# Boundedness of stochastic singular integral operators and its application to
stochastic partial differential equations

Ildoo Kim  Center for Mathematical Challenges, Korea Institute for Advanced
Study, 85 Hoegiro Dongdaemun-gu, Seoul 130-722, Republic of Korea
<EMAIL_ADDRESS>and  Kyeong-Hun Kim  Department of Mathematics, Korea
University, 1 Anam-dong, Sungbuk-gu, Seoul, 136-701, Republic of Korea
<EMAIL_ADDRESS>
###### Abstract.

In this article we present a stochastic counterpart of the Hörmander condition
and Calderón-Zygmund theorem. Let $W_{t}$ be a Wiener process defined on a
probability space $(\Omega,\mathbb{P})$ and $K(r,t,x,y)$ be a random kernel
which is stochastically singular in the sense that

$\mathbb{E}\left|\int_{0}^{t}\int_{|x-y|<\varepsilon}|K(s,t,x,y)|dydW_{s}\right|^{p}=\infty,\quad\forall\,t,p,\varepsilon>0,\,x\in\mathbf{R}^{d}.$

We prove that the stochastic singular integral of the type

$\displaystyle\mathbb{T}g(t,x):=\int_{0}^{t}\int_{\mathbf{R}^{d}}K(t,s,x,y)g(s,y)dydW_{s}$
(0.1)

is a bounded operator on
$\mathbb{L}_{p}=L_{p}(\Omega\times(0,\infty);L_{p}(\mathbf{R}^{d}))$ for any
$p\geq 2$ if it is bounded on $\mathbb{L}_{2}$ and the following (which we
call stochastic Hörmander condition) holds: there exists a quasi-metric $\rho$
on $(0,\infty)\times\mathbf{R}^{d}$ and a positive constant $C_{0}$ such that
for $X=(t,x),Y=(s,y),Z=(r,z)\in(0,\infty)\times\mathbf{R}^{d}$,

$\sup_{\omega\in\Omega,X,Y}\int_{0}^{\infty}\left[\int_{\rho(X,Z)\geq
C_{0}\rho(X,Y)}|K(r,t,z,x)-K(r,s,z,y)|~{}dz\right]^{2}dr<\infty.$

As a consequence of our result on stochastic singular integral operators, we
obtain the maximal regularity for a very wide class of stochastic partial
differential equations.

###### Key words and phrases:

Stochastic Singular Integral Operator, $L_{p}$-estimate, Pseudo-differential
operator, Stochastic Partial Differential Equation

###### 2010 Mathematics Subject Classification:

60H15, 42B20, 35S10, 35K30, 35B45

The first author was supported by the TJ Park Science Fellowship of POSCO TJ
Park Foundation

The second author was supported by Samsung Science and Technology Foundation
under Project Number SSTF-BA1401-02

##  1\. introduction

Since Calderón and Zygmund’s work, the singular integral theory has been one
of most important fields in Mathematics and it has been developed considerably
in various directions (see e.g. [2, 11]). In particular, due to Hörmander the
singular integral

$\displaystyle Tf(x):=\int_{\mathbf{R}^{d}}K(x,y)f(y)dy$ (1.1)

becomes a bounded operator on $L_{p}(\mathbf{R}^{d})$ if the kernel $K$
satisfies the Hörmander condition (see [11, Theorem I.5.3])

$\displaystyle\sup_{x,y\in\mathbf{R}^{d}}\int_{|x-z|>2|x-y|}|K(x,z)-K(y,z)|dz<\infty.$
(1.2)

Hörmander’s condition is considered as one of most general conditions in the
theory of the singular integral, and there is a huge number of applications to
partial differential equations. For instance, consider the heat equation

$\displaystyle u_{t}=\Delta u+f,\quad(t,x)\in(0,T]\times\mathbf{R}^{d},\quad
u(0)=0.$ (1.3)

As is well known, for the solution $u$ we have

$u_{x^{i}x^{j}}(t,x)=\int_{0}^{t}\int_{\mathbf{R}^{d}}p_{x^{i}x^{j}}(s,t,x-y)f(s,y)dyds,$

where $p(t,x)$ is the heat kernel. One can prove that the kernel
$K(s,t,x,y)=1_{s<t}p_{x^{i}x^{j}}(t-s,x-y)$ is singular but satisfies (1.2) on
$\mathbf{R}^{d+1}$. Consequently this leads to

$\displaystyle\|u_{xx}\|_{L_{p}((0,T)\times\mathbf{R}^{d})}\leq
C\|f\|_{L_{p}((0,T)\times\mathbf{R}^{d})}.$

Regarding the $L_{p}$-theory for stochastic PDEs, Krylov [6, 7] firstly
introduced the maximal $L_{p}$-regularity of the stochastic heat equation

$\displaystyle du=\Delta udt+gdW_{t},\,\,t>0,\quad u(0,\cdot)=0.$ (1.4)

In particular, he proved the $L_{p}$-boundedness of

$\displaystyle\nabla u(t,x)=\int_{0}^{t}\int_{\mathbf{R}^{d}}\nabla
p(t-s,x-y)g(s,y)dydW_{s}.$ (1.5)

The right hand side of (1.5) becomes a stochastic singular integral in the
sense that

$\mathbb{E}\left|\int_{0}^{t}\int_{\mathbf{R}^{d}}|\nabla
p(s,t,x-y)|dydW_{s}\right|^{p}=\infty,\quad\forall\,t,p>0.$

Lately, $L_{p}$-theory has been further developed for high-order stochastic
PDEs, stochastic integro-differential equations and certain stochastic pseudo-
differential equations. For related works, we refer to [4, 6, 7, 9] (Krylov’s
analytic approach) and [12, 13] ($H^{\infty}$-calculus). Krylov’s approach
requires differentiability of the kernel, and $H^{\infty}$-calculus approach
works only if the corresponding operator is a generator of bounded analytic
semigroup and does not depend on the time variable.

Our primary goal is to introduce a theory with which one can investigate the
maximal regularity for very large classes of stochastic partial differential
equations. The stochastic singular integral of type (0.1) naturally appears if
one tries to obtain the maximal $L_{p}$-regularity of solutions to stochastic
partial differential equations. We prove that the stochastic Hörmander
condition is sufficient for the $L_{p}$-boundedness of the stochastic integral
and demonstrate that our result on stochastic singular integral (0.1) leads to
the maximal $L_{p}$-regularity of large classes of stochastic partial
differential equations.

Here is a brief comment on our approach. We noticed that some key techniques
in Krylov’s approach, e.g. integration by parts, are not applicable for
general kernels. Hence we combined Krylov’s idea with some tools used for the
deterministic singular integral theory and Calderón-Zygmund theorem.

The article is organized as follows. The main theorem is given Section 2 and
the related parabolic Littlewood-Paley inequality is introduced and proved in
Section 3. In section 4, the main theorem is proved on the basis of the
parabolic Littlewood-Paley inequality. Finally, the maximal $L_{p}$-regularity
result for SPDEs is given in Section 5.

We finish the introduction with the notation used in the article. $\mathbb{N}$
and $\mathbb{Z}$ denote the natural number system and the integer number
system, respectively. As usual $\mathbf{R}^{d}$ stands for the Euclidean space
of points $x=(x^{1},...,x^{d})$. For $i=1,...,d$, multi-indices
$\alpha=(\alpha_{1},...,\alpha_{d})$, $\alpha_{i}\in\\{0,1,2,...\\}$, and
functions $u(x)$ we set

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

We also use the notation $D^{m}$ for a partial derivative of order $m$ with
respect to $x$. For $p\in[1,\infty)$, a normed space $F$, and a measure space
$(X,\mathcal{M},\mu)$, $L_{p}(X,\mathcal{M},\mu;F)$ denotes the space of all
$F$-valued $\mathcal{M}^{\mu}$-measurable functions $u$ so that

$\left\|u\right\|_{L_{p}(X,\mathcal{M},\mu;F)}:=\left(\int_{X}\left\|u(x)\right\|_{F}^{p}\mu(dx)\right)^{1/p}<\infty,$

where $\mathcal{M}^{\mu}$ denotes the completion of $\mathcal{M}$ with respect
to the measure $\mu$.

For $p=\infty$, we write $u\in L_{\infty}(X,\mathcal{M},\mu;F)$ iff

$\sup_{x}|u(x)|:=\|u\|_{L_{\infty}(X,\mathcal{M},\mu;F)}:=\inf\left\\{\nu\geq
0:\mu(\\{x:\|u(x)\|_{F}>\nu\\})=0\right\\}<\infty.$

If there is no confusion for the given measure and $\sigma$-algebra, we
usually omit the measure and the $\sigma$-algebra. In particular, for a domain
$\mathcal{O}\subset\mathbf{R}^{d}$ we denote
$L_{p}(\mathcal{O})=L_{p}(\mathcal{O},\mathcal{L},\ell;\mathbf{R})$ and
$L_{p}(l_{2})=L_{p}(\mathcal{O},\mathcal{L},\ell;l_{2})$, where $\mathcal{L}$
is the Lebesgue measurable sets, $\ell$ is the Lebesuge measure, and $l_{2}$
is the space of sequences $a=(a_{n})$ so that

$|a|^{2}_{l_{2}}=\sum_{n=1}^{\infty}|a_{n}|^{2}<\infty.$

We use “$:=$” to denote a definition. For $a,b\in\mathbf{R}$, $a\wedge
b:=\min\\{a,b\\}$, $a\vee b:=\max\\{a,b\\}$, 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 d-dimensional 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 Lebesgue measurable set $A\subset\mathbf{R}^{d}$, we use
$|A|$ to denote its Lebesgue measure. For a set $B$, $1_{B}$ is the indicator
of $B$, i.e. $1_{B}(b)=1$ if $b\in B$ and $1_{B}(b)=0$ otherwise. For a
complex number $z$, $\overline{z}$ is the complex conjugate of $z$ and
$\Re[z]$ is the real part of $z$. For functions depending on $\omega$, $t$,
and $x$, the argument $\omega\in\Omega$ will be usually omitted. Usually
$X_{0}$, $X$, $Y$, $Z$ denote the vectors in $(0,\infty)\times\mathbf{R}^{d}$
and are represented by

$X_{0}=(t_{0},x_{0}),\quad X=(t,x),\quad Y=(s,y),\quad Z=(r,z),$

where $t_{0}$, $t$, $s$, $r$ are positive numberes and $x_{0}$, $x$, $y$, $z$
are vectors in $\mathbf{R}^{d}$. Finally, $N$ denotes a generic constant which
can differ from line to line and if we write $N=N(a,b,\ldots)$, then this
means that the constant $N$ depends only on $a,b,\ldots$.

##  2\. main result

Let $(\Omega,\mathcal{F},P)$ be a probability space and
$\\{\mathcal{F}_{t},t\geq 0\\}$ be an increasing filtration of $\sigma$-fields
on $\Omega$ satisfying the usual condition, i.e.
$\mathcal{F}_{t}\subset\mathcal{F}$ contains all $(\mathcal{F},P)$-null sets
and $\mathcal{F}_{t}=\bigcap_{s>t}\mathcal{F}_{s}$. By $\mathcal{P}$ we denote
the predictable $\sigma$-algebra, that is, $\mathcal{P}$ is the smallest
$\sigma$-algebra containing the collection of all sets $A\times(s,t]$, where
$0\leq s\leq t<\infty$ and $A\in\mathcal{F}_{s}$. Let
$W^{1}_{t},W^{2}_{t},\cdots$ be an infinite sequence of independent one-
dimensional Wiener processes defined on $\Omega$, each of which is a Wiener
process relative to $\\{\mathcal{F}_{t},t\geq 0\\}$. For $T\in(0,\infty]$ and
a domain $\mathcal{O}\subset\mathbf{R}^{d}$, we denote

$\mathcal{O}_{T}:=(0,T)\times\mathcal{O}.$

Define

$\mathbb{L}_{p}(\mathcal{O}_{T})=L_{p}(\Omega\times(0,T),\mathcal{P};L_{p}(\mathcal{O})),\quad\mathbb{L}_{p}(\mathcal{O}_{T},l_{2})=L_{p}(\Omega\times(0,T),\mathcal{P};L_{p}(\mathcal{O};l_{2})),$

and

$\|g\|_{\mathbb{L}_{p}(\mathcal{O}_{T},l_{2})}=\left(\mathbb{E}\int^{T}_{0}\int_{\mathcal{O}}|g|^{p}_{l_{2}}dxdt\right)^{1/p}.$

If $\mathcal{O}_{T}=(0,\infty)\times\mathbf{R}^{d}$, we simply put
$\mathbb{L}_{p}(\mathcal{O}_{T})=\mathbb{L}_{p}$ and
$\mathbb{L}_{p}(\mathcal{O}_{T},l_{2})=\mathbb{L}_{p}(l_{2})$.

Denote $\mathbf{R}_{+}=(0,\infty)$ and let $K(r,t,z,x)=K(\omega,r,t,z,x)$ be a
$\mathcal{P}\otimes\mathcal{B}(\mathbf{R}_{+})\otimes\mathcal{B}(\mathcal{O})\otimes\mathcal{B}(\mathcal{O})$-measurable
function such that $K(r,t,z,x)=0$ if $r\geq t$. For
$g=(g^{1},g^{2},\ldots)\in\mathbb{L}_{p}(\mathcal{O},l_{2})$ and
$(t,x)\in\mathcal{O}_{T}$, define

$\displaystyle\mathbb{T}_{\varepsilon}g(t,x):=\int_{0}^{t-\varepsilon}\int_{\mathcal{O}}K(r,t,z,x)g^{k}(r,z)dzdW_{r}^{k}$

and

$\displaystyle\mathbb{T}g(t,x)$
$\displaystyle:=\int_{0}^{t}\int_{\mathcal{O}}K(r,t,z,x)g^{k}(r,z)dzdW_{r}^{k}$
$\displaystyle:=\lim_{\varepsilon\downarrow 0}\mathbb{T}_{\varepsilon}g(t,x),$

where the sense of convergence will be specified in Assumption 2.3.

######  Definition 2.1.

Let $D$ be a subset of $\mathbf{R}^{d+1}$. A function $\rho(X,Y)$ defined on
$D\times D$ is called a quasi-metric iff the following four properties hold:

(i) $\rho(X,Y)\geq 0$ for all $X,Y\in D$

(ii) $\rho(X,Y)=0$ iff $X=Y$

(iii) $\rho(X,Y)=\rho(Y,X)$ for all $X,Y\in D$

(iv) There exists a constant $N_{\rho}\geq 1$ such that $\rho(X,Y)\leq
N_{\rho}\left(\rho(X,Z)+\rho(Z,Y)\right)$ for all $X,Y,Z\in D$.

Define balls related to the quasi-metric $\rho$ as

$\displaystyle B_{c}(X):=\\{Z\in D:\rho(X,Z)<c\\},\quad X\in D,~{}c>0.$

Note that the center $X$ of the ball $B_{c}(X)$ is always in $D$.

Throughout the article we assume that the quasi-metric $\rho$ satisfies the
doubling ball condition on $D$, that is, for any $\gamma>0$ there exists a
constant $N_{\gamma}$ so that

$\displaystyle|B_{\gamma c}(X)|\leq N_{\gamma}|B_{c}(X)|\quad\forall
c>0,~{}X\in D.$ (2.1)

For a locally integrable function $f$ on $D$, define its sharp function as

$\displaystyle f^{\sharp}(t,x)$
$\displaystyle:=\sup-\int_{B_{c}(Y)}|f(r,z)-f_{B_{c}(Y)}|~{}drdz$
$\displaystyle:=\sup\frac{1}{|B_{c}(Y)|}\int_{B_{c}(Y)}|f(r,z)-f_{B_{c}(Y)}|~{}drdz$
$\displaystyle\approx\sup\frac{1}{|B_{c}(Y)|^{2}}\int_{B_{c}(Y)}\int_{B_{c}(Y)}|f(Z)-f(Z^{\prime})|~{}dZdZ^{\prime}$

where the sup is taken over all $B_{c}(Y)$ containing $X=(t,x)$ and

$f_{B_{c}(Y)}=-\int_{B_{c}(Y)}f(r,z)~{}drdz.$

Similarly, the maximal function $\mathcal{M}f(t,x)$ is defined as

$\mathcal{M}f(t,x):=\sup-\int_{B_{c}(Y)}|f(r,z)|~{}drdz,$

where the sup is taken over all $B_{c}(Y)$ containing $X=(t,x)$.

Below is a version of Hardy-Littlewood and Fefferman-Stein theorems.

######  Theorem 2.2.

For any $p>1$,

$\displaystyle\|\mathcal{M}f\|_{L_{p}(D)}\leq
N(p)\|f\|_{L_{p}(D)},\quad\forall f\in L_{p}(D).$ (2.2)

Furthermore, if $|D|=\infty$, then

$\|f\|_{L_{p}(D)}\leq N_{p}\|f^{\sharp}\|_{L_{p}(D)}$ (2.3)

For the proof of this theorem, see e.g. [1, Theorem 2.2 and Theorem 2.4]. If
(2.3) holds, we say $\rho$ admits the Feffreman-Stein theorem (FS), which
obviously holds if $|D|=\infty$.

######  Assumption 2.3 ($L_{2}$-boundedness).

For each $g\in\mathbb{L}_{2}(\mathcal{O}_{T},l_{2})$,
$\mathbb{T}_{\varepsilon}g(t,x)$ converges in $\mathbb{L}_{2}(O_{T})$ as
$\varepsilon\downarrow 0$. Moreover the operator $g\mapsto\mathbb{T}g$ is
bounded from $\mathbb{L}_{2}(\mathcal{O}_{T},l_{2})$ to
$\mathbb{L}_{2}(\mathcal{O}_{T})$, i.e., there exists a constant $N_{0}$ such
that for all $g\in\mathbb{L}_{2}(\mathcal{O}_{T},l_{2})$,

$\displaystyle\|\mathbb{T}g\|_{\mathbb{L}_{2}(\mathcal{O}_{T})}\leq
N_{0}\|g\|_{\mathbb{L}_{2}(\mathcal{O}_{T},l_{2})}.$ (2.4)

######  Assumption 2.4 (A stochastic Hörmander condition).

There exist positive constants $C_{0}$ and $N_{1}$ such that for all
$X=(t,x),Y=(s,y),Z=(r,z)\in\mathcal{O}_{T}$,

$\displaystyle\sup_{\omega\in\Omega}\sup_{X,Y}\int_{0}^{T}\left[\int_{\rho(X,Z)\geq
C_{0}\rho(X,Y)}|K(r,t,z,x)-K(r,s,z,y)|~{}dz\right]^{2}dr\leq N_{1},$ (2.5)

where $\rho$ is a quasi-metric admitting FS.

By $\mathbb{L}(\mathcal{O}_{T},l_{2})$ (simply $\mathbb{L}(l_{2})$ if
$\mathcal{O}_{T}=(0,\infty)\times\mathbf{R}^{d}$), we denote the space of the
processes $g=(g^{1},g^{2},\ldots)$ such that $g^{k}=0$ for all large $k$ and
each $g^{k}$ is of the type

$g^{k}(t,x)=\sum_{i=1}^{j(k)}1_{(\tau_{i-1},\tau_{i}]}(t)g^{ik}(x),$

where $g^{ik}\in C_{0}^{\infty}(\mathcal{O})$, and $\tau_{i}$ are stopping
times so that $\tau_{i}\leq T$. It is known that
$\mathbb{L}(\mathcal{O}_{T},l_{2})$ is dense in
$\mathbb{L}_{p}(\mathcal{O}_{T},l_{2})$ for all $p\geq 1$ (for instance, see
[7, Theorem 3.10]) if $\mathcal{O}_{T}=(0,T)\times\mathbf{R}^{d}$. The idea of
[7, Theorem 3.10] is easily applied even for general $\mathcal{O}_{T}$.

Here is our main result.

######  Theorem 2.5.

Suppose that Assumptions 2.3 and 2.4 hold. Then for any $p>2$, the operator
$\mathbb{T}$ can be continuously extended from
$\mathbb{L}(\mathcal{O}_{T},l_{2})$ to
$\mathbb{L}_{p}(\mathcal{O}_{T},l_{2})$. Moreover, for any
$g\in\mathbb{L}_{p}(\mathcal{O}_{T},l_{2})$,

$\displaystyle\|\mathbb{T}g\|_{\mathbb{L}_{p}(\mathcal{O}_{T})}\leq
N(d,p,C_{0},N_{0},N_{1})\|g\|_{\mathbb{L}_{p}(\mathcal{O}_{T},l_{2})}.$

The proof of this theorem will be given in Section 4.

##  3\. Parabolic Littlewood-Paley inequality

For $l_{2}$-valued measurable functions $f=(f^{1},f^{2},\cdots)$ on
$\mathcal{O}_{T}$, denote

$\displaystyle\mathcal{G}f(t,x)$
$\displaystyle:=\left[\int_{0}^{t}\left|\int_{\mathcal{O}}K(r,t,z,x)f(r,z)dz\right|^{2}_{l_{2}}~{}dr\right]^{1/2}$
$\displaystyle:=\lim_{\varepsilon\downarrow
0}\left[\int_{0}^{t-\varepsilon}\left|\int_{\mathcal{O}}K(r,t,z,x)f(r,z)dz\right|^{2}_{l_{2}}~{}dr\right]^{1/2}.$
(3.1)

In this section we study the boundedness of operator $\mathcal{G}$ in
$L_{p}(\mathcal{O}_{T};l_{2})$. Since the integral above is deterministic one
may assume that the kernel $K$ is nonrandom throughout this section.

Theorem 3.1 below is the main result of this section which we call “Parabolic
Littlewood-Paley inequality”. This inequality was first proved by Krylov for
$K(r,t,x,y)=\nabla_{x}p(t-r,x-y)$, where $p(t,x)=\frac{1}{(4\pi
t)^{d/2}}e^{-|x|^{2}/(4t)}$ is the heat kernel. If $f$ is independent of $t$
then parabolic Littlewood-Paley inequality with $K=\nabla_{x}p(t-r,x-y)$ leads
to the classical (elliptic) Littlewood-Paley inequality (for instance, see [4,
Section 1]).

######  Theorem 3.1 (Parabolic Littlewood-Paley inequality).

Let $p\geq 2$. Suppose Assumptions 2.3 and 2.4 hold. Then for any $f\in
L_{2}(\mathcal{O}_{T};l_{2})\cap L_{\infty}(\mathcal{O}_{T};l_{2})$,

$\displaystyle\|\mathcal{G}f\|_{L_{p}(\mathcal{O}_{T})}\leq
N\|f\|_{L_{p}(\mathcal{O}_{T};l_{2})},$

where $N$ depends only on $d$, $p$, $C_{0}$, $N_{1}$, and $N_{2}$.

The proof of this theorem will be given at the end of this section.

######  Lemma 3.2.

Suppose Assumption 2.3 holds. Then for each $f\in
L_{2}(\mathcal{O}_{T};l_{2})$, $\mathcal{G}f(t,x)$ is finite almost
everywhere, and moreover the operator $f\rightarrow\mathcal{G}f$ is a bounded
operator from $L_{2}(\mathcal{O}_{T};l_{2})$ to $L_{2}(\mathcal{O}_{T})$.

###### Proof.

Obviously, since $f$ is nonrandom, $f\in\mathbb{L}_{2}(l_{2})$. By Itô’s
isometry and (2.4),

$\displaystyle\int_{0}^{T}\int_{\mathcal{O}}|\mathcal{G}f(t,x)|^{2}dxdt$
$\displaystyle=\mathbb{E}\int_{0}^{T}\int_{\mathcal{O}}|\mathbb{T}f(t,x)|^{2}dxdt$
$\displaystyle\leq
N_{0}\mathbb{E}\int_{0}^{T}\int_{\mathcal{O}}|f(t,x)|_{l_{2}}^{2}dxdt$
$\displaystyle=N_{0}\int_{0}^{T}\int_{\mathcal{O}}|f(t,x)|_{l_{2}}^{2}dxdt.$

Thus the lemma is proved. $\Box$

Denote

$\mathcal{K}f(r,t,x)=\int_{\mathcal{O}}K(r,t,z,x)f(r,z)dz$

and

$\displaystyle
Gf(t,s,x,y)=\left[\int_{0}^{T}|\mathcal{K}f(r,t,x)-\mathcal{K}f(r,s,y)|_{l_{2}}^{2}~{}dr\right]^{1/2}.$

Observe

$\mathcal{G}f(t,x)=\left[\int_{0}^{t}\left|\mathcal{K}f(r,t,x)\right|^{2}_{l_{2}}~{}dr\right]^{1/2}$

and

$Gf(t,s,x,y)=\left[\int_{0}^{T}|1_{r<t}\mathcal{K}f(r,t,x)-1_{r<s}\mathcal{K}f(r,s,y)|_{l_{2}}^{2}~{}dr\right]^{1/2},$

where the last equality is due to the assumption that $K(r,t,z,x)=0$ if $t\leq
r$.

######  Lemma 3.3.

Let $(t_{1},x_{1})\in B_{c}(X_{0})$ and suppose Assumption 2.3 holds. Then for
any $f_{1},f_{2}\in L_{2}(\mathcal{O}_{T};l_{2})$,

$\displaystyle-\int_{B_{c}(X_{0})}-\int_{B_{c}(X_{0})}|\mathcal{G}(f_{1}+f_{2})(t,x)-\mathcal{G}(f_{1}+f_{2})(s,y)|~{}dtdxdsdy$
$\displaystyle\leq
2\mathcal{M}(\mathcal{G}f_{1})(t_{1},x_{1})+-\int_{B_{c}(X_{0})}-\int_{B_{c}(X_{0})}Gf_{2}(t,s,x,y)~{}dtdxdsdy.$
(3.2)

###### Proof.

Set $f=f_{1}+f_{2}$ and let $(t,x),(s,y)\in B_{c}(X_{0})$. By Lemma 3.2 we may
assume

$\mathcal{G}(f_{1})(t,x)+\mathcal{G}(f_{2})(t,x)+\mathcal{G}(f_{1})(s,y)+\mathcal{G}(f_{2})(s,y)<\infty.$

Then by Minkowski’s inequality,

$\displaystyle|\mathcal{G}f(t,x)-\mathcal{G}f(s,y)|$
$\displaystyle=\left|\left[\int_{0}^{T}1_{r<t}|\mathcal{K}f(r,t,x)|_{l_{2}}^{2}~{}dr\right]^{1/2}-\left[\int_{0}^{T}1_{r<s}|\mathcal{K}f(r,s,y)|_{l_{2}}^{2}~{}dr\right]^{1/2}\right|$
$\displaystyle\leq\left[\int_{0}^{T}\Big{|}1_{r<t}\mathcal{K}f(r,t,x)-1_{r<s}\mathcal{K}f(r,s,y)\Big{|}_{l_{2}}^{2}~{}dr\right]^{1/2}$
$\displaystyle\leq\left[\int_{0}^{T}1_{r<t}|\mathcal{K}f_{1}(r,t,x)|_{l_{2}}^{2}~{}dr\right]^{1/2}+\left[\int_{0}^{T}1_{r<s}|\mathcal{K}f_{1}(r,s,y)|_{l_{2}}^{2}~{}dr\right]^{1/2}$
$\displaystyle\quad+\left[\int_{0}^{T}|1_{r<t}\mathcal{K}f_{2}(r,t,x)-1_{r<s}\mathcal{K}f_{2}(r,s,y)|_{l_{2}}^{2}~{}dr\right]^{1/2}.$

Taking mean average to the above inequality, we get (3.2). $\Box$

Take constants $N_{\rho}$ and $C_{0}$ from Definition 2.1 and Assumption 2.4
respectively, and denote

$\gamma_{0}=\gamma_{0}(N_{\rho},C_{0}):=(2C_{0}N_{\rho}+1)N_{\rho}.$

######  Lemma 3.4.

Suppose Assumption 2.3 holds, $f$ belongs to $L_{2}(\mathcal{O}_{T};l_{2})\cap
L_{\infty}(\mathcal{O}_{T};l_{2})$ and vanishes outside of
$Q_{\gamma_{0}c}(t_{0},x_{0})$. Then

$\displaystyle-\int_{B_{c}(X_{0})}-\int_{B_{c}(X_{0})}Gf(t,s,x,y)~{}dtdxdsdy\leq
N\|f\|_{L_{\infty}(\mathcal{O}_{T};l_{2})},$ (3.3)

where $N$ depends only on $d$, $\gamma_{0}$, and $N_{0}$.

###### Proof.

Let $(t,x),(s,y)\in B_{c}(X_{0})$ and assume
$\mathcal{G}f(t,x)+\mathcal{G}f(s,y)<\infty$. Then by Minkowski’s inequality,

$Gf(t,s,x,y)\leq\mathcal{G}f(t,x)+\mathcal{G}f(s,y).$

Therefore the left side of (3.3) is less than or equal to

$2-\int_{B_{c}(X_{0})}\mathcal{G}f(t,x)~{}dtdx.$

Moreover by Hölder’s inequality and Lemma 3.2,

$\displaystyle-\int_{B_{c}(X_{0})}\mathcal{G}f(t,x)~{}dtdx$
$\displaystyle\leq\frac{1}{|B_{c}(X_{0})|^{1/2}}\left[\int_{B_{c}(X_{0})}|\mathcal{G}f(t,x)|^{2}~{}dtdx\right]^{1/2}$
$\displaystyle\leq\frac{1}{|B_{c}(X_{0})|^{1/2}}\left[\int_{\mathcal{O}_{T}}|\mathcal{G}f(t,x)|^{2}~{}dtdx\right]^{1/2}$
$\displaystyle\leq\frac{N_{0}}{|B_{c}(X_{0})|^{1/2}}\left[\int_{\mathcal{O}_{T}}|f(t,x)|^{2}~{}dtdx\right]^{1/2}$
$\displaystyle\leq
N(d,\gamma_{0},N_{0})\|f\|_{L_{\infty}(\mathcal{O}_{T};l_{2})},$

where the last inequality is due to the assumption that $f=0$ outside of
$B_{\gamma_{0}c}(X_{0})$ and (2.1). Thus the lemma is proved. $\Box$

######  Lemma 3.5.

Let $f\in L_{2}(\mathcal{O}_{T};l_{2})\cap L_{\infty}(\mathcal{O}_{T};l_{2})$
and $f=0$ on $Q_{\gamma_{0}c}(t_{0},x_{0})$. Suppose that Assumption 2.3 and
Assumption 2.4 hold. Then

$\displaystyle-\int_{B_{c}(X_{0})}-\int_{B_{c}(X_{0})}Gf(t,s,x,y)~{}dtdxdsdy\leq
N\|f\|_{L_{\infty}(\mathcal{O}_{T};l_{2})},$

where $N$ depends only on $N_{1}$.

###### Proof.

If $X=(t,x)$, $Y=(s,y)\in B_{c}(X_{0})$, and
$Z=(r,z)\in\mathcal{O}_{T}\setminus Q_{\gamma_{0}c}(t_{0},x_{0})$, then

$\displaystyle\rho(Z,X)\geq\frac{\rho(Z,X_{0})}{N_{\rho}}-\rho(X,X_{0})\geq
2C_{0}N_{\rho}c\geq C_{0}\rho(X,Y).$ (3.4)

Thus recalling the definition of $Gf$ and the assumptions on $f$, we have

$\displaystyle|Gf(t,s,x,y)|^{2}$
$\displaystyle\leq\int_{0}^{T}\left[\int_{\mathcal{O}}|K(r,t,z,x)-K(r,s,z,y)||f(r,z)|_{l_{2}}dz\right]^{2}~{}dr$
$\displaystyle\leq\|f\|^{2}_{L_{\infty}(\mathcal{O}_{T};l_{2})}\int_{0}^{T}\left[\int_{A(t,r,s,x,y)}|K(r,t,z,x)-K(r,s,z,y)|dz\right]^{2}~{}dr,$

where $A(t,r,s,x,y)$ is the set of all $z\in\mathbf{R}^{d}$ for which
inequality (3.4) holds. Therefore by (2.5),

$|Gf(t,s,x,y)|\leq N_{1}^{1/2}\|f\|_{L_{\infty}(\mathcal{O}_{T};l_{2})}$

and

$\displaystyle-\int_{B_{c}(X_{0})}-\int_{B_{c}(X_{0})}Gf(t,s,x,y)~{}dtdxdsdy\leq
N_{1}^{1/2}\|f\|_{L_{\infty}(\mathcal{O}_{T};l_{2})}.$

The lemma is proved. $\Box$

######  Lemma 3.6.

Let $f_{1}\in L_{2}(\mathcal{O}_{T};l_{2})$, $f_{2}\in
L_{2}(\mathcal{O}_{T};l_{2})\cap L_{\infty}(\mathcal{O}_{T};l_{2})$, and
suppose Assumptions 2.3 and 2.4 hold. Then for each
$(t_{1},x_{1})\in\mathcal{O}_{T}$,

$\displaystyle[\mathcal{G}(f_{1}+f_{2})]^{\sharp}(t_{1},x_{1})\leq
2\mathcal{M}(\mathcal{G}f_{1})(t_{1},x_{1})+N\|f_{2}\|_{L_{\infty}(\mathcal{O}_{T};l_{2})},$
(3.5)

where $N$ depends only on $d$, $\gamma$, $N_{0},C_{0}$, and $N_{1}$.

###### Proof.

Let $(t_{1},x_{1})\in B_{c}(X_{0})$. Then by Lemma 3.3,

$\displaystyle-\int_{B_{c}(X_{0})}-\int_{B_{c}(X_{0})}|\mathcal{G}(f_{1}+f_{2})(t,x)-\mathcal{G}(f_{1}+f_{2})(s,y)|~{}dtdxdsdy$
$\displaystyle\leq
2\mathcal{M}(\mathcal{G}f_{1})(t_{1},x_{1})+-\int_{B_{c}(X_{0})}-\int_{B_{c}(X_{0})}Gf_{2}(t,s,x,y)~{}dtdxdsdy.$

Moreover, defining
$f_{2,1}(t,x):=f_{2}(t,x)1_{Q_{\gamma_{0}c}(t_{0},x_{0})}(t,x)$ and
$f_{2,2}(t,x):=f_{2}(t,x)-f_{2,1}(t,x)$, we have

$\displaystyle-\int_{B_{c}(X_{0})}-\int_{B_{c}(X_{0})}Gf_{2}~{}dtdxdsdy$
$\displaystyle\leq-\int_{B_{c}(X_{0})}-\int_{B_{c}(X_{0})}Gf_{2,1}~{}dtdxdsdy+-\int_{B_{c}(X_{0})}-\int_{B_{c}(X_{0})}Gf_{2,2}~{}dtdxdsdy.$

Therefore we obtain (3.5) by applying Lemma 3.4 and Lemma 3.5. The lemma is
proved. $\Box$

Proof of Theorem 3.1

Since the case $p=2$ is already proved in Lemma 3.2, we assume $p>2$. Let
$f\in L_{2}(\mathcal{O}_{T};l_{2})\cap L_{\infty}(\mathcal{O}_{T};l_{2})$. For
$\lambda>0$, we put

$f_{1,\lambda}(t,x)=f(t,x)1_{|f|>\delta\lambda}(t,x)\quad\text{and}\quad
f_{2,\lambda}(t,x)=f(t,x)1_{|f|\leq\delta\lambda}(t,x),$

where $\delta$ is a positive constant which will be specified later.
Obviously,

$f=f_{1,\lambda}+f_{2,\lambda}.$

Assume

$\lambda\leq[\mathcal{G}(f)]^{\sharp}(t,x).$

Then by Lemma 3.6,

$\displaystyle\lambda\leq[\mathcal{G}(f)]^{\sharp}(t,x)$ $\displaystyle\leq
2\mathcal{M}(\mathcal{G}f_{1,\lambda})(t,x)+N\|f_{2,\lambda}\|_{L_{\infty}(\mathcal{O}_{T};l_{2})}$
$\displaystyle\leq
2\mathcal{M}(\mathcal{G}f_{1,\lambda})(t,x)+N\delta\lambda,$

where $N$ is independent of $\lambda$ and $\delta$. Take $\delta>0$ so that
$N\delta<1/2$. Then the above inequality implies that

$\lambda\leq 4\mathcal{M}(\mathcal{G}f_{1,\lambda})(t,x).$

Thus

$\displaystyle|\\{(t,x)\in\mathcal{O}_{T}:\lambda\leq[\mathcal{G}(f)]^{\sharp}(t,x)\\}|\leq|\\{(t,x)\in\mathcal{O}_{T}:\lambda\leq
4\mathcal{M}(\mathcal{G}f_{1,\lambda})(t,x)\\}|.$ (3.6)

By (2.2),

$\|\mathcal{G}f\|_{L_{p}(\mathcal{O}_{T})}\leq
N_{p}\|[\mathcal{G}f]^{\sharp}\|_{L_{p}(\mathcal{O}_{T})}.$

Observe that

$\displaystyle\|[\mathcal{G}f]^{\sharp}\|^{p}_{L_{p}(\mathcal{O}_{T})}=p\int_{0}^{\infty}\lambda^{p-1}|\\{(t,x)\in\mathcal{O}_{T}:\lambda\leq[\mathcal{G}(f)]^{\sharp}(t,x)\\}|~{}d\lambda.$

Therefore by (3.6), Chebyshev’s inequality, (2.2), and Lemma 3.2,

$\displaystyle\|[\mathcal{G}f]^{\sharp}\|^{p}_{L_{p}(\mathcal{O}_{T})}$
$\displaystyle\leq
p\int_{0}^{\infty}\lambda^{p-1}|\\{(t,x)\in\mathcal{O}_{T}:\lambda\leq
4\mathcal{M}(\mathcal{G}f_{1,\lambda})(t,x)\\}|d\lambda$ $\displaystyle\leq
N\int_{0}^{\infty}\lambda^{p-3}\int_{\mathcal{O}_{T}}|\mathcal{M}(\mathcal{G}f_{1,\lambda})(t,x)|^{2}dtdxd\lambda$
$\displaystyle\leq
N\int_{0}^{\infty}\lambda^{p-3}\int_{\mathcal{O}_{T}}|\mathcal{G}f_{1,\lambda}(t,x)|^{2}dtdxd\lambda$
$\displaystyle\leq
N\int_{0}^{\infty}\lambda^{p-3}\int_{\mathcal{O}_{T}}|f_{1,\lambda}(t,x)|^{2}dtdxd\lambda$
$\displaystyle\leq
N\int_{0}^{\infty}\lambda^{p-3}\int_{\mathcal{O}_{T}\cap\\{|f|>\delta\lambda\\}}|f(t,x)|^{2}dtdxd\lambda$
$\displaystyle=N\int_{\mathcal{O}_{T}}\left(\int^{|f|/\delta}_{0}\lambda^{p-3}d\lambda\right)\,|f(t,x)|^{2}dtdx$
$\displaystyle\leq N\|f\|_{L_{p}(\mathcal{O}_{T};l_{2})}.$

The last inequality is due to $p>2$. The theorem is proved. $\Box$

##  4\. Proof of Theorem 2.5

Let $g\in\mathbb{L}(O_{T})$. Then for each $\omega$, $g\in
L_{2}(\mathcal{O}_{T};l_{2})\cap L_{\infty}(\mathcal{O}_{T};l_{2})$. Therefore
by Fubini’s Theorem, Burkholder-Davis-Gundy’s inequality, and Theorem 3.1,

$\displaystyle\mathbb{E}\int_{0}^{T}\|\mathbb{T}g(t,\cdot)\|_{L_{p}}^{p}dt$
$\displaystyle=\int_{0}^{T}\int_{\mathbf{R}^{d}}\mathbb{E}\left|\int_{0}^{t}\int_{\mathbf{R}^{d}}K(r,t,z,x)g^{k}(r,z)dzdW_{r}^{k}\right|^{p}dxdt$
$\displaystyle\leq
N(p)\int_{0}^{T}\int_{\mathbf{R}^{d}}\mathbb{E}\left(\int_{0}^{t}\left|\int_{\mathbf{R}^{d}}K(r,t,z,x)g(r,z)dz\right|^{2}_{l_{2}}dr\right)^{p/2}dxdt$
$\displaystyle\leq
N\mathbb{E}\int_{0}^{T}\int_{\mathbf{R}^{d}}|\mathcal{G}g|^{p}dxdt\leq
N\mathbb{E}\int_{0}^{T}\int_{\mathbf{R}^{d}}|g|^{p}dxdt.$

The theorem is proved. $\Box$

##  5\.  Application to SPDE: Maximal $L_{p}$-regularity

We study the maximal $L_{p}$-regularity of SPDEs of the type

$\displaystyle
du(t,x)=A(t)u(t,x)dt+\sum_{k=1}^{\infty}g^{k}(t,x)dW^{k}_{t},\quad(t,x)\in(0,\infty)\times\mathbf{R}^{d};\quad
u(0)=0,$ (5.1)

where $W^{k}_{t}$ are independent one-dimensional Wiener process defined on
$\Omega$.

###  5.1. Time measureable pseudo-differential operator

Assume that $A(t)$ is a pseudo differential operator with the symbol
$\psi(t,\xi)$, that is,

$A(t)f(x):=\mathcal{F}^{-1}\left[\psi(t,\xi)\mathcal{F}(f)(\xi)\right](x).$

We set

$d_{0}:=\left\lfloor\frac{d}{2}\right\rfloor+1,$

and assume there exists a constant $\nu>0$ such that

$\displaystyle\Re[-\psi(t,\xi)]\geq\nu|\xi|^{\gamma},$ (5.2)

and

$\displaystyle\int_{R\leq|\xi|<2R}\prod_{i=1}^{d_{0}}\left|D^{\alpha_{i}}_{\xi}\psi(t,\xi)\right|^{k_{i}}d\xi$
$\displaystyle\quad\leq\nu^{-1}R^{\left(d+k_{1}(\gamma-|\alpha_{1}|)+k_{2}(\gamma-|\alpha_{2}|)+\cdots+k_{d_{0}}(\gamma-|\alpha_{d_{0}}|)\right)}$
(5.3)

for any $R>0$, multi-indexes $\alpha_{i}\in(\mathbb{Z}_{+})^{d}$ and
$k_{i}\in\mathbb{Z}_{+}$ ($i=1,2,\cdots,d_{0}$) such that

$\sum_{i=1}^{d_{0}}|\alpha_{i}|+\sum_{i=1}^{d_{0}}k_{i}\leq
d_{0}+\sum_{i=1}^{d_{0}}1_{k_{i}>0}.$

######  Remark 5.1.

Here is a sufficient condition for (5.3): $\exists\,c>0$ such that

$\displaystyle|D^{\alpha}_{\xi}\psi(t,\xi)|\leq
c|\xi|^{\gamma-|\alpha|},\quad\quad\forall\xi\in\mathbf{R}^{d}\setminus\\{0\\},\quad\forall\,|\alpha|\leq
d_{0}.$ (5.4)

Indeed, if this holds then, for $R\leq|\xi|<2R$,

$\prod_{i=1}^{d_{0}}|D^{\alpha_{i}}_{\xi}\psi(t,\xi)|^{k_{i}}\leq
N(c,d)R^{k_{1}(\gamma-|\alpha_{1}|)+k_{2}(\gamma-|\alpha_{2}|)+\cdots+k_{d_{0}}(\gamma-|\alpha_{d_{0}})|}$

Thus by integrating on $\\{\xi\in\mathbf{R}^{d}:R\leq|\xi|<2R\\}$ we certainly
get (5.3).

Define

$p(s,t,x):=1_{0<s<t}\mathcal{F}^{-1}\left[\exp\left(\int_{s}^{t}\psi(r,\xi)dr\right)\right](x).$

and

$(-\Delta)^{\gamma/4}p(s,t,x):=1_{0<s<t}\mathcal{F}^{-1}\left[|\xi|^{\gamma/2}\exp\left(\int_{s}^{t}\psi(r,\xi)dr\right)\right](x).$

Then for any $g\in\mathbb{L}_{2}(l_{2})$, the (weak) solution to (5.1) is
given by

$u(t,x)=\int_{0}^{t}\int_{\mathbf{R}^{d}}p(s,t,x-y)g^{k}(s,y)dydW_{s}^{k}.$
(5.5)

See e.g. [7, Theorem 4.2] for details. Actually in [7] the representation
formula of the weak solution is derived only for $\psi(t,\xi)=-|\xi|^{2}$, but
one can easily check the argument there works for the general case.

Due to (5.2) we may say $A(t)$ is a linear operator of order $\gamma$.
Applying the Ito’s formula to $|u(t,x)|^{2}$, taking the expectation, and then
integrating over $\mathbf{R}^{d}$, we get for any $t>0$,

$\mathbb{E}\|u(t)\|^{2}_{L_{2}(\mathbf{R}^{d})}-2\Re\left[\mathbb{E}\int^{t}_{0}\int_{\mathbf{R}^{d}}u\overline{Au}dxds\right]=\mathbb{E}\int^{t}_{0}\|g\|^{2}_{L_{2}(\mathbf{R}^{d};l_{2})}ds.$

By Plancherel’s theorem and (5.2),

$\displaystyle-\Re\left[\int_{\mathbf{R}^{d}}u\overline{Au}dxds\right]=-\Re\left[\int_{\mathbf{R}^{d}}\overline{\psi(s,\xi)}|\mathcal{F}(u)|^{2}d\xi\right]$
$\displaystyle=\int_{\mathbf{R}^{d}}\Re[-\psi(s,\xi)]|\mathcal{F}(u)|^{2}d\xi\geq\nu\int_{\mathbf{R}^{d}}|\xi|^{\gamma}|\mathcal{F}(u)|^{2}d\xi=\nu\|(-\Delta)^{\gamma/4}u\|^{2}_{L_{2}(\mathbf{R}^{d})}.$

It follows that

$\mathbb{E}\|u(t)\|^{2}_{L_{2}(\mathbf{R}^{d})}+2\nu\mathbb{E}\int^{t}_{0}\|(-\Delta)^{\gamma/4}u\|^{2}_{L_{2}(\mathbf{R}^{d})}ds\leq\mathbb{E}\int^{t}_{0}\|g\|^{2}_{L_{2}(\mathbf{R}^{d};l_{2})}ds,$

and above calculations suggest that $(-\Delta)^{\gamma/4}u$ is the maximal
regularity of solutions if there is no smoothness condition on $g$.

The following theorem extends the above $L_{2}$-estimate to $L_{p}$-estimate.

######  Theorem 5.2.

Let $p\geq 2$ and assume (5.2) and (5.3) hold. Then for any
$g\in\mathbb{L}(l_{2})$ and $u$ defined as in (5.5), we have

$\displaystyle\mathbb{E}\int_{0}^{\infty}\|(-\Delta)^{\gamma/4}u(t,\cdot)\|^{p}_{\mathbb{L}_{p}}dt\leq
N(d,p,\gamma,\nu)\mathbb{E}\int_{0}^{\infty}\|g(t,\cdot)\|^{p}_{L_{p}(l_{2})}dt.$
(5.6)

######  Remark 5.3.

A proof of (5.6) is given in [4] with a stronger condition than (5.4), that is

$\displaystyle|D^{\alpha}_{\xi}\psi(t,\xi)|\leq\nu^{-1}|\xi|^{\gamma-|\alpha|},\quad\forall\,|\alpha|\leq
d_{0}+1.$

The proof of [4] highly depends on the integration by parts, which requires
the stronger assumption on $\psi(t,\xi)$.

######  Example 5.4.

Let $m\in\mathbb{N}$ and
$A(t)=(-1)^{m-1}\sum_{|\alpha|=|\beta|=m}a^{\alpha\beta}(t)D^{\alpha+\beta}$
be a $2m$-order differential operator. Assume that $a^{\alpha\beta}(t)$ are
bounded complex-valued measurable functions and satisfy an ellipticity
condition, i.e.,

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

Then $A(t)$ is the pseudo-differential operator whose symbol is given by
$\psi(t,\xi)=(-1)^{m}\sum_{|\alpha|=|\beta|=m}a^{\alpha\beta}(t)\xi^{\alpha}\xi^{\beta}$.
Obviously $\psi(t,\xi)$ satisfies (5.2) and (5.4) with $\gamma=2m$.

######  Example 5.5.

The class of pseudo-differential operators we are considering in this article
covers a certain class of non-local operators. Let $\gamma\in(0,2)$ and denote

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

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

(i) If $\gamma=1$ then

$\displaystyle\int_{\partial B_{1}}m(t,w)w~{}S_{1}(dw)=0,\quad\forall t>0,$
(5.7)

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 $d_{0}=\lfloor\frac{d}{2}\rfloor+1$.

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

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

(iv) 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.

Using (i)-(iv) one can check that $A(t)$ is a pseudo differential operator
with the symbol $\psi(t,\xi)$ satisfying (5.2) and (5.4), where

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

and $c_{1}(\gamma,d)$, $c_{2}(\gamma,d)$ are certain positive constants (see
[9] for the detail).

To apply Theorem 2.5 we set $T=\infty$, $\mathcal{O}=\mathbf{R}^{d}$, and

$\rho(X,Y)=|t-s|^{1/\gamma}+|x-y|,$

where $X=(t,x)$ and $Y=(s,y)$. Since $\rho$ is a quasi-metric with the
doubling ball condition and $|(0,\infty)\times\mathbf{R}^{d}|=\infty$, $\rho$
admits the Fefferman-Stein theorem. Define

$\displaystyle\mathbb{T}_{\psi,\varepsilon}g:=\int_{0}^{t-\varepsilon}\int_{\mathbf{R}^{d}}(-\Delta)^{\gamma/4}p(r,t,x-y)g^{k}(r,y)dydW_{r}^{k}$

and

$\displaystyle\mathbb{T}_{\psi}g:=\lim_{\varepsilon\downarrow
0}\int_{0}^{t-\varepsilon}\int_{\mathbf{R}^{d}}(-\Delta)^{\gamma/4}p(r,t,x-y)g^{k}(r,y)dydW_{r}^{k},$

where the limit is in the sense of $\mathbb{L}_{2}$-norm.

In the next lemma, we first show that $\mathbb{T}_{\psi,\varepsilon}g$
converges with respect to the norm in $\mathbb{L}_{2}$ and $\mathbb{T}_{\psi}$
is a bounded operator from $\mathbb{L}_{2}(l_{2})$ to $\mathbb{L}_{2}$.

######  Lemma 5.6.

For each $g\in\mathbb{L}_{2}(l_{2})$, $\mathbb{T}_{\varepsilon}g(t,x)$
converges in $\mathbb{L}_{2}$ as $\varepsilon\downarrow 0$. Moreover the
operator $g\mapsto\mathbb{T}g$ is bounded from $\mathbb{L}_{2}(l_{2})$ to
$\mathbb{L}_{2}$, i.e., there exists a constant $N_{0}$ such that for all
$g\in\mathbb{L}_{2}(l_{2})$,

$\displaystyle\mathbb{E}\int_{0}^{\infty}\|\mathbb{T}_{\psi}g(t,\cdot)\|_{L_{2}}^{2}dt\leq
N_{0}\mathbb{E}\int_{0}^{\infty}\|g(t,\cdot)\|_{L_{2}(l_{2})}^{2}dt.$ (5.8)

###### Proof.

Let $\varepsilon_{1}>\varepsilon_{2}>0$. Then by Fubini’s theorem, Itô’s
isometry, and Plancherel’s theorem,

$\displaystyle\mathbb{E}\int_{0}^{\infty}\|\mathbb{T}_{\psi,\varepsilon_{1}}g(t,\cdot)-\mathbb{T}_{\psi,\varepsilon_{1}}g(t,\cdot)\|_{L_{2}}^{2}dt$
$\displaystyle=\int_{\mathbf{R}^{d}}\int_{0}^{\infty}\mathbb{E}\left|\int_{t-\varepsilon_{1}}^{t-\varepsilon_{2}}\int_{\mathbf{R}^{d}}(-\Delta)^{\gamma/4}p(r,t,x-y)g^{k}(r,y)dydW_{r}^{k}\right|^{2}dtdx$
$\displaystyle=\int_{0}^{\infty}\mathbb{E}\int_{t-\varepsilon_{1}}^{t-\varepsilon_{2}}\int_{\mathbf{R}^{d}}\left|\int_{\mathbf{R}^{d}}(-\Delta)^{\gamma/4}p(r,t,x-y)g^{k}(r,y)dy\right|_{l_{2}}^{2}dxdrdt$
$\displaystyle=N(d)\int_{0}^{\infty}\mathbb{E}\int_{t-\varepsilon_{1}}^{t-\varepsilon_{2}}\int_{\mathbf{R}^{d}}|\xi|^{\gamma}\exp\left(2\int_{r}^{t}\Re[\psi(\rho,\xi)]d\rho\right)|\mathcal{F}(g)(r,\cdot)|^{2}_{l_{2}}d\xi
drdt$ $\displaystyle\leq
N(d)\mathbb{E}\int_{0}^{\infty}\int_{\mathbf{R}^{d}}\int_{\varepsilon_{2}}^{\varepsilon_{1}}|\xi|^{\gamma}\exp\left(-2t\nu|\xi|^{\gamma}\right)dt|\mathcal{F}(g)(r,\cdot)|^{2}_{l_{2}}d\xi
dr$ $\displaystyle\leq
N(d)\mathbb{E}\int_{0}^{\infty}\int_{\mathbf{R}^{d}}\left|\exp\left(-2\nu\varepsilon_{1}|\xi|^{\gamma}\right)-\exp\left(-2t\nu\varepsilon_{2}|\xi|^{\gamma}\right)\right||\mathcal{F}(g)(r,\cdot)|^{2}_{l_{2}}d\xi
dr.$

The last term goes to zero as $\varepsilon_{1},\varepsilon_{2}\to 0$ by the
Lebesgue dominated convergence theorem. Therefore $\mathbb{T}_{\psi}g$ is
well-defined and using Fubini’s theorem, Itô’s isometry, and Plancherel’s
theorem again, we get (5.8). The lemma is proved. $\Box$

Due to Lemma 5.6, to prove (5.6) it suffices to show that Assumption 2.4 holds
with

$K(r,t,z,x)=1_{0<r<t}(-\Delta)^{\gamma/4}p(r,t,x-z).$

For $0<s<t$ and $x\in\mathbf{R}^{d}$, denote

$\displaystyle
q_{1}(s,t,x)=\mathcal{F}^{-1}\left[\exp\left(\int_{s}^{t}\psi(r,(t-s)^{-1/\gamma}\xi)dr\right)\right](x),$

and

$\displaystyle q_{2}(s,t,x)$
$\displaystyle=(t-s)\mathcal{F}^{-1}\left[\psi(t,(t-s)^{-1/\gamma}\xi)|\xi|^{\gamma/2}\exp\left(\int_{s}^{t}\psi(r,(t-s)^{-1/\gamma}\xi)dr\right)\right](x).$

By the change of variables,

$(t-s)^{d/\gamma}p(s,t,(t-s)^{1/\gamma}x)=q_{1}(s,t,x),$
$\displaystyle(t-s)^{d/\gamma}(t-s)^{1/2}(-\Delta)^{\gamma/4}p(s,t,(t-s)^{1/\gamma}x)=(-\Delta)^{\gamma/4}q_{1}(s,t,x),$
(5.9)

and

$\displaystyle\frac{\partial}{\partial
t}(-\Delta)^{\gamma/4}p(s,t,x)=(t-s)^{-d/\gamma}(t-s)^{-1}q_{2}(s,t,(t-s)^{-1/\gamma}x).$
(5.10)

######  Lemma 5.7.

There exists a constant $N=N(d,\nu,\gamma)$ so that for any multi-index
$\alpha$ with $|\alpha|\leq d_{0}$, $0<s<t$, and $i=1,\ldots,d$,

$\displaystyle\int_{\mathbf{R}^{d}}\left|D^{\alpha}_{\xi}\left(|\xi|^{\gamma/2}\mathcal{F}(q_{1}(t,s,\cdot)(\xi)\right)\right|d\xi+\int_{\mathbf{R}^{d}}\left|D^{\alpha}_{\xi}\left(\xi^{i}|\xi|^{\gamma/2}\mathcal{F}(q_{1}(t,s,\cdot)(\xi)\right)\right|d\xi$
$\displaystyle\quad+\int_{\mathbf{R}^{d}}\left|D^{\alpha}_{\xi}\left(\mathcal{F}(q_{2}(t,s,\cdot)(\xi)\right)\right|d\xi\leq
N.$

###### Proof.

Because of the similarity, we only show

$\int_{\mathbf{R}^{d}}\left|D^{\alpha}_{\xi}\left(|\xi|^{\gamma/2}\mathcal{F}(q_{1}(t,s,\cdot)(\xi)\right)\right|d\xi\leq
N.$

This is an easy conesequence of (5.2) and (5.3). Indeed,

$\displaystyle\int_{\mathbf{R}^{d}}\left|D^{\alpha}_{\xi}\left(|\xi|^{\gamma/2}\mathcal{F}(q_{1}t,s,\cdot)(\xi)\right)\right|d\xi$
$\displaystyle\leq\sum_{n\in\mathbb{Z}}\int_{2^{n}\leq|\xi|<2^{n+1}}\left|D^{\alpha}_{\xi}\left(|\xi|^{\gamma/2}\mathcal{F}(q_{1}t,s,\cdot)(\xi)\right)\right|d\xi$
$\displaystyle=\sum_{n\in\mathbb{Z}}\int_{2^{n}\leq|\xi|<2^{n+1}}\left|D^{\alpha}_{\xi}\left(|\xi|^{\gamma/2}\exp\left(\int_{s}^{t}\psi(r,(t-s)^{-1/\gamma}\xi)dr\right)\right)\right|d\xi$
$\displaystyle\leq
N\sum_{n\in\mathbb{Z}}\sum_{k=1}^{|\alpha|}2^{n(d+\frac{3\gamma}{2}-k)}e^{-\nu
2^{n}}\leq N(d,\nu,\gamma).$

The lemma is proved. $\Box$

Note that for any $f\in L_{1}(\mathbf{R}^{d})$,

$\displaystyle\sup_{x\in\mathbf{R}^{d}}\left|\mathcal{F}^{-1}(f)(x)\right|\leq
N(d)\|f\|_{L_{1}(\mathbf{R}^{d})}.$

Thus by Lemma 5.7, there exists a constant $N=N(d,\nu,\gamma)$ so that for any
$t>s$ and $x\in\mathbf{R}^{d}$

$\displaystyle\left|(-\Delta)^{\gamma/4}q_{1}(s,t,x)\right|+\left|\frac{\partial}{\partial
x^{i}}(-\Delta)^{\gamma/4}q_{1}(s,t,x)\right|+|q_{2}(s,t,x)|\leq N.$ (5.11)

######  Lemma 5.8.

Let $\varepsilon\in\left[0,\frac{d+3\gamma-2(d_{0}-1)}{2}\right)$. Then, there
exists a constant $N=N(d,\nu,\gamma,\varepsilon)$ so that for any multi-index
$\alpha$ with $|\alpha|\leq d_{0}-1$, $0<s<t$, and $i=1,\ldots,d$,

$\displaystyle\int_{\mathbf{R}^{d}}\left||\xi|^{-\varepsilon}D^{\alpha}_{\xi}\left(|\xi|^{\gamma/2}\mathcal{F}(q_{1}(t,s,\cdot)(\xi)\right)\right|^{2}d\xi+\int_{\mathbf{R}^{d}}\left||\xi|^{-\varepsilon}D^{\alpha}_{\xi}\left(\mathcal{F}(q_{2}(t,s,\cdot)(\xi)\right)\right|^{2}d\xi$
$\displaystyle\quad+\int_{\mathbf{R}^{d}}\left||\xi|^{-\varepsilon}D^{\alpha}_{\xi}\left(\xi^{i}|\xi|^{\gamma/2}\mathcal{F}(q_{1}(t,s,\cdot)(\xi)\right)\right|^{2}d\xi\leq
N.$

###### Proof.

Because of the similarity, we only show

$\int_{\mathbf{R}^{d}}\left||\xi|^{-\varepsilon}D^{\alpha}_{\xi}\left(|\xi|^{\gamma/2}\mathcal{F}(q_{1}(t,s,\cdot)(\xi)\right)\right|^{2}d\xi\leq
N.$

Since $d+3\gamma-2\varepsilon-2(d_{0}-1)>0$,

$\displaystyle\sum_{n-\infty}^{1}2^{n(d+3\gamma-2\varepsilon-2(d_{0}-1))}<\infty.$

Therefore by (5.2) and (5.3),

$\displaystyle\int_{\mathbf{R}^{d}}\left||\xi|^{-\varepsilon}D^{\alpha}_{\xi}\left(|\xi|^{\gamma/2}\mathcal{F}(q_{1}t,s,\cdot)(\xi)\right)\right|^{2}d\xi$
$\displaystyle\leq\sum_{n\in\mathbb{Z}}\int_{2^{n}\leq|\xi|<2^{n+1}}\left||\xi|^{-\varepsilon}D^{\alpha}_{\xi}\left(|\xi|^{\gamma/2}\mathcal{F}(q_{1}t,s,\cdot)(\xi)\right)\right|^{2}d\xi$
$\displaystyle=\sum_{n\in\mathbb{Z}}\int_{2^{n}\leq|\xi|<2^{n+1}}\left||\xi|^{-\varepsilon}D^{\alpha}_{\xi}\left(|\xi|^{\gamma/2}\exp\left(\int_{s}^{t}\psi(r,(t-s)^{-1/\gamma}\xi)dr\right)\right)\right|^{2}d\xi$
$\displaystyle\leq
N\sum_{n\in\mathbb{Z}}\sum_{k=1}^{|\alpha|}2^{n(d+3\gamma-2\varepsilon-2k)}e^{-\nu
2^{n}}\leq N(d,\nu,\gamma,\varepsilon).$

The lemma is proved. $\Box$

######  Lemma 5.9.

There exists a constant $N=N(d,\nu,\gamma)$ so that for all $c>0$, multi-index
$|\alpha|\leq d_{0}$, $0<s<t$, and $i=1,\ldots,d$,

$\displaystyle\int_{|\xi|\geq
c}\left|D^{\alpha}_{\xi}\left(|\xi|^{\gamma/2}\mathcal{F}(q_{1}(t,s,\cdot)(\xi)\right)\right|^{2}d\xi+\int_{|\xi|\geq
c}\left|D^{\alpha}_{\xi}\left(\xi^{i}|\xi|^{\gamma/2}\mathcal{F}(q_{1}(t,s,\cdot)(\xi)\right)\right|^{2}d\xi$
$\displaystyle\quad+\int_{|\xi|\geq
c}\left|D^{\alpha}_{\xi}\left(\mathcal{F}(q_{2}(t,s,\cdot)(\xi)\right)\right|^{2}d\xi\leq
N\left(1+1_{c<1}c^{d+3\gamma-2d_{0}}\right).$

###### Proof.

As in the proofs of the previous lemmas, we only show

$\int_{|\xi|\geq
c}\left|D^{\alpha}_{\xi}\left(|\xi|^{\gamma/2}\mathcal{F}(q_{1}(t,s,\cdot)(\xi)\right)\right|^{2}d\xi\leq
N.$

By (5.2) and (5.3),

$\displaystyle\int_{|\xi|\geq
c}\left|D^{\alpha}_{\xi}\left(|\xi|^{\gamma/2}\mathcal{F}(q_{1}t,s,\cdot)(\xi)\right)\right|^{2}d\xi$
$\displaystyle\leq\sum_{n\in\mathbb{Z}}\int_{2^{n}\leq|\xi|<2^{n+1}}1_{|\xi|\geq
c}\left|D^{\alpha}_{\xi}\left(|\xi|^{\gamma/2}\mathcal{F}(q_{1}t,s,\cdot)(\xi)\right)\right|^{2}d\xi$
$\displaystyle=\sum_{2^{n}\geq c/2}\int_{2^{n}\leq|\xi|<2^{n+1}}1_{|\xi|\geq
c}\left|D^{\alpha}_{\xi}\left(|\xi|^{\gamma/2}\exp\left(\int_{s}^{t}\psi(r,(t-s)^{-1/\gamma}\xi)dr\right)\right)\right|^{2}d\xi$
$\displaystyle\leq N\sum_{2^{n}\geq
c/2}\sum_{k=1}^{|\alpha|}2^{n(d+3\gamma-2k)}e^{-\nu 2^{n}}\leq
N(d,\nu,\gamma)\left(1+1_{c<1}c^{d+3\gamma-2d_{0}}\right).$

The lemma is proved. $\Box$

######  Lemma 5.10.

Let $0<\delta<\left(\frac{\gamma}{2}\wedge\frac{1}{2}\right)$. Then there
exists a constant $N=N(d,\nu,\gamma,\delta)$ so that for any $0<s<t$

$\displaystyle\int_{\mathbf{R}^{d}}\left||x|^{\frac{d}{2}+\delta}(-\Delta)^{\gamma/4}q_{1}(s,t,x)\right|^{2}~{}dx\leq
N,$ (5.12)
$\displaystyle\int_{\mathbf{R}^{d}}\left||x|^{\frac{d}{2}+\delta}\frac{\partial}{\partial
x^{i}}(-\Delta)^{\gamma/4}q_{1}(s,t,x)\right|^{2}~{}dx\leq N,$ (5.13)

and

$\displaystyle\int_{\mathbf{R}^{d}}\left||x|^{\frac{d}{2}+\delta}q_{2}(s,t,x)\right|^{2}~{}dx\leq
N.$ (5.14)

###### Proof.

We only prove (5.12). The proofs of (5.13) and (5.14) are similar.

Note that it suffices to show that for each $j=1,\ldots,d$,

$\displaystyle\int_{\mathbf{R}^{d}}\left|(ix^{j})^{\frac{d}{2}+\delta}(-\Delta)^{\gamma/4}q_{1}(s,t,x)\right|^{2}~{}dx\leq
N,$ (5.15)

where $i$ is the imaginary number, i.e. $i^{2}=-1$. Set
$\varepsilon=\varepsilon(\delta)=\frac{d}{2}+\delta-(d_{0}-1)$ and

$\hat{q}(s,t,\xi)=\mathcal{F}\left((-\Delta)^{\gamma/4}q_{1}(s,t,x)\right)(\xi).$

By the property of the Fourier inverse transform,

$(ix^{j})^{d_{0}-1}\mathcal{F}^{-1}\left(f(\xi)\right)(x)=(-1)^{d_{0}-1}\mathcal{F}^{-1}\left(D^{d_{0}-1}_{\xi^{j}}f(\xi)\right)(x).$

The left hand side of (5.15) is equal to

$\displaystyle\int_{\mathbf{R}^{d}}\left|(ix^{j})^{\frac{d}{2}+\delta-(d_{0}-1)}\mathcal{F}^{-1}\left(D_{\xi^{j}}^{d_{0}-1}\hat{q}(s,t,\xi)\right)(x)\right|^{2}~{}dx$
$\displaystyle\leq\int_{\mathbf{R}^{d}}\left||x|^{\frac{d}{2}+\delta-(d_{0}-1)}\mathcal{F}^{-1}\left(D_{\xi^{j}}^{d_{0}-1}\hat{q}(s,t,\xi)\right)(x)\right|^{2}~{}dx.$
(5.16)

Moreover by Plancherel’s theorem, the last term above equals to

$\displaystyle
N(d)\int_{\mathbf{R}^{d}}\left|(-\Delta)^{\varepsilon/2}\left(D_{\xi^{j}}^{d_{0}-1}\hat{q}(s,t,\xi)\right)\right|^{2}~{}d\xi.$
(5.17)

Obviously,
$\varepsilon\in\left(0,1\wedge\frac{d+\gamma-2(d_{0}-1)}{2}\right)$. Using the
integral representation of the Fractional Laplacian operator
$(-\Delta)^{\varepsilon/2}$ we get

$\displaystyle(-\Delta)^{\varepsilon/2}(D_{\xi^{j}}^{d_{0}-1}\hat{q}(s,t,\xi))=N\int_{\mathbf{R}^{d}}\frac{D_{\xi^{j}}^{d_{0}-1}\hat{q}(s,t,\xi+\eta)-D_{\xi^{j}}^{d_{0}-1}\hat{q}(s,t,\xi)}{|\eta|^{d+\varepsilon}}d\eta.$

We divide $(-\Delta)^{\varepsilon/2}(D_{\xi^{j}}^{d_{0}-1}\hat{q}(s,t,\xi))$
into two terms:

$\displaystyle N\int_{|\eta|\geq
1}\frac{D_{\xi^{j}}^{d_{0}-1}\hat{q}(s,t,\xi+\eta)-D_{\xi^{j}}^{d_{0}-1}\hat{q}(s,t,\xi)}{|\eta|^{d+\varepsilon}}d\eta$
$\displaystyle\quad+N\int_{|\eta|<1}\frac{D_{\xi^{j}}^{d_{0}-1}\hat{q}(s,t,\xi+\eta)-D_{\xi^{j}}^{d_{0}-1}\hat{q}(s,t,\xi)}{|\eta|^{d+\varepsilon}}d\eta=:\mathcal{I}_{1}(s,t,\xi)+\mathcal{I}_{2}(s,t,\xi).$

By Minkowski’s inequality and Lemma 5.8,

$\displaystyle\left[\int_{\mathbf{R}^{d}}\left|\mathcal{I}_{1}(s,t,\xi)\right|^{2}d\xi\right]^{1/2}$
$\displaystyle\leq
2\left\|D_{\xi^{j}}^{d_{0}-1}\hat{q}(s,t,\cdot)\right\|_{L_{2}(\mathbf{R}^{d})}\int_{|\eta|\geq
1}\frac{1}{|\eta|^{d+\varepsilon}}d\eta$ $\displaystyle\leq N(d,\nu,\gamma).$

We split $\mathcal{I}_{2}$ into $\mathcal{I}_{2,1}$, $\mathcal{I}_{2,2}$, and
$\mathcal{I}_{2,3}$, where

$\displaystyle\mathcal{I}_{2,1}(s,t,\xi):=\int_{|\eta|<1}1_{|\eta|<\frac{|\xi|}{2}}\frac{D_{\xi^{j}}^{d_{0}-1}\hat{q}(s,t,\xi+\eta)-D_{\xi^{j}}^{d_{0}-1}\hat{q}(s,t,\xi)}{|\eta|^{d+\varepsilon}}d\eta$
$\displaystyle\mathcal{I}_{2,2}(s,t,\xi):=\int_{|\eta|<1}1_{|\eta|\geq\frac{|\xi|}{2}}\frac{D_{\xi^{j}}^{d_{0}-1}\hat{q}(s,t,\xi+\eta)}{|\eta|^{d+\varepsilon}}d\eta,$

and

$\displaystyle\mathcal{I}_{2,3}(s,t,\xi):=-\int_{|\eta|<1}1_{|\eta|\geq\frac{|\xi|}{2}}\frac{D_{\xi^{j}}^{d_{0}-1}\hat{q}(s,t,\xi)}{|\eta|^{d+\varepsilon}}d\eta.$

By the fundamental theorem of calculus,

$\displaystyle|\mathcal{I}_{2,1}(s,t,\xi)|\leq\int_{0}^{1}\int_{|\eta|<1}1_{|\eta|<\frac{|\xi|}{2}}\frac{\left|\nabla
D_{\xi^{j}}^{d_{0}-1}\hat{q}(s,t,\xi+\theta\eta)\right|}{|\eta|^{d+\varepsilon-1}}d\eta
d\theta.$

Hence by Minkowski’s inequality and Lemma 5.9,

$\displaystyle\|\mathcal{I}_{2,1}(s,t,\cdot)\|^{2}_{L_{2}(\mathbf{R}^{d})}$
$\displaystyle\leq\left[\int_{|\eta|<1}\left(\int_{|\eta|<|\xi|}\left|\nabla
D_{\xi^{j}}^{d_{0}-1}\hat{q}(s,t,\xi)\right|^{2}d\xi\right)^{1/2}\frac{1}{|\eta|^{d+\varepsilon-1}}d\eta\right]^{2}$
$\displaystyle\leq
N\left[\int_{|\eta|<1}\frac{1+|\eta|^{(d+3\gamma-2d_{0})/2}}{|\eta|^{d+\varepsilon-1}}d\eta\right]^{2}\leq
N(d,\nu,\gamma)$

since

$(d+3\gamma-2d_{0})/2-d-\varepsilon+1>-d.$

On the other hand, if $|\xi|\geq 2$, then
$\mathcal{I}_{2,2}(s,t,\xi)=\mathcal{I}_{2,3}(s,t,\xi)=0$ and thus we may
assume $|\xi|\leq 2$. Recalling the range of $\varepsilon$, we have

$\varepsilon+\gamma<\frac{d+3\gamma-2(d_{0}-1)}{2}.$

Hence by Hölder’s inequality and Lemma 5.8,

$\displaystyle|\mathcal{I}_{2,2}(s,t,\xi)|$
$\displaystyle\leq\left[\int_{|\eta|<1}1_{|\eta|\geq\frac{|\xi|}{2}}\frac{|\xi+\eta|^{2\varepsilon+2\gamma}}{|\eta|^{2d+2\varepsilon}}d\eta\right]^{1/2}\left[\int_{\mathbf{R}^{d}}\left||\xi+\eta|^{-\varepsilon-\gamma}D_{\xi^{j}}^{d_{0}-1}\hat{q}(s,t,\xi+\eta)\right|^{2}d\eta\right]^{1/2}$
$\displaystyle\leq
N\left[\int_{|\eta|<1}1_{|\eta|\geq\frac{|\xi|}{2}}|\eta|^{-2d+2\gamma}d\eta\right]^{1/2}\left[\int_{\mathbf{R}^{d}}\left||\eta|^{-\varepsilon-\gamma}D_{\xi^{j}}^{d_{0}-1}\hat{q}(s,t,\eta)\right|^{2}d\eta\right]^{1/2}$
$\displaystyle\leq N\left(1+|\xi|^{-\frac{d}{2}+\gamma}\right).$

Therefore we have

$\displaystyle\|\mathcal{I}_{2,2}(s,t,\cdot)\|^{2}_{L_{2}(\mathbf{R}^{d})}\leq
N\int_{|\xi|<2}\left(1+|\xi|^{-d+2\gamma}\right)d\xi\leq N(d,\nu,\gamma).$

Finally by Lemma 5.8 again,

$\displaystyle\|\mathcal{I}_{2,3}(s,t,\cdot)\|^{2}_{L_{2}(\mathbf{R}^{d})}\leq
N.$

Due to (5.16) and (5.17), combining all estimates for
$\mathcal{I}_{1},\mathcal{I}_{2,1},\mathcal{I}_{2,2},\mathcal{I}_{2,3}$, we
have (5.15). The lemma is proved. $\Box$

######  Remark 5.11.

If $\gamma$ is not small, Lemma 5.10 is easily obtained from properties of the
Fourier tansform. Indeed,

$\displaystyle\int_{\mathbf{R}^{d}}\left||x|^{d_{0}}\mathcal{F}^{-1}\left(D_{\xi^{j}}^{d_{0}-1}\hat{q}(s,t,\xi)\right)(x)\right|^{2}dx$
$\displaystyle\leq
N\sum_{j=1}^{d}\int_{\mathbf{R}^{d}}\left|(ix^{j})^{d_{0}}\mathcal{F}^{-1}\left(\hat{q}(s,t,\xi)\right)(x)\right|^{2}~{}dx$
$\displaystyle=N\sum_{j=1}^{d}\int_{\mathbf{R}^{d}}\left|D_{\xi^{j}}^{d_{0}}\hat{q}(s,t,\xi)\right|^{2}~{}d\xi.$

Due to (5.2) and (5.3), the above term is finite if $3\gamma+d>2d_{0}$.

######  Lemma 5.12.

Let $\delta\in\left(0,\frac{1}{2}\wedge\frac{\gamma}{2}\right)$. Then there
exists a constant $N(d,\nu,\gamma,\delta)$ such that for all $0<s<t$, $c>0$,
$a\in\mathbf{R}$,

$\displaystyle\int_{s}^{t}\left|\int_{|z|\geq
c}|(-\Delta)^{\gamma/4}p(r,t,z)|~{}dz\right|^{2}dr\leq
N\left((t-s)^{1/\gamma}c^{-1}\right)^{2\delta},$ (5.18)
$\displaystyle\int_{0}^{a}\left[\int_{\mathbf{R}^{d}}\big{|}(-\Delta)^{\gamma/4}p(r,t,z+h)-(-\Delta)^{\gamma/4}p(r,t,z)\big{|}~{}dz\right]^{2}dr\leq
N\left(|h|(t-a)^{-1/\gamma}\right)^{2},$ (5.19)

and

$\displaystyle\int_{0}^{a}\left[\int_{\mathbf{R}^{d}}|(-\Delta)^{\gamma/4}p(r,t,z)-(-\Delta)^{\gamma/4}p(r,s,z)|~{}dz\right]^{2}dr\leq
N\left((t-s)(s-a)^{-1}\right)^{2}.$ (5.20)

###### Proof.

First we prove (5.18). By (5.9), Hölder’s inequality, and (5.12),

$\displaystyle\left|\int_{|z|\geq
c}|(-\Delta)^{\gamma/4}p(r,t,z)|~{}dz\right|^{2}$
$\displaystyle=(t-r)^{-1}\left|\int_{(t-r)^{1/\gamma}|z|\geq
c}|(-\Delta)^{\gamma/4}q_{1}(r,t,z)|~{}dz\right|^{2}$
$\displaystyle\leq(t-r)^{-1}\int_{(t-r)^{1/\gamma}|z|\geq
c}|z|^{-d-2\delta}~{}dz\int_{(t-r)^{1/\gamma}|z|\geq
c}\left||z|^{\frac{d}{2}+\delta}(-\Delta)^{\gamma/4}q_{1}(r,t,z)\right|^{2}~{}dz$
$\displaystyle\leq(t-r)^{-1+(2\delta)/\gamma}c^{-2\delta}.$

Hence we have

$\displaystyle\int_{s}^{t}\left|\int_{|z|\geq
c}(-\Delta)^{\gamma/4}p(r,t,z)~{}dz\right|^{2}dr$ $\displaystyle\leq
N\left((t-s)^{1/\gamma}c^{-1}\right)^{2\delta}.$

Next we prove (5.19). From (5.9),

$\displaystyle\frac{\partial}{\partial
x^{i}}(-\Delta)^{\gamma/4}p(r,t,z)=(t-r)^{-d/\gamma}(t-r)^{-1/2-1/\gamma}\frac{\partial}{\partial
x^{i}}(-\Delta)^{\gamma/4}q_{1}(r,t,(t-r)^{-1/\gamma}z),$

and by Hölder’s inequality, (5.11), and (5.13),

$\displaystyle\left[\int_{\mathbf{R}^{d}}\Big{|}\frac{\partial}{\partial
x^{i}}\Delta^{\gamma/2}q_{1}(r,t,z)\Big{|}dz\right]^{2}$ $\displaystyle\leq
N+\int_{|z|\geq
1}|z|^{-d-2\delta}dz\int_{\mathbf{R}^{d}}\left||z|^{\frac{d}{2}+\delta}\frac{\partial}{\partial
x^{i}}\Delta^{\gamma/2}q_{1}(r,t,z)\right|^{2}dz\leq N,$

where $N$ is independent of $t$ and $r$. Therefore, by the fundamental theorem
of calculus,

$\displaystyle\int_{0}^{a}\left[\int_{\mathbf{R}^{d}}\big{|}(-\Delta)^{\gamma/4}p(r,t,z+h)-(-\Delta)^{\gamma/4}p(r,t,z)\big{|}~{}dz\right]^{2}dr$
$\displaystyle\leq|h|^{2}\int_{0}^{a}\left[\int_{\mathbf{R}^{d}}|\nabla\Delta^{\gamma/2}p(r,t,z)|~{}dz\right]^{2}dr$
$\displaystyle\leq|h|^{2}\int_{0}^{a}(t-r)^{-1-2/\gamma}\left(\int_{\mathbf{R}^{d}}\Big{|}\frac{\partial}{\partial
x^{i}}(-\Delta)^{\gamma/4}q_{1}(r,t,z)\Big{|}~{}dz\right)^{2}dr$
$\displaystyle\leq N|h|^{2}\int_{0}^{a}(t-r)^{-1-2/\gamma}dr\leq
N|h|\int_{t-a}^{\infty}r^{-1-2/\gamma}dr=N\left(|h|(t-a)^{-1/\gamma}\right)^{2}.$

It only remains to prove (5.20). By the mean-value theorem and (5.10),

$\displaystyle|(-\Delta)^{\gamma/4}p(r,t,z)-(-\Delta)^{\gamma/4}p(r,s,z)|$
$\displaystyle\leq|t-s|(\theta
t+(1-\theta)s-r)^{-d/\gamma-3/2}\left|q_{2}(r,\theta t+(1-\theta)s,(\theta
t+(1-\theta)s-r)^{-1/\gamma}z)\right|,$

where $\theta\in[0,1]$. Moreover by Hölder’s inequality, (5.11), and (5.14),

$\int_{\mathbf{R}^{d}}|q_{2}(r,\theta t+(1-\theta)s,z)|~{}dz<N,$

where $N$ is independent of $t$, $s$, $r$, and $\theta$. Therefore,

$\displaystyle\int_{0}^{a}\left[\int_{\mathbf{R}^{d}}\left|(-\Delta)^{\gamma/4}p(r,t,z)-(-\Delta)^{\gamma/4}p(r,s,z)\right|~{}dz\right]^{2}dr$
$\displaystyle\leq\int_{0}^{a}\frac{|t-s|^{2}}{\big{(}\theta
t+(1-\theta)s-r\big{)}^{3}}dr\leq|t-s|^{2}(s-a)^{-2}.$

The lemma is proved. $\Box$

In the following corollary, we finally prove that the kerenel

$K(r,t,z,x):=1_{0<r<t}(-\Delta)^{\gamma/4}p(r,t,x-z)$

satisfies Assumption 2.4. Recall

$\rho(X,Y)=|t-s|^{1/\gamma}+|x-y|.$

For $r>0$ and $X=(t,x),Y=(s,y)\in(0,\infty)\times\mathbf{R}^{d}$, set

$\displaystyle A(r,X,Y)$
$\displaystyle:=\left\\{z\in\mathbf{R}^{d}:\rho(X,Z)\geq 4\cdot
2^{1/\gamma}\rho(X,Y)\right\\}$
$\displaystyle=\left\\{z\in\mathbf{R}^{d}:|t-r|^{1/\gamma}+|x-z|\geq 4\cdot
2^{1/\gamma}(|t-s|^{1/\gamma}+|x-y|)\right\\},$

where $Z=(r,z)$.

######  Corollary 5.13.

There is a constant $N=N(d,\nu,\gamma)$ so that for any
$X=(t,x),Y=(s,y)\in(0,\infty)\times\mathbf{R}^{d}$ and $r>0$,

$\displaystyle\int_{0}^{\infty}\left[\int_{A(r,X,Y)}\left|1_{0<r<t}(-\Delta)^{\gamma/4}p(r,t,x-z)-1_{0<r<s}(-\Delta)^{\gamma/4}p(r,s,y-z)\right|~{}dz\right]^{2}dr$
$\displaystyle\leq N.$

###### Proof.

We use the notation $(-\Delta)^{\gamma/4}p(r,t,x)$ instead of
$1_{0<r<t}(-\Delta)^{\gamma/4}p(r,t,x)$. In other words, we assume that
$(-\Delta)^{\gamma/4}p(r,t,x)=0$ unless $0<r<t$. Moreover we may assume $t\geq
s$ without loss of generality. Since the proof of the case $t=s$ is simpler,
we only prove the case $t>s$.

Fix a constant $\delta\in\left(0,\frac{1}{2}\wedge\frac{\gamma}{2}\right)$.
Denote

$\displaystyle\mathcal{I}(r,X,Y)$
$\displaystyle=\left[\int_{A(r,X,Y)}\left|(-\Delta)^{\gamma/4}p(r,t,x-z)-(-\Delta)^{\gamma/4}p(r,s,y-z)\right|~{}dz\right]^{2}.$

Obviously $\mathcal{I}(r,X,Y)=0$ if $r\geq t$. Thus

$\displaystyle\int_{0}^{\infty}\mathcal{I}(r,X,Y)dr$
$\displaystyle=\int_{2s-t}^{t}\mathcal{I}(r,X,Y)dr+\int_{0}^{2s-t}\mathcal{I}(r,X,Y)dr$
$\displaystyle=:\mathcal{I}_{1}(X,Y)+\mathcal{I}_{2}(X,Y).$

First we estimate $\mathcal{I}_{1}(X,Y)$. By (5.18),

$\displaystyle\mathcal{I}_{1}(X,Y)$
$\displaystyle\leq\int_{2s-t}^{t}\left[\int_{|z|\geq|t-s|^{1/\gamma}}\left|(-\Delta)^{\gamma/4}p(r,t,z)\right|~{}dz\right]^{2}dr$
$\displaystyle\quad+\int_{2s-t}^{s}\left[\int_{|z|\geq|t-s|^{1/\gamma}}\left|(-\Delta)^{\gamma/4}p(r,s,z)\right|~{}dz\right]^{2}dr\leq
N.$

We split $\mathcal{I}_{2}$. Observe

$\displaystyle\mathcal{I}_{2}\leq\mathcal{I}_{2,1}+\mathcal{I}_{2,2}$
$\displaystyle:=\int_{0}^{2s-t}\left[\int_{A(r,X,Y)}\left|(-\Delta)^{\gamma/4}p(r,t,x-z)-(-\Delta)^{\gamma/4}p(r,t,y-z)\right|~{}dz\right]^{2}dr$
$\displaystyle+\int_{0}^{2s-t}\left[\int_{A(r,X,Y)}\left|(-\Delta)^{\gamma/4}p(r,t,y-z)-(-\Delta)^{\gamma/4}p(r,s,y-z)\right|~{}dz\right]^{2}dr.$

If $|x-y|\leq(t-s)^{1/\gamma}$ then by (5.19),

$\displaystyle\mathcal{I}_{2,1}\leq
N\left(|x-y|(t-s)^{-1/\gamma}\right)^{2}\leq N.$

On the other hand, if $|x-y|>(t-s)^{1/\gamma}$, then

$\displaystyle\mathcal{I}_{2,1}\leq 2\mathcal{I}_{2,1,1}+\mathcal{I}_{2,1,2},$

where

$\displaystyle\mathcal{I}_{2,1,1}:=\int_{s-|x-y|^{\gamma}}^{t}\left[\int_{|z|\geq|t-s|^{1/\gamma}+|x-y|}\left|(-\Delta)^{\gamma/4}p(r,t,z)\right|~{}dz\right]^{2}dr,$

and

$\displaystyle\mathcal{I}_{2,1,2}$
$\displaystyle:=\int_{0}^{s-|x-y|^{\gamma}}\left[\int_{\mathbf{R}^{d}}\left|(-\Delta)^{\gamma/4}p(r,t,x-z)-(-\Delta)^{\gamma/4}p(r,t,y-z)\right|~{}dz\right]^{2}dr.$

By (5.18) again,

$\mathcal{I}_{2,1,1}\leq
N\left(\left(t-s+|x-y|^{\gamma}\right)^{-1/\gamma}\left((t-s)^{1/\gamma}+|x-y|\right)\right)^{2\delta}\leq
N$

and by (5.19)

$\displaystyle\mathcal{I}_{2,1,2}\left(|x-y|\left(t-s+|x-y|^{\gamma}\right)^{-1/\gamma}\right)^{2}\leq
N.$

It only remains to estimate $\mathcal{I}_{2,2}$. However, this is an easy
consequence of (5.20) since $2s-t<t$. Indeed,

$\displaystyle\mathcal{I}_{2,2}\leq N\left((t-s)|t-s|^{-1}\right)^{2}\leq N.$

The corollary is proved. $\Box$

Finally, applying Theorem 2.5 with

$\mathbb{T}_{\varepsilon}=\mathbb{T}_{\psi,\varepsilon}\quad\text{and}\quad\mathbb{T}=\mathbb{T}_{\psi},$

we obtain (5.6).

###  5.2. Infinitesimal generators of subordinate Brownian motions

In this subsection we consider the infinitesimal generators of subordinate
Brownian motions. In general the symbols of such operators do not satisfy
(5.2) which is assumed in the previous subsection.

Let $S_{t}$ be a subordinator, that is, an increasing Lévy process taking
values in $[0,\infty)$ with $S_{0}=0$. A subordinator $S$ is completely
characterized by its Laplace exponent $\phi$, i.e. $\mathbb{E}e^{-\lambda
S_{t}}=e^{-t\phi(\lambda)}$ for $\lambda>0$. Actually a function function
$\phi:(0,\infty)\to(0,\infty)$ with $\phi(0+)=0$ is a Laplace exponent of a
subordinator if and only if it is a Bernstein function (i.e.
$(-1)^{n}D^{n}\phi\leq 0,\forall n$). Also it is of the form

$\phi(\lambda)=b\lambda+\int_{(0,\infty)}(1-e^{-\lambda
t})\,\mu(dt)\,,\quad\lambda>0\,,$

where $b\geq 0$ and $\mu$ is a measure on $(0,\infty)$ satisfying
$\int_{(0,\infty)}(1\wedge t)\,\mu(dt)<\infty$, called the Lévy measure. Let
$B_{t}$ be a $d$-dimensional Brownian motion independent of $S_{t}$. Then
$\phi(\Delta)$ can be defined as the infinitesimal generator of the
subordinate Brownian motion $B_{S_{t}}$:

$\phi(\Delta)f(x)=\lim_{t\to 0}\frac{\mathbb{E}f(x+B_{S_{t}})-f(x)}{t},\quad
f\in C^{2}_{b}(\mathbf{R}^{d}),$

and its integral version is

$b\Delta f(x)+\int_{{\mathbb{R}}^{d}}\left(f(x+y)-f(x)-\nabla f(x)\cdot
y{\mathbf{1}}_{\\{|y|\leq 1\\}}\right)\,J(y)\,dy\,,$ (5.21)

where $J(x)=j(|x|)$ with $j:(0,\infty)\to(0,\infty)$ given by

$j(r)=\int_{0}^{\infty}(4\pi t)^{-d/2}e^{-r^{2}/(4t)}\,\mu(dt)\,.$

See e.g. [10] for more details. In general for $n=0,1,2,\cdots$, we define
$\phi(\Delta)^{n/2}$ on the Schwartz space $\mathcal{S}$ as the operator with
symbol $-\phi(|\xi|^{2})$, i.e.

$\phi(\Delta)^{n/2}f(x):=-\phi(-\Delta)^{n/2}f(x):=\mathcal{F}^{-1}\left[-\phi(|\xi|^{2})^{n/2}\mathcal{F}f(\xi)\right](x).$

Consider the operator $A(t)=\phi(\Delta)$. Then (5.1) has a solution $u$ given
by

$u(t,x)=\int_{0}^{t}\int_{\mathbf{R}^{d}}p(t-s,x-y)g^{k}(s,y)dydW_{s}^{k},$

where

$p(t,x)=\mathcal{F}^{-1}\left[\exp\left(-t\phi(|\xi|^{2})\right)\right](x).$

Let $\phi^{-1}$ denote the generalized inverse of $\phi$, i.e.

$\phi^{-1}(t):=\inf\\{s>0:\phi(s)\geq t\\}.$

######  Assumption 5.14.

(i) There exists a constant $N$ such that for all $t\leq T\in(0,\infty]$ and
$x\in\mathbf{R}^{d}$

$\displaystyle\left|\phi(\Delta)^{1/2}p(t,\cdot)(x)\right|\leq
N\left(t^{-1/2}(\phi^{-1}(t^{-1}))^{d/2}\wedge\frac{\phi(|x|^{-2})^{1/2}}{|x|^{d}}\right),$
(5.22) $\displaystyle\left|\phi(\Delta)^{1/2}\nabla p(t,\cdot)(x)\right|\leq
N\left(t^{-1/2}(\phi^{-1}(t^{-1}))^{(d+1)/2}\wedge\frac{\phi(|x|^{-2})^{1/2}}{|x|^{d+1}}\right),$
(5.23)

and

$\displaystyle\left|\phi(\Delta)^{3/2}p(t,\cdot)(x)\right|\leq
N\left(t^{-3/2}(\phi^{-1}(t^{-1}))^{d/2}\wedge
t^{-1}\frac{\phi(|x|^{-2})^{1/2}}{|x|^{d}}\right).$ (5.24)

(ii) $\phi$ satisfies the following scaling property: there exist positive
constants $N_{1}$, $N_{2}$, $\delta_{1}$, and $\delta_{2}$ so that

$\displaystyle
N_{1}\left(\frac{b}{a}\right)^{\delta_{1}}\leq\frac{\phi(b)}{\phi(a)}\leq
N_{2}\left(\frac{b}{a}\right)^{\delta_{2}},\quad\quad\forall\,\,0<a\leq b.$
(5.25)

Using (5.25) one can find constants $\bar{\delta}_{1}$, $\bar{\delta}_{2}$,
$\bar{N}_{1}$, and $\bar{N}_{2}$ depending only on $\delta_{1}$, $\delta_{2}$,
$N_{1}$, and $N_{2}$ so that

$\displaystyle\bar{N}_{1}\left(\frac{b}{a}\right)^{\bar{\delta}_{1}}\leq\frac{\phi^{-1}(b)}{\phi^{-1}(a)}\leq\bar{N}_{2}\left(\frac{b}{a}\right)^{\bar{\delta}_{2}},\quad\quad
0<a\leq b.$ (5.26)

Furthermore,

$\lim_{t\uparrow\infty}\phi(t)=\lim_{t\uparrow\infty}\phi^{-1}(t)=\infty,\quad\quad\lim_{t\downarrow
0}\phi(t)=\lim_{t\downarrow 0}\phi^{-1}(t)=0,$

and

$\displaystyle\phi(\phi^{-1}(t))=t.$ (5.27)

######  Example 5.15.

A sufficient condition to (5.22)-(5.25) can be founded e.g. in [3]:

(H1): $\exists$ constants $0<\delta_{1}\leq\delta_{2}<1$ and $c_{1},c_{2}>0$
such that

$c_{1}\lambda^{\delta_{1}}\phi(t)\leq\phi(\lambda t)\leq
c_{2}\lambda^{\delta_{2}}\phi(t),\quad\lambda\geq 1,t\geq 1\,;$

(H2): $\exists$ constants $0<\delta_{3}\leq 1$ and $c_{3}>0$ such that

$\phi(\lambda t)\leq c_{3}\lambda^{\delta_{3}}\phi(t),\quad\lambda\leq 1,t\leq
1\,.$

Actually using (H1) and (H2) one can prove (see [3])

$\displaystyle\left|\phi(\Delta)^{n/2}D^{\beta}p(t,\cdot)(x)\right|\leq
N\left(t^{-n/2}(\phi^{-1}(t^{-1}))^{(d+|\beta|)/2}\wedge
t^{-(n-1)/2}\frac{\phi(|x|^{-2})^{1/2}}{|x|^{d+|\beta|}}\right)$

for any $n\leq 3$ and multi-index $\beta$ with $|\beta|\leq 2$.

Here are some examples of Bernstein functions satisfying (H1) and (H2):

  * (1)

$\phi(\lambda)=\lambda^{\alpha}+\lambda^{\beta}$, $0<\alpha<\beta<1$;

  * (2)

$\phi(\lambda)=(\lambda+\lambda^{\alpha})^{\beta}$, $\alpha,\beta\in(0,1)$;

  * (3)

$\phi(\lambda)=\lambda^{\alpha}(\log(1+\lambda))^{\beta}$, $\alpha\in(0,1)$,
$\beta\in(0,1-\alpha)$;

  * (4)

$\phi(\lambda)=\lambda^{\alpha}(\log(1+\lambda))^{-\beta}$, $\alpha\in(0,1)$,
$\beta\in(0,\alpha)$;

  * (5)

$\phi(\lambda)=(\log(\cosh(\sqrt{\lambda})))^{\alpha}$, $\alpha\in(0,1)$;

  * (6)

$\phi(\lambda)=(\log(\sinh(\sqrt{\lambda}))-\log\sqrt{\lambda})^{\alpha}$,
$\alpha\in(0,1)$.

For example, the subordinate Brownian motion corresponding to the example (1)
$\phi(\lambda)=\lambda^{\alpha}+\lambda^{\beta}$ is the sum of two independent
symmetric $\alpha$ and $\beta$ stable processes, and its infinitesimal
generator is $-(-\Delta)^{\beta/2}-(-\Delta)^{\alpha/2}$.

Define

$\displaystyle\mathbb{T}_{\phi,\varepsilon}g:=\int_{0}^{t-\varepsilon}\int_{\mathbf{R}^{d}}\phi(\Delta)^{1/2}p(t-r,x-y)g^{k}(r,y)dydW_{r}^{k}$

and

$\displaystyle\mathbb{T}_{\phi}g:=\lim_{\varepsilon\downarrow
0}\int_{0}^{t-\varepsilon}\int_{\mathbf{R}^{d}}\phi(\Delta)^{1/2}p(t-r,x-y)g^{k}(r,y)dydW_{r}^{k},$

where the limit is in the sense of $\mathbb{L}_{2}$-norm.

Here is the main result of this subsection.

######  Theorem 5.16.

Let $p\in[2,\infty)$ and Assumption 5.14 hold. Then

$\displaystyle\|\mathbb{T}_{\phi}g\|_{\mathbb{L}_{p}(\mathcal{O}_{T})}\leq
N\|g\|_{\mathbb{L}_{p}(\mathcal{O}_{T},l_{2})}\quad\forall
g\in\mathbb{L}_{p}(\mathcal{O}_{T},l_{2}),$

where $N$ depends only on $d$ and the constants appearing in Assumption 5.14.

To apply Theorem 2.5, we set $\mathcal{O}=\mathbf{R}^{d}$ and

$\rho(X,Y)=\left(\phi^{-1}\left(|t-s|^{-1}\right)\right)^{-1/2}+|x-y|,$

where $X=(t,x)$, $Y=(s,y)$, and
$\left(\phi^{-1}\left(0^{-1}\right)\right)^{-1/2}:=0$. Due to (5.26), one can
easily check that $\rho$ is a quasi-metric and satisfies the doubling ball
condition. Thus, we only need to check that

$K(t-r,z,x):=1_{0<r<t<T}\phi(\Delta)^{1/2}p(t-r,x-z)$

satisfies Assumption 2.4 since the proof of $L_{2}$-boundedness of
$\mathcal{T}_{\phi}g$ can be easily proved as the proof of Lemma 5.6.

######  Lemma 5.17.

There exists a constant $N$ such that for all $0<a<s<t<T$, $c>0$,

$\displaystyle\int_{s}^{t}\left[\int_{|z|\geq
c}\left|\phi(\Delta)^{1/2}p(t-r,z)\right|~{}dz\right]^{2}dr\leq
N(t-s)\phi(c^{-2}),$ (5.28)
$\displaystyle\int_{0}^{a}\left[\int_{\mathbf{R}^{d}}\left|\phi(\Delta)^{1/2}p(t-r,z+h)-\phi(\Delta)^{1/2}p(t-r,z)\right|~{}dz\right]^{2}dr$
$\displaystyle\leq N|h|^{2}\phi^{-1}\left((t-a)^{-1}\right),$ (5.29)

and

$\displaystyle\int_{0}^{a}\left[\int_{\mathbf{R}^{d}}\left|\phi(\Delta)^{1/2}p(t-r,z)-\phi(\Delta)^{1/2}p(s-r,z)\right|~{}dz\right]^{2}dr\leq
N(t-s)^{2}(s-a)^{-2},$ (5.30)

where $N$ depends only on $d$ and the constants appearing in Assumption 5.14.

###### Proof.

First we prove (5.28). This is an easy consequence of assumptions on $\phi$.
Indeed, by (5.22) and (5.25),

$\displaystyle\int_{|z|\geq c}|\phi(\Delta)^{1/2}p(t-r,z)|~{}dz$
$\displaystyle\leq\int_{|z|\geq c}\frac{\phi(|z|^{-2})^{1/2}}{|z|^{d}}~{}dz$
$\displaystyle\leq\int_{|z|\geq 1}\frac{\phi(|cz|^{-2})^{1/2}}{|z|^{d}}~{}dz$
$\displaystyle\leq N\phi(c^{-2})^{1/2}.$

Next we prove (5.29). By the fundamental theorem of calculus,

$\displaystyle\int_{\mathbf{R}^{d}}\left|\phi(\Delta)^{1/2}p(t-r,z+h)-\phi(\Delta)^{1/2}p(t-r,z)\right|~{}dz$
$\displaystyle\leq|h|\int_{\mathbf{R}^{d}}\big{|}\phi(\Delta)^{1/2}\nabla
p(t-r,z)dz.$

Denote

$a_{t}:=\left(\phi^{-1}(t^{-1})\right)^{1/2}.$

Then by (5.23), (5.25), and (5.27),

$\displaystyle\int_{\mathbf{R}^{d}}\left|\phi(\Delta)^{1/2}\nabla
p(t-r,z)\right|dz$
$\displaystyle=\int_{|z|<1/a_{t-r}}\left|\phi(\Delta)^{1/2}\nabla
p(t-r,z)\right|dz+\int_{|z|\geq 1/a_{t-r}}\left|\phi(\Delta)^{1/2}\nabla
p(t-r,z)\right|dz$ $\displaystyle\leq
N\left((1/a_{t-r})^{d}(t-r)^{-1/2}(\phi^{-1}((t-r)^{-1}))^{(d+1)/2}+\int_{|z|\geq
1/a_{t-r}}\frac{\phi(|z|^{-2})^{1/2}}{|z|^{d+1}}dx\right)$ $\displaystyle\leq
N\left((t-r)^{-1/2}(\phi^{-1}((t-r)^{-1}))^{1/2}\right).$

Hence by (5.26),

$\displaystyle\int_{0}^{a}\left[\int_{\mathbf{R}^{d}}\big{|}\phi(\Delta)^{1/2}p(t-r,z+h)-\phi(\Delta)^{1/2}p(t-r,z)\big{|}~{}dz\right]^{2}dr$
$\displaystyle\leq
N|h|^{2}\left(\int_{0}^{a}(t-r)^{-1}\phi^{-1}\left((t-r)^{-1}\right)dr\right)$
$\displaystyle\leq N|h|^{2}\left(\phi^{-1}\left((t-a)^{-1}\right)\right).$

Finally we prove (5.30). By the mean-value theorem, (5.24), and (5.25),

$\displaystyle\int_{\mathbf{R}^{d}}\left|\phi(\Delta)^{1/2}p(t-r,z)-\phi(\Delta)^{1/2}p(s-r,z)\right|~{}dz$
$\displaystyle=(t-s)\int_{\mathbf{R}^{d}}\left|\frac{d}{dr}\phi(\Delta)^{1/2}p(\theta
t+(1-\theta)s-r,z)\right|~{}dz$
$\displaystyle=(t-s)\int_{\mathbf{R}^{d}}\left|\phi(\Delta)^{3/2}p(\theta
t+(1-\theta)s-r,z)\right|~{}dz$ $\displaystyle\leq N(t-s)(\theta
t+(1-\theta)s-r)^{-3/2}$ $\displaystyle\quad+N(t-s)\int_{|z|\geq 1/a_{\theta
t+(1-\theta)s-r}}(\theta
t+(1-\theta)s-r)^{-1}\frac{\phi(|z|^{-2})^{1/2}}{|z|^{d}}dz$
$\displaystyle\leq N(t-s)(\theta t+(1-\theta)s-r)^{-3/2},$

where $\theta\in[0,1]$. Therefore,

$\displaystyle\int_{0}^{a}\left[\int_{\mathbf{R}^{d}}|\phi(\Delta)^{1/2}p(t-r,z)-\phi(\Delta)^{1/2}p(s-r,z)|~{}dz\right]^{2}dr$
$\displaystyle\leq N(t-s)^{2}\int_{0}^{a}(\theta
t+(1-\theta)s-r)^{-3}dr\leq(t-s)^{2}(s-a)^{-2}.$

The lemma is proved. $\Box$

In the following corollary, we finally prove that the kerenel

$K(t-r,z,x):=1_{0<r<t<T}\phi(\Delta)^{1/2}p(t-r,x-z)$

satisfies Assumption 2.4. Recall

$\rho(X,Y)=\left(\phi^{-1}\left(|t-s|^{-1}\right)\right)^{-1/2}+|x-y|.$

Due to (5.26), there exists a constant $N_{\phi}\geq 1$ so that

$\phi^{-1}\left(a\right)\leq
N_{\phi}\phi^{-1}\left(2^{-1}a\right)\qquad\forall a>0.$

For $r>0$, $X=(t,x),Y=(s,y)\in(0,\infty)\times\mathbf{R}^{d}$ set

$\displaystyle A(r,X,Y):=\left\\{z\in\mathbf{R}^{d}:\rho(X,Z)\geq
4N_{\phi}\rho(X,Y)\right\\},$

where $Z=(r,z)$. For the notational convenience, we use

$\phi(\Delta)^{1/2}p(t-r,x-z)$

to denote

$1_{0<r<t<T}\phi(\Delta)^{1/2}p(t-r,x-z),$

that is, we assume

$\phi(\Delta)^{1/2}p(t-r,x-z)=0$

unless $0<r<t<T$.

######  Corollary 5.18.

There exists a constant $N$ so that for all
$X=(t,x),Y=(s,y)\in(0,T)\times\mathbf{R}^{d}$,

$\displaystyle\int_{0}^{T}\left[\int_{A(r,X,Y)}\left|\phi(\Delta)^{1/2}p(t-r,x-z)-\phi(\Delta)^{1/2}p(s-r,y-z)\right|~{}dz\right]^{2}dr\leq
N,$

where $N$ depends only on $d$ and the constants appearing in Assumption 5.14.

###### Proof.

Without loss of generality, we assume $t\geq s$. We only focus on proving the
case $t>s$ since the proof of the case $t=s$ is simpler. Denote

$\displaystyle\mathcal{I}(r,X,Y)=\left[\int_{A(r,X,Y)}\left|\phi(\Delta)^{1/2}p(t-r,x-z)-\phi(\Delta)^{1/2}p(s-r,y-z)\right|~{}dz\right]^{2}.$

If $r\geq t$, then $\mathcal{I}(r,X,Y)=0$. Thus

$\displaystyle\int_{0}^{T}\mathcal{I}(r,X,Y)dr$
$\displaystyle=\int_{2s-t}^{t}\mathcal{I}(r,X,Y)dr+\int_{0}^{2s-t}\mathcal{I}(r,X,Y)dr$
$\displaystyle=:\mathcal{I}_{1}(X,Y)+\mathcal{I}_{2}(X,Y).$

First we estimate $\mathcal{I}_{1}(X,Y)$. By (5.28),

$\displaystyle\mathcal{I}_{1}(X,Y)$
$\displaystyle\leq\int_{2s-t}^{t}\left[\int_{|z|\geq\left(\phi^{-1}(|t-s|^{-1})\right)^{-1/2}}\left|\phi(\Delta)^{1/2}p(t-r,z)\right|~{}dz\right]^{2}dr$
$\displaystyle\quad+\int_{2s-t}^{s}\left[\int_{|z|\geq\left(\phi^{-1}(|t-s|^{-1})\right)^{-1/2}}\left|\phi(\Delta)^{1/2}p(s-r,z)\right|~{}dz\right]^{2}dr\leq
N.$

We split $\mathcal{I}_{2}$. Observe

$\displaystyle\mathcal{I}_{2}$
$\displaystyle\leq\mathcal{I}_{2,1}+\mathcal{I}_{2,2}$
$\displaystyle:=\int_{0}^{2s-t}\left[\int_{A(r,X,Y)}\left|\phi(\Delta)^{1/2}p(t-r,x-z)-\phi(\Delta)^{1/2}p(t-r,y-z)\right|~{}dz\right]^{2}dr$
$\displaystyle+\int_{0}^{2s-t}\left[\int_{A(r,X,Y)}\left|\phi(\Delta)^{1/2}p(t-r,y-z)-\phi(\Delta)^{1/2}p(s-r,y-z)\right|~{}dz\right]^{2}dr.$

If $|x-y|\leq\left(\phi^{-1}(|t-s|^{-1})\right)^{-1/2}$ then by (5.29),

$\displaystyle\mathcal{I}_{2,1}\leq
N|x-y|^{2}\left(\phi^{-1}\left(|t-s|^{-1}\right)\right)\leq N.$

On the other hand, if

$\displaystyle|x-y|>\left(\phi^{-1}(|t-s|^{-1})\right)^{-1/2},$

then

$\displaystyle\mathcal{I}_{2,1}\leq 2\mathcal{I}_{2,1,1}+\mathcal{I}_{2,1,2},$

where

$\displaystyle\mathcal{I}_{2,1,1}:=\int_{s-\left(\phi\left(|x-y|^{-2}\right)\right)^{-1}}^{t}\left[\int_{|z|\geq\left(\phi^{-1}(|t-s|^{-1})\right)^{-1/2}+|x-y|}\left|\phi(\Delta)^{1/2}p(t-r,z)\right|~{}dz\right]^{2}dr$
$\displaystyle\mathcal{I}_{2,1,2}$
$\displaystyle:=\int_{0}^{s-\left(\phi\left(|x-y|^{-2}\right)\right)^{-1}}\left[\int_{\mathbf{R}^{d}}\left|\phi(\Delta)^{1/2}p(t-r,x-z)-\phi(\Delta)^{1/2}p(t-r,y-z)\right|~{}dz\right]^{2}dr.$

By (5.28),

$\displaystyle\mathcal{I}_{2,1,1}$ $\displaystyle\leq
N\left[\left(t-s+\left(\phi\left(|x-y|^{-2}\right)\right)^{-1}\right)\phi\left(\left(\left(\phi^{-1}(|t-s|^{-1})\right)^{-1/2}+|x-y|\right)^{-2}\right)\right]$
$\displaystyle\leq N$

and by (5.29),

$\displaystyle\mathcal{I}_{2,1,2}$ $\displaystyle\leq
N|x-y|^{2}\phi^{-1}\left(\left(t-s+\left(\phi\left(|x-y|^{-2}\right)\right)^{-1}\right)^{-1}\right)\leq
N.$

It only remains to estimate $\mathcal{I}_{2,2}$. However, this is an easy
consequence of (5.30). Indeed,

$\displaystyle\mathcal{I}_{2,2}\leq N\left((t-s)|t-s|^{-1}\right)^{2}\leq N.$

The corollary is proved. $\Box$

Consequently, to prove Theorem 5.16 it is enough to apply Theorem 2.5 with

$\mathbb{T}_{\varepsilon}=\mathbb{T}_{\phi,\varepsilon}\quad\text{and}\quad\mathbb{T}=\mathbb{T}_{\phi}.$

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