}^{0}\int_{B_{4R^{\delta_{0}}}(x)}|\mathcal{A}_{1}(s,y)|_{H}^{2}dsdy$
$\displaystyle\leq
NR^{\delta_{0}d}\int_{-10R}^{0}\mathbb{M}_{x}|\mathcal{A}_{1}(s,x)|_{H}^{2}dsdy$
$\displaystyle\leq
NR^{1+\delta_{0}d}\mathbb{M}_{t}\mathbb{M}_{x}|\mathcal{A}_{1}(t,x)|_{H}^{2}$
$\displaystyle\leq
NR^{1+\delta_{0}d}\mathbb{M}_{t}\mathbb{M}_{x}|f(t,x)|_{H}^{2}.$

Hence it only remains to show (4.5) for $\mathcal{G}\mathcal{A}_{2}$ instead
of $\mathcal{G}\mathcal{A}$.

Due to (4.1), (3.3) holds for $\mu=\mu_{1}$ and
$(\sigma,\kappa)=(\sigma_{1},\kappa_{1})$. Thus from Lemma 3.4 with
$(r_{2},r_{1})=(R,R^{\delta_{0}})$ we have

$\displaystyle\int_{Q_{R}}|\mathcal{G}\mathcal{A}_{2}(s,y)|^{2}~{}dsdy$
$\displaystyle\leq
NR^{\delta_{0}(2d+2-\mu_{1})-2{\sigma_{1}}+{\kappa_{1}}(\mu_{1}+d)+2}\mathbb{M}_{t}\mathbb{M}_{x}|f|^{2}_{H}(t,x)$
$\displaystyle\leq
NR^{\Theta(2\delta_{0}+\kappa_{1},\sigma_{1}-\delta_{0})+2-(\delta_{0}-\kappa_{1})\mu_{1}}\mathbb{M}_{t}\mathbb{M}_{x}|f|^{2}_{H}(t,x).$

Moreover due to (2.25) and (2.15),

$\Theta(2\delta_{0}+\kappa_{1},\sigma_{1}-\delta_{0})+2-(\delta_{0}-\kappa_{1})\mu_{1}=\Theta(\delta_{0},0)+1=\delta_{0}d+1$

and so (4.5) is obtained. To go further, recall (2.25) and (4.2),

$\Theta(\kappa_{2}+\delta_{0},\sigma_{2}-\delta_{0})-(\delta_{0}-\kappa_{2})\mu_{2}=\Theta(2\delta_{0},c_{2}-\delta_{0})=-2\delta_{0}-1<-1$

so (3.7) holds with $\mu=\mu_{2}$ and
$(\sigma,\kappa,c)=(\sigma_{2},\kappa_{2},c_{2})$. Hence applying Lemma 3.5
with $(r_{2},r_{1})=(R,R^{\delta_{0}})$,

$\displaystyle\sup_{Q_{R}}|\nabla\mathcal{G}\mathcal{B}|^{2}$
$\displaystyle\leq
N\Big{(}R^{\Theta(\kappa_{2}+\delta_{0},\sigma_{2}-\delta_{0})-(\delta_{0}-\kappa_{2})\mu_{2}+1}\wedge
R^{\Theta(2\delta_{0},c_{2}-\delta_{0})+1}\Big{)}\mathbb{M}_{t}\mathbb{M}_{x}|\mathcal{B}|_{H}^{2}(t,x)$
$\displaystyle\leq
NR^{-2\delta_{0}}\mathbb{M}_{t}\mathbb{M}_{x}|\mathcal{B}|_{H}^{2}(t,x).$

Hence

$\displaystyle\sup_{Q_{R}}|R^{\delta_{0}}\nabla\mathcal{G}\mathcal{B}|^{2}\leq
N\mathbb{M}_{t}\mathbb{M}_{x}|\mathcal{B}|_{H}^{2}(t,x).$ (4.6)

Similarly Lemma 3.6 with $(r_{2},r_{1})=(R,R^{\delta_{0}})$,
$(\mu,\delta,\sigma,\kappa,c)=(\mu_{3},\delta_{0},\sigma_{3},\kappa_{3},c_{3})$
gives

$\displaystyle\sup_{Q_{R}}|R\frac{\partial}{\partial
t}(\mathcal{G}\mathcal{B})|^{2}\leq
N\mathbb{M}_{t}\mathbb{M}_{x}|\mathcal{B}|_{H}^{2}(t,x).$ (4.7)

To apply Lemma 3.5 and Lemma 3.6 above we used the fact that
$\mathcal{G}\mathcal{B}(s,y)=\mathcal{G}(I_{(-\infty,0)}\mathcal{B})(s,y)$ on
$Q_{R}$. Next by (4.4),

$\displaystyle\frac{1}{|Q_{R}|^{2}}\int_{Q_{R}}$
$\displaystyle\int_{Q_{R}}\big{|}\mathcal{G}f(s,y)-\mathcal{G}f(r,z)\big{|}^{2}~{}dsdydrdz$
$\displaystyle\leq
2\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}|\mathcal{G}f-c|^{2}~{}dsdy\leq
4\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}|\mathcal{G}\mathcal{A}|^{2}~{}dsdy+4\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}|\mathcal{G}\mathcal{B}-c|^{2}~{}dsdy.$

Taking $c=\mathcal{G}\mathcal{B}(t,x)$, from (4.5), (4.6), and (4.7) we get

$\displaystyle\frac{1}{|Q_{R}|^{2}}\int_{Q_{R}}$
$\displaystyle\int_{Q_{R}}\big{|}\mathcal{G}f(s,y)-\mathcal{G}f(r,z)\big{|}^{2}~{}dsdydrdz$
$\displaystyle\leq
4\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q_{R}}|\mathcal{G}\mathcal{A}|^{2}~{}dsdy+4\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q_{R}}|\mathcal{G}\mathcal{B}-\mathcal{G}\mathcal{B}(t,x)|^{2}~{}dsdy$
$\displaystyle\leq
N\mathbb{M}_{t}\mathbb{M}_{x}|f|_{H}^{2}(t,x)+4\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q_{R}}|\mathcal{G}\mathcal{B}-\mathcal{G}\mathcal{B}(t,x)|^{2}~{}dsdy$
$\displaystyle\leq
N\mathbb{M}_{t}\mathbb{M}_{x}|f|_{H}^{2}(t,x)+N\sup_{Q_{R}}\big{(}|RD_{s}\mathcal{G}\mathcal{B}|^{2}+|R^{\delta_{0}}\nabla_{\mathcal{G}}\mathcal{B}|^{2}\big{)}$
$\displaystyle\leq N\mathbb{M}_{t}\mathbb{M}_{x}|f|_{H}^{2}(t,x).$

The lemma is proved. $\Box$

We continue the proof of the theorem. For measurable functions $h(t,x)$ on
$\mathbf{R}^{d+1}$, we define the sharp function $h^{\sharp}(t,x)$

$\displaystyle
h^{\sharp}(t,x)=\sup_{Q}\frac{1}{|Q|}\int_{Q}|f(r,z)-f_{Q}|~{}drdz,$

where $f_{Q}:=\frac{1}{|Q|}\int_{Q}f(r,z)~{}drdz$, and the sup is taken all
$Q$ containing $(t,x)$ of the type

$Q=(s-R,s+R)\times B_{R^{\delta_{0}}}(y),\quad R>0.$

By Fefferman-Stein Theorem [12, Theorem 4.2.2], for any $h\in
L_{p}(\mathbf{R}^{d+1})$,

$\|h\|_{L_{p}(\mathbf{R}^{d+1})}\leq
N\|h^{\sharp}\|_{L_{p}(\mathbf{R}^{d+1})}.$

Now we claim

$(\mathcal{G}f)^{\sharp}(t,x)\leq
N(\mathbb{M}_{t}\mathbb{M}_{x}|f|^{2}_{H})^{1/2}(t,x).$ (4.8)

By Jensen’s inequality, to prove (4.8) it suffices to prove that for each
$Q\in\mathcal{Q}$ and $(t,x)\in Q$,

$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}|\mathcal{G}f-(\mathcal{G}f)_{Q}|^{2}~{}dyds\leq
N\mathbb{M}_{t}\mathbb{M}_{x}|f|_{H}^{2}(t,x).$

Note that for any $h_{1}\in\mathbf{R}$ and $h_{2}\in\mathbf{R}^{d}$,

$\displaystyle\mathcal{G}f(t-h_{1},x-h_{2})=\mathcal{G}\tilde{f}(t,x)=\Big{(}\int_{-\infty}^{t}\big{|}\tilde{K}(t,s,\cdot)\ast\tilde{f}(s,\cdot)(x)\big{|}_{H}^{2}ds\Big{)}^{1/2},$

where $\tilde{f}(t,x)=f(t-h_{1},x-h_{2})$ and
$\tilde{K}(t,s,y)=K(t-h_{1},s-h_{1},y)$. Since $\tilde{K}$ also satisfies
Assumptions 2.1 and 2.2 with the same constnats, we may assume
$Q=[-2R,0]\times B_{R^{\delta_{0}}}$. Thus Lemma 4.1 proves (4.8) because

$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}|\mathcal{G}f-(\mathcal{G}f)_{Q}|^{2}~{}dyds\leq\frac{1}{|Q_{R}|^{2}}\int_{Q_{R}}\int_{Q_{R}}\big{|}\mathcal{G}f(s,y)-\mathcal{G}f(r,z)\big{|}^{2}~{}dsdydrdz.$

Finally, combining the Fefferman-Stein theorem and Hardy-Littlewood maximal
theorem [12, Theorem 1.3.1], we conclude (recall $p/2>1$)

$\displaystyle\|u\|_{L_{p}(\mathbf{R}^{d+1})}^{p}\leq
N\|(\mathbb{M}_{t}\mathbb{M}_{x}|f|_{H}^{2})^{1/2}\|_{L_{p}(\mathbf{R}^{d+1})}^{p}$
$\displaystyle=N\int_{\mathbf{R}^{d}}\int_{\mathbf{R}}(\mathbb{M}_{t}\mathbb{M}_{x}|f|_{H}^{2})^{p/2}dtdx$
$\displaystyle\leq
N\int_{\mathbf{R}^{d}}\int_{\mathbf{R}}(\mathbb{M}_{x}|f|_{H}^{2})^{p/2}dtdx$
$\displaystyle=N\int_{\mathbf{R}}\int_{\mathbf{R}^{d}}(\mathbb{M}_{x}|f|_{H}^{2})^{p/2}dxdt$
$\displaystyle\leq N\|f\|_{L_{p}(\mathbf{R}^{d+1},H)}^{p}.$

Therefore, the theorem is proved. $\Box$

##  5\.  Proof of Theorem 2.5

Denote

$K(t,s,x)=(-\Delta)^{\gamma/4}p(t,s,x)=I_{0\leq
s<t}\mathcal{F}^{-1}\Big{(}|\xi|^{\gamma/2}\exp\big{\\{}\int^{t}_{s}\psi(r,\xi)dr\big{\\}}\Big{)}(x).$

We prove that Assumptions 2.1 and 2.2 hold with

$F_{1}(t,s,x)=I_{0\leq
s<t}\sum_{i}\left|\mathcal{F}^{-1}\Big{(}\xi^{i}|\xi|^{\gamma/2}\exp\big{\\{}M(t,s,\xi)\big{\\}}\Big{)}(x)\right|,$
$F_{2}(t,s,x)=I_{0\leq
s<t}\sum_{i,j}\left|\mathcal{F}^{-1}\Big{(}\xi^{i}\xi^{j}|\xi|^{\gamma/2}\exp\big{\\{}M(t,s,\xi)\big{\\}}\Big{)}(x)\right|,$

and

$\displaystyle F_{3}(t,s,x)=I_{0\leq
s<t}\sum_{i}\left|\mathcal{F}^{-1}\Big{(}(t-s)\psi\Big{(}t,\frac{\xi}{(t-s)^{1/\gamma}}\Big{)}\xi^{i}|\xi|^{\gamma/2}\exp\big{\\{}M(t,s,\xi)\big{\\}}\Big{)}(x)\right|,$

where
${M}(t,s,\xi):=\int_{s}^{t}\psi\Big{(}r,\frac{\xi}{(t-s)^{1/\gamma}}\Big{)}dr.$

In the the following lemma we first prove (2.1)-(2.5) with

$\kappa_{1}=\kappa_{2}=\kappa_{3}=\gamma^{-1},\quad\sigma_{1}=\frac{d+1}{\gamma}+\frac{1}{2},\quad\sigma_{2}=c_{2}=\frac{d+2}{\gamma}+\frac{1}{2},\quad\sigma_{3}=c_{3}=\frac{d+1}{\gamma}+\frac{3}{2}.$

######  Lemma 5.1.

There exists a constant $N=N(d,\gamma,\nu)>0$ such that

$\displaystyle\int_{0}^{\infty}\Big{|}\mathcal{F}\Big{(}K(t,s,\cdot)\Big{)}(\xi)\Big{|}^{2}dt<N,$
$\displaystyle\big{|}D_{x}K(t,s,x)\big{|}\leq
N(t-s)^{-\frac{d}{\gamma}-\frac{1}{2}-\frac{1}{\gamma}}\Big{(}|F_{1}(t,s,x)|\wedge
1\Big{)},$ $\displaystyle\big{|}D^{2}_{x}K(t,s,x)\big{|}\leq
N(t-s)^{-\frac{d}{\gamma}-\frac{1}{2}-\frac{2}{\gamma}}\Big{(}|F_{2}(t,s,x)|\wedge
1\Big{)},$

and

$\displaystyle\big{|}\frac{\partial}{\partial t}D_{x}K(t,s,x)\big{|}\leq
N(t-s)^{-\frac{d}{\gamma}-\frac{3}{2}-\frac{1}{\gamma}}\Big{(}|F_{3}(t,s,x)|\wedge
1\Big{)}.$

###### Proof.

The first assertion comes from (2.27). Indeed, since
$\Re\psi(t,\xi)\leq-\nu|\xi|^{\gamma}$,

$\displaystyle\int_{0}^{\infty}\Big{|}\mathcal{F}\Big{(}K(t,s,\cdot)\Big{)}(\xi)\Big{|}^{2}dt=\int^{\infty}_{s}\left||\xi|^{\gamma/2}\exp\big{\\{}\int^{t}_{s}\psi(r,\xi)dr\big{\\}}\right|^{2}dt$
$\displaystyle\leq N\int_{0}^{\infty}|\xi|^{\gamma}e^{-2\nu
t|\xi|^{\gamma}}dt\leq N.$

Next because of the similarity, we only prove the last assertion. From the
definition of $K(t,s,x)$ and $M(t,s,\xi)$,

$\displaystyle\Big{|}\frac{\partial}{\partial x^{i}}\frac{\partial}{\partial
t}K(t,s,x)\Big{|}=I_{0\leq
s<t}\Big{|}\mathcal{F}^{-1}\Big{(}\psi(t,\xi)\xi^{i}|\xi|^{\gamma/2}\exp\big{(}\int_{s}^{t}\psi(r,\xi)dr\big{)}\Big{)}(x)\Big{|}$
$\displaystyle=I_{0\leq
s<t}(t-s)^{-\frac{d}{\gamma}-\frac{3}{2}-\frac{1}{\gamma}}$
$\displaystyle\quad\quad\quad\Big{|}\mathcal{F}^{-1}\Big{(}(t-s)\psi\Big{(}t,\frac{\xi}{(t-s)^{1/\gamma}}\Big{)}\xi^{i}|\xi|^{\gamma/2}\exp\big{\\{}M(t,s,\xi)\big{\\}}\Big{)}\Big{(}\frac{x}{(t-s)^{1/\gamma}}\Big{)}\Big{|}$
$\displaystyle\leq(t-s)^{-\frac{d}{\gamma}-\frac{3}{2}-\frac{1}{\gamma}}F_{3}(t,s,(t-s)^{-1/{\gamma}}x).$

Furthermore, by (2.27) and (2.28),

$\displaystyle\Big{|}\mathcal{F}^{-1}\Big{(}(t-s)\psi\Big{(}t,\frac{\xi}{(t-s)^{1/\gamma}}\Big{)}\xi^{i}|\xi|^{\gamma/2}\exp\big{\\{}M(t,s,\xi)\big{\\}}\Big{)}(x)\Big{|}$
$\displaystyle\leq
N\int_{\mathbf{R}^{d}}\Big{|}(t-s)\psi\Big{(}t,\frac{\xi}{(t-s)^{1/\gamma}}\Big{)}\xi^{i}|\xi|^{\gamma/2}\exp\big{(}\int_{s}^{t}\psi(r,\frac{\xi}{(t-s)^{1/\gamma}})dr\big{)}\Big{|}~{}d\xi$
$\displaystyle\leq
N\int_{\mathbf{R}^{d}}|\xi|^{\frac{3\gamma}{2}+1}\exp\big{(}-\nu|\xi|^{\gamma}\big{)}~{}d\xi\leq
N.$

Hence the assertion is proved. $\Box$

######  Lemma 5.2.

Let $h\in C^{2}(\mathbf{R}^{d}\setminus\\{0\\})$ satisfy

$\displaystyle|h(x)|\leq N_{0}|x|^{\varsigma}e^{-c|x|^{\gamma}},\quad\forall
x\in\mathbf{R}^{d}\setminus\\{0\\},$

with some constants $c,N_{0}>0$, $\varsigma>\eta-\frac{d}{2}$ and $\gamma>0$.
Further assume that either

$\displaystyle\eta\in[0,1)\quad\text{and}\quad\big{|}Dh(x)\big{|}\leq
N_{0}|x|^{\varsigma-1}e^{-c|x|^{\gamma}},\quad\forall
x\in\mathbf{R}^{d}\setminus\\{0\\}$

or

$\eta\in[1,2)\quad\text{and}\quad\big{|}D^{2}h(x)\big{|}\leq
N_{0}|x|^{\varsigma-2}e^{-c|x|^{\gamma}},\quad\forall
x\in\mathbf{R}^{d}\setminus\\{0\\}$

holds. Then

$\|(-\Delta)^{\eta/2}h\|_{L_{2}(\mathbf{R}^{d})}<N<\infty,$

where $N=N(N_{0},\eta,c,\varsigma,\gamma)$.

###### Proof.

See [3, Lemma 5.1]. $\Box$

######  Corollary 5.3.

Suppose

$\begin{cases}&2\lfloor\frac{\mu}{4}\rfloor+1\leq\lfloor\frac{d}{2}\rfloor+2\quad\text{if}\quad\frac{\mu}{2}-2\lfloor\frac{\mu}{4}\rfloor\in[0,1)\\\
&2\lfloor\frac{\mu}{4}\rfloor+2\leq\lfloor\frac{d}{2}\rfloor+2\quad\text{if}\quad\frac{\mu}{2}-2\lfloor\frac{\mu}{4}\rfloor\in[1,2).\end{cases}$

Then,

$\displaystyle\sup_{s<t}\int_{\mathbf{R}^{d}}|x|^{\mu}|F_{1}(t,s,x)|^{2}dx<\infty,\quad\text{if}\quad\mu<\gamma+d+2;$
$\displaystyle\sup_{s<t}\int_{\mathbf{R}^{d}}|x|^{\mu}|F_{2}(t,s,x)|^{2}dx<\infty,\quad\text{if}\quad\mu<(\gamma+d+4);$
$\displaystyle\sup_{s<t}\int_{\mathbf{R}^{d}}|x|^{\mu}|F_{3}(t,s,x)|^{2}dx<\infty,\quad\text{if}\quad\mu<(3\gamma+d+2).$

###### Proof.

Because of the similarity of proofs, we only prove the last assertion. By
Parseval’s identity, it suffices to show

$\displaystyle\sup_{s<t}\int_{\mathbf{R}^{d}}\Big{|}(-\Delta)^{\mu/4}\tilde{F}^{i}_{3}(t,s,\xi)\Big{|}^{2}d\xi<\infty,\quad\forall\,i,$

where

$\displaystyle\tilde{F}^{i}_{3}(t,s,\cdot)\big{)}(\xi)=I_{0\leq
s<t}(t-s)\psi\Big{(}t,\frac{\xi}{(t-s)^{1/\gamma}}\Big{)}\xi^{i}|\xi|^{\gamma/2}\exp\big{\\{}M(t,s,\xi)\big{\\}}.$

Using (2.27) and (2.28), one can check that there exists a constant
$N=N(\nu,m)$ such for each $0<s<t$, $\xi\neq 0$, and
$\mu<\lfloor\frac{d}{2}\rfloor+2$

$|(-\Delta)^{\lfloor\mu/4\rfloor}\tilde{F}^{i}_{3}(s,t,\xi)|\leq
N|\xi|^{\frac{3\gamma}{2}+1-2\lfloor\mu/4\rfloor}e^{-\nu|\xi|^{\gamma}}.$

Moreover

$|\frac{\partial^{2}}{\partial\xi^{i}}(-\Delta)^{\lfloor\mu/4\rfloor}\tilde{F}_{3}^{i}(s,t,\xi)|\leq
N|\xi|^{\frac{3\gamma}{2}-2\lfloor\mu/4\rfloor}e^{-\nu|\xi|^{\gamma}},$

if $\frac{\mu}{4}-\lfloor\frac{\mu}{4}\rfloor\in[0,1)$, and

$|\frac{\partial^{2}}{\partial\xi^{i}\partial\xi^{j}}(-\Delta)^{\lfloor\mu/4\rfloor}\tilde{F}_{3}^{i}(s,t,\xi)|\leq
N|\xi|^{\frac{3\gamma}{2}-1-2\lfloor\mu/4\rfloor}e^{-\nu|\xi|^{\gamma}}.$

if $\frac{\mu}{4}-\lfloor\frac{\mu}{4}\rfloor\in[1,2)$. Finally we set

$\eta=\mu/2-2\lfloor\mu/4\rfloor,\quad\varsigma=\frac{3\gamma}{2}+1-2\lfloor\mu/4\rfloor.$

Then, for $\mu<3\gamma+d+2$, we have

$\eta-\frac{d}{2}<\varsigma.$

Therefore Lemma 5.2 is applicable, and the assertion is proved. $\Box$

We continue the proof of the theorem. Recall that we defined

${\kappa_{1}}={\kappa_{2}}={\kappa_{3}}=\frac{1}{\gamma},\quad{\sigma_{1}}=\frac{d}{\gamma}+\frac{1}{2}+\frac{1}{\gamma},$
${c_{2}}={\sigma_{2}}=\frac{d}{\gamma}+\frac{1}{2}+\frac{2}{\gamma},\quad{c_{3}}={\sigma_{3}}=\frac{d}{\gamma}+\frac{3}{2}+\frac{1}{\gamma}.$

So obviously

$\displaystyle\delta_{0}={c_{2}}-c_{3}+1=\frac{1}{\gamma},\quad
c_{2}>\frac{1}{2},$

$\displaystyle\Theta(\kappa_{1}+\delta_{0},\sigma_{1}-\delta_{0})+1=\frac{2d}{\gamma}-2\left(\frac{d}{\gamma}+\frac{1}{2}\right)+1=0,$

$\displaystyle\Theta(\kappa_{2}-\delta_{0},\sigma_{2}-c_{2})=\Theta(0,0)=0,$

and

$\displaystyle\Theta(\kappa_{3}-\delta_{0},\sigma_{3}-c_{3})=\Theta(0,0)=0.$

Thus (2.14) (or equivalently (2.25)) is satisfied for any
$(\mu_{1},\mu_{2},\mu_{3})\in\mathbf{R}^{3}$. Next we choose
$(\mu_{1},\mu_{2},\mu_{3})$ such that

$d+2<\mu_{1}<\gamma+d+2,$ $d+2<\mu_{2}<\gamma+d+4,$

and

$d+2<\mu_{3}<3\gamma+d+2$

so that for all $1\leq i\leq 3$

$\begin{cases}&2\lfloor\frac{\mu_{i}}{4}\rfloor+1\leq\lfloor\frac{d}{2}\rfloor+2\quad\text{if}\quad\frac{\mu_{i}}{2}-2\lfloor\frac{\mu_{i}}{4}\rfloor\in[0,1)\\\
&2\lfloor\frac{\mu_{i}}{4}\rfloor+2\leq\lfloor\frac{d}{2}\rfloor+2\quad\text{if}\quad\frac{\mu_{i}}{2}-2\lfloor\frac{\mu_{i}}{4}\rfloor\in[1,2).\end{cases}$

Then due to Corollary 5.3, we see that (2.6), (2.7), and (2.8) hold for these
$\mu_{1}$, $\mu_{2}$, and $\mu_{3}$ hence Assumption 2.2 holds. The theorem is
proved.

## References

  * [1] T. Chang and K. Lee, On a stochastic partial differential equation with a fractional Laplacian operator, Stochastic Process. Appl., 122 (2012), 3288-3311.
  * [2] I. Kim and K. Kim, A generalization of the Littlewood-paley inequality for the fractional Laplacian $\Delta^{\alpha/2}$, J. Math. Anal. Appl., 388 (2012), no.1, 175-190.
  * [3] I. Kim, K. Kim, and S. Lim, Parabolic BMO estimates for pseudo-differential operators of arbitrary order, to apper in J. Math. Anal. Appl., doi:10.1016/j.jmaa.2015.02.065.
  * [4] I. Kim, K. Kim, and P. Kim, Parabolic Littlewood-Paley inequality for $\phi(\Delta)$-type operators and applications to Stochastic integro-differential equations, Advances in Math, 249 (2013), 161-203.
  * [5] N.V. Krylov, A generalization of the Littlewood-Paley inequality and some other results related to stochastic partial differential equations, Ulam Quaterly, 2 (1994), no.4, 16-26.
  * [6] N.V. Krylov, Introduction to the theory of diffusion processes, Translations in Mathematical Monographs, 142, AMS, Providence, RI, 1995.
  * [7] N.V. Krylov, On the foundation of the $L_{p}$-Theory of SPDEs, Stochastic partial differential equations and applications—VII, 179-191, Lect. Notes Pure Appl. Math., 245, Chapman & Hall/CRC, Boca Raton, FL, 2006.
  * [8] N.V. Krylov, An analytic approach to SPDEs, pp. 185-242 in Stochastic Partial Differential Equations: Six Perspectives, Mathematical Surveys and Monographs, 64 (1999), AMS, Providence, RI.
  * [9] R. Mikulevicius and H. Pragarauskas, On the cauchy problem for certain integro-differential operators in Sobolev and Hölder spaces, Lithuanina Math. J, 32 (1992), no.2, 238-264.
  * [10] R. Mikulevicius and H. Pragarauskas, On Lp-estimates of some singular integrals related to jump processes, SIAM J. Math. Anal., 44 (2012), no.4, 2305-2328.
  * [11] J. Neerven, M. Veraar, and L. Weis, Maximal $L^{p}$-regularity for stochastic evolution equations, SIAM J. Math. Anal., 44 (2012), no.3, 1372-1414.
  * [12] E. Stein, Harmonic analysis : real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, 1993.