A Sobolev space theory for the stochastic partial differential equations with space-time non-local operators

Abstract

We deal with the Sobolev space theory for the stochastic partial differential equation (SPDE) driven by Wiener processes

$$\begin{aligned} \partial _{t}^{\alpha }u=\left( \phi (\varDelta ) u +f(u) \right) + \partial _t^\beta \sum _{k=1}^\infty \int _0^t g^k(u)\,dw_s^k, \quad t>0, \,x\in {\mathbb {R}}^d \end{aligned}$$

as well as the SPDE driven by space-time white noise

$$\begin{aligned} \partial ^{\alpha }_{t}u=\phi (\varDelta )u + f(u) + \partial ^{\beta -1}_{t}h(u) {\dot{W}}, \quad t>0,\, x\in {\mathbb {R}}^d. \end{aligned}$$

Here, $\alpha \in (0,1), \beta < \alpha +1/2$, $\{w_t^k : k=1,2,\ldots \}$ is a family of independent one-dimensional Wiener processes and ${\dot{W}}$ is a space-time white noise defined on $[0,\infty )\times {\mathbb {R}}^d$. The time non-local operator $\partial _{t}^{\alpha }$ denotes the Caputo fractional derivative of order $\alpha$, the function $\phi$ is a Bernstein function, and the spatial non-local operator $\phi (\varDelta )$ is the integro-differential operator whose symbol is $-\phi (|\xi |^2)$. In other words, $\phi (\varDelta )$ is the infinitesimal generator of the d-dimensional subordinate Brownian motion. We prove the uniqueness and existence results in Sobolev spaces and obtain the maximal regularity results of solutions.

Auxiliary results

In this section, we obtain some sharp upper bounds of space-time fractional derivatives of the fundamental solution q(t, x) related to the equation

$$\begin{aligned} \partial _t^\alpha u = \phi (\varDelta )u ,\quad t>0; \quad u(0,\cdot )=u_0. \end{aligned}$$

First we record some elementary facts on Bernstein functions.

Lemma A.1

Let $\phi$ be a Bernstein function satisfying Assumption 3.1.

(i) There exists a constant $c=c(\gamma ,\kappa _0,\delta _0)$ such that for any $\lambda >0$,

$$\begin{aligned} \int _{\lambda ^{-1}}^\infty r^{-1}\phi (r^{-2}) dr \le c \phi (\lambda ^2). \end{aligned}$$

(A.1)

(ii) For any $\gamma \in (0,1)$, the function $\phi ^\gamma :=\left( \phi (\cdot )\right) ^\gamma$ is also a Bernstein function with no drift, and it satisfies Assumption 3.1 with $\gamma \delta _0$ and $\kappa _0^{\gamma }$, in place of $\delta _0$ and $\kappa _0$, respectively.

(iii) Let $\mu _\gamma$ be the Lévy measure of $\phi ^\gamma$ (i.e., $\phi ^{\gamma }(\lambda )=\int _{(0,\infty )} (1-e^{-\lambda t})\mu _{\gamma }(dt)$), and set

$$\begin{aligned} j_{\gamma ,d}(r):=\int _0^\infty (4\pi t)^{-d/2} \exp (-r^2/4t)\mu _\gamma (dt), \qquad r>0. \end{aligned}$$

(A.2)

Then

$$\begin{aligned} j_{\gamma ,d} (r) \le c(d) \,r^{-d}\phi (r^{-2})^\gamma , \qquad \forall r>0, \end{aligned}$$

(A.3)

and for any $f\in C_b^2({\mathbb {R}}^d)$ and $r>0$, it holds that

$$\begin{aligned} \phi (\varDelta )^\gamma f(\cdot )(x) =&\int _{{\mathbb {R}}^d} \left( f(x+y)-f(x)-\nabla f(x)\cdot y {\mathbf {1}}_{|y|\le r} \right) j_{\gamma ,d} (|y|) dy \end{aligned}$$

(A.4)

Proof

(i) By (3.4) this and the change of variables,

(ii) $\phi ^\gamma$ is a Bernstein function due to [42, Corollary 3.8 (iii)], and (3.3) easily yields

$$\begin{aligned} \kappa _0^\gamma \left( \frac{R}{r}\right) ^{\gamma \delta _0}\le \frac{\phi (R)^\gamma }{\phi (r)^\gamma }, \qquad 0<r<R<\infty . \end{aligned}$$

If we denote drift of $\phi ^{\gamma }$ by $b_{\gamma }$ (see (2.4)), it follows that

$$\begin{aligned} \lim _{\lambda \rightarrow \infty } \frac{\phi (\lambda )}{\lambda }=b, \quad \lim _{\lambda \rightarrow \infty } \frac{\phi (\lambda )^{\gamma }}{\lambda }=b_{\gamma }. \end{aligned}$$

Hence, we have

$$\begin{aligned} b_\gamma =\lim _{\lambda \rightarrow \infty } \frac{\phi (\lambda )^\gamma }{\lambda }=\lim _{\lambda \rightarrow \infty } \left( \frac{\phi (\lambda )}{\lambda }\right) ^\gamma \lambda ^{\gamma -1}=0. \end{aligned}$$

(iii) (A.3) follows from [24, Lemma 3.3] (recall that $\phi ^{\gamma }$ is a Bernstein function with no drift), and the second assertion is a consequence of (2.7). $\square$

Recall that p(t, x) is the transition density of the subordinate Brownian motion $X_t$ with characteristic exponent $\phi (|\xi |^2)$. Also, for any $t>0$, and $x\in {\mathbb {R}}^d$,

$$\begin{aligned} p(t,x)= & {} \frac{1}{(2\pi )^d} \int _{{\mathbb {R}}^d} e^{i \xi \cdot x} e^{-t\phi (|\xi |^2)} \,d\xi \nonumber \\= & {} \int _{(0,\infty )} (4\pi s)^{-d/2} \exp \left( -\frac{|x|^2}{4s}\right) \eta _t (ds) \end{aligned}$$

(A.5)

where $\eta _t(ds)$ is the distribution of $S_t$ (see [2, Sect. 5.3.1]). Thus $X_t$ is rotationally invariant.

Lemma A.2

(i) There exists a constant $C=C(d,\delta _0,\kappa _0)$ such that for $(t,x)\in (0,\infty )\times {\mathbb {R}}^d$,

$$\begin{aligned} p(t,x)\le C\left( \left( \phi ^{-1}(t^{-1})\right) ^{d/2}\wedge t\frac{\phi (|x|^{-2})}{|x|^d}\right) . \end{aligned}$$

(ii) For any $m\in {\mathbb {N}}$, there exists a constant $C=C(d,\delta _0,\kappa _0, m)$ so that for any $(t,x)\in (0,\infty )\times {\mathbb {R}}^d$,

$$\begin{aligned}&|D^m_x p(t,x)| \le C \sum _{m-2n\ge 0,n\in {\mathbb {N}}_0} |x|^{m-2n}\left( (\phi ^{-1}(t^{-1}))^{d/2+m-n}\wedge t\frac{\phi (|x|^{-2})}{|x|^{d+2(m-n)}}\right) . \end{aligned}$$

Proof

See [27, Lemma 3.4, Lemma 3.6]. $\square$

The following lemma is an extension of [24, Lemma 4.2]. The main difference is that our estimate holds for all $t>0$. Such result is needed for us to prove estimates of solutions to SPDEs (see, e.g., (3.2)).

Lemma A.3

Let $\gamma \in (0,1)$ and $m\in {\mathbb {N}}_0$. Then for any $(t,x)\in (0,\infty )\times {\mathbb {R}}^d$,

$$\begin{aligned}&|\phi (\varDelta )^\gamma D^m_x p(t,\cdot )(x)| \\&\quad \le C(d,\delta _0, \kappa _0, m, \gamma ) \left( t^{-\gamma }(\phi ^{-1}(t^{-1}))^{(d+m)/2}\wedge \frac{\phi (|x|^{-2})^\gamma }{|x|^{d+m}}\right) . \end{aligned}$$

Proof

Note first that for any given $a>0$,

$$\begin{aligned} s^{a}e^{-s}\le c(a)e^{-s/2}, \quad \forall s>0. \end{aligned}$$

Also note that by (3.3), if $a^2 \ge \phi ^{-1}(t^{-1})$, then

$$\begin{aligned} \kappa _0 \left( \frac{a^2}{\phi ^{-1}(t^{-1})}\right) ^{\delta _0} \le \frac{\phi (a^2)}{t^{-1}}=t\phi (a^2). \end{aligned}$$

Therefore, by (A.5),

$$\begin{aligned}&\left| \phi (\varDelta )^\gamma D^{m}_{x} p(t,x)\right| =\left| {\mathcal {F}}^{-1} \left[ {\mathcal {F}}(\phi (\varDelta )^\gamma D^{m}_{x} p(t,\cdot )) (\xi ) \right] (x)\right| \\&\quad \le C\int _{{\mathbb {R}}^d} \left| t^{-\gamma }\left( t\phi (|\xi |^2)\right) ^\gamma e^{-t\phi (|\xi |^2)}\right| |\xi |^{m} d\xi \\&\quad \le Ct^{-\gamma } \left( \int _{|\xi |^2>\phi ^{-1}(t^{-1})} |\xi |^{m} e^{-t\frac{\phi (|\xi |^2)}{2}}d\xi + \int _{|\xi |^2\le \phi ^{-1}(t^{-1})} |\xi |^{m} d\xi \right) \\&\quad \le Ct^{-\gamma }\left( \int _{|\xi |^2>\phi ^{-1}(t^{-1})} |\xi |^{m} e^{-\frac{\kappa _0}{2} \left( \frac{|\xi |^2}{\phi ^{-1}(t^{-1})}\right) ^{\delta _0}}d\xi +\left( \phi ^{-1}(t^{-1})\right) ^{\frac{d+m}{2}} \right) \\&\quad \le C t^{-\gamma }\left( \phi ^{-1}(t^{-1})\right) ^{\frac{d+m}{2}} \left( \int _{|\xi |^2>1} e^{-\frac{\kappa _0}{4}|\xi |^{2\delta _0}} d\xi +1 \right) \\&\quad \le C t^{-\gamma }\left( \phi ^{-1}(t^{-1})\right) ^{\frac{d+m}{2}}. \end{aligned}$$

Hence, to finish the proof, we may assume $t^{-\gamma }(\phi ^{-1}(t^{-1}))^{\frac{d+m}{2}} \ge \frac{\phi (|x|^{-2})^\gamma }{|x|^{d+m}}$ (equivalently, $t \phi (|x|^{-2})\le 1$) and prove

$$\begin{aligned} |\phi (\varDelta )^\gamma D^m_x p(t,\cdot )(x)| \le C \frac{\phi (|x|^{-2})^\gamma }{|x|^{d+m}}. \end{aligned}$$

By (A.4) with $r=|x|/2$,

$$\begin{aligned}&\left| \phi (\varDelta )^\gamma D^{m}_{x} p(t,\cdot )(x)\right| \\&\quad = \left| \int _{{\mathbb {R}}^d} \left( D^{m}_{x}p(t,x+y)-D^{m}_{x}p(t,x)-\nabla D^{m}_{x}p(t,x)\cdot y {\mathbf {1}}_{|y|\le \frac{|x|}{2}}\right) j_{\gamma ,d} (|y|)dy \right| \\&\quad \le |D^{m}_{x}p(t,x)|\int _{|y|>|x|/2} j_{\gamma ,d} (|y|)dy \\&\qquad + \left| \int _{|y|>|x|/2} D^{m}_{x}p(t,x+y)j_{\gamma ,d} (|y|)dy\right| \\&\qquad + \int _{|x|/2>|y|} \int _0^1 \left| D^{m+1}_{x}p(t,x+sy)- D^{m+1}_{x}p(t,x)\right| |y|j_{\gamma ,d}(|y|) dsdy \\&\quad =: |D^{m}_{x}p(t,x)|\times I + II + III. \end{aligned}$$

By (A.3), (A.1) with $\phi ^{\gamma }$, and (3.4), we have

$$\begin{aligned} I \le C \int _{r>|x|/2} r^{-1}\phi (r^{-2})^\gamma dr \le C \phi (4|x|^{-2})^{\gamma } \le C \phi (|x|^{-2})^{\gamma }. \end{aligned}$$

This together with Lemma A.2 (ii) yields (recall we assume $t\phi (|x|^{-2})\le 1$)

$$\begin{aligned} |D^{m}_{x}p(t,x)|\times I \le C t\frac{\phi (|x|^{-2})^{1+\gamma }}{|x|^{d+m}}\le C \frac{\phi (|x|^{-2})^{\gamma }}{|x|^{d+m}} . \end{aligned}$$

For III, by the fundamental theorem of calculus and Lemma A.2 (ii),

$$\begin{aligned} III \le&C(d)\int _{|x|/2>|y|} \int _0^1 \int _0^1 \left| D^{m+2}_{x}p(t,x+usy)\right| |y|^2j_{\gamma ,d}(|y|) dudsdy \\ \le&C\int _{|x|/2>|y|} \int _0^1\int _0^1 t\frac{\phi (|x+usy|^{-2})}{|x+usy|^{d+m+2}} |y|^2j_{\gamma ,d}(|y|) dudsdy \\ \le&C t\frac{\phi (|x|^{-2})}{|x|^{d+m+2}}\int _{|x|/2>|y|} |y|^2j_{\gamma ,d}(|y|) dy. \end{aligned}$$

For the last inequality above, we used $|x+usy|\ge |x|/2$. By (A.3), and (3.4) with $r=|x|^{-2}$ and $R=\rho ^{-2}$,

$$\begin{aligned} \int _{|x|/2>|y|} |y|^2j_{\gamma ,d}(|y|)dy \le&C \int _0^{|x|} \rho \phi (\rho ^{-2})^\gamma d\rho \\ \le&C |x|^{2\gamma }\phi (|x|^{-2})^\gamma \int _0^{|x|} \rho ^{1-2\gamma } d\rho \\ \le&C |x|^2 \phi (|x|^{-2})^\gamma . \end{aligned}$$

Therefore, it follows that for $t \phi (|x|^{-2})\le 1$,

$$\begin{aligned} III\le C t\frac{\phi (|x|^{-2})^{1+\gamma }}{|x|^{d+m}} \le C \frac{\phi (|x|^{-2})^{\gamma }}{|x|^{d+m}}. \end{aligned}$$

Now we estimate II. By using the integration by parts m-times, we have

$$\begin{aligned} II&\le \sum _{k=0}^{m-1} \int _{|y|=\frac{|x|}{2}} \left| \left( \frac{d^{k}}{d\rho ^{k}} j_{\gamma ,d}\right) (|y|) D^{m-1-k}_{x}p(t,x+y) \right| dS \\&\quad +\int _{|y|>\frac{|x|}{2}} \left| \left( \frac{d^{m}}{d\rho ^{m}}j_{\gamma ,d}\right) (|y|)p(t,x+y) \right| dy. \end{aligned}$$

Differentiating $j_{\gamma ,d}(\rho )$, and then using (A.2) and (A.3), for $k\in {\mathbb {N}}_{0}$ we get

$$\begin{aligned} \left| \frac{d^{k}}{d\rho ^{k}} j_{\gamma ,d}(\rho ) \right| \le C \sum _{k-2l\ge 0,l\in {\mathbb {N}}_{0}} \rho ^{k-2l}|j_{\gamma ,d+2(k-l)}(\rho )| \le C \rho ^{-d-k}\phi (\rho ^{-2})^{\gamma }. \end{aligned}$$

This and Lemma A.2 (ii) yield that

$$\begin{aligned} II&\le C\sum _{k=0}^{m-1} \Big ( |x|^{d-1}|x|^{-d-k} |x|^{-d-m+1+k} t \phi (|x|^{-2})^{\gamma +1} \Big ) + C \frac{\phi (|x|^{-2})^{\gamma }}{|x|^{d+m}}\\&\le C \frac{\phi (|x|^{-2})^{\gamma }}{|x|^{d+m}}. \end{aligned}$$

for $t\phi (|x|^{-2})\le 1$. Hence, the lemma is proved. $\square$

Below we provide the proof of Lemma 3.2. The proof is mainly based on [27, Lemma 3.7, Lemma 3.8].

Proof of Lemma 3.2

(i) See [27, Lemma 3.7 (iii)].

(ii) See [27, Lemma 3.8] for (3.6) with arbitrary $\beta$ and for (3.7) when $\beta \notin {\mathbb {N}}$. Hence, we only prove (3.7) when $\beta \in {\mathbb {N}}$. Let $\beta \in {\mathbb {N}}$, then by [27, Lemma 3.8], we have

$$\begin{aligned} |q_{\alpha ,\beta } (t,x)|&\le C \int _{(\phi (|x|^{-2}))^{-1}}^{2t^{\alpha }} (\phi ^{-1}(r^{-1}))^{(d+m)/2} r t^{-\alpha -\beta } dr \\&\le C \int _{(\phi (|x|^{-2}))^{-1}}^{2t^{\alpha }} (\phi ^{-1}(r^{-1}))^{(d+m)/2} t^{-\beta } dr. \end{aligned}$$

For the last inequality, we used $rt^{-\alpha }\le 2$ whenever $r\le 2t^{\alpha }$.

(iii) We follow the proof of [27, Lemma 3.8]. By [27, Lemma 3.7], there exist constants $c,C>0$ depending only on $\alpha ,\beta$ such that

$$\begin{aligned} |\varphi _{\alpha ,\beta }(t,r)|\le C t^{-\beta }e^{-c(rt^{-\alpha })^{1/(1-\alpha )}} \end{aligned}$$

(A.6)

for $rt^{-\alpha }\ge 1$, and

$$\begin{aligned} |\varphi _{\alpha ,\beta }(t,r)|\le \left\{ \begin{array}{ll} C rt^{-\alpha -\beta }~&{}:\, \beta \in {\mathbb {N}}\\ C t^{-\beta }~&{}:\, \beta \notin {\mathbb {N}}\end{array} \right. \end{aligned}$$

(A.7)

for $rt^{-\alpha }\le 1$. Therefore, we have

$$\begin{aligned} \int _0^\infty |\varphi _{\alpha ,\beta }(t,r)|dr<\infty . \end{aligned}$$

(A.8)

Let $x\in {\mathbb {R}}^d\setminus \{0\}$. Then for any $r>0$ and $y\ne 0$ sufficiently close to x, we have

$$\begin{aligned} |\phi (\varDelta )^\gamma D^{\sigma }p(r,y)| \le C(\phi ,x,d,m,\gamma ) , \quad |\sigma |\le m \end{aligned}$$

due to Lemma A.3. Using (A.8) and the dominated convergence theorem, we get

$$\begin{aligned} D^m_x q^\gamma _{\alpha ,\beta } (t,x)=\int _0^\infty \phi (\varDelta )^\gamma D^m_x p(r,x)\varphi _{\alpha ,\beta }(t,r) dr. \end{aligned}$$

Hence, by (A.6) and (A.7) (also recall $rt^{-\alpha }\le 2$ whenever $r\le 2t^{\alpha }$),

$$\begin{aligned} |D^m_x q^\gamma _{\alpha ,\beta } (t,x)|&\le C \int _0^{t^\alpha } |\phi (\varDelta )^\gamma D^m_x p(r,x)|t^{-\beta } dr \nonumber \\&\quad + C \int _{t^\alpha }^\infty |\phi (\varDelta )^\gamma D^m_x p(r,x)| t^{-\beta }e^{-c(rt^{-\alpha })^{1/(1-\alpha )}} dr \nonumber \\&=: I+II. \end{aligned}$$

(A.9)

By Lemma A.3,

$$\begin{aligned} I \le C \int _0^{t^\alpha } t^{-\beta } \frac{\phi (|x|^{-2})^{\gamma }}{|x|^{d+m}}dr \le C t^{\alpha -\beta } \frac{\phi (|x|^{-2})^{\gamma }}{|x|^{d+m}}. \end{aligned}$$

Also, by the change of variables $rt^{-\alpha }\rightarrow r$,

$$\begin{aligned} II&\le C t^{-\beta } \int _{t^\alpha }^\infty \frac{\phi (|x|^{-2})^{\gamma }}{|x|^{d+m}} e^{-c(rt^{-\alpha })^{1/(1-\alpha )}} dr \\&\le C t^{\alpha -\beta }\frac{\phi (|x|^{-2})^{\gamma }}{|x|^{d+m}} \int _1^\infty e^{-cr^{1/(1-\alpha )}} dr \\&\le C t^{\alpha -\beta }\frac{\phi (|x|^{-2})^{\gamma }}{|x|^{d+m}}. \end{aligned}$$

Hence, (3.8) is proved.

Next we prove (3.9). Assume $t^\alpha \phi (|x|^{-2})\ge 1$. Again we consider I and II defined in (A.9). For I we have

$$\begin{aligned} I&=t^{-\beta } \int _0^{(\phi (|x|^{-2}))^{-1}}|\phi (\varDelta )^\gamma D^m_x p(r,x)|dr \\&\quad +t^{-\beta } \int _{(\phi (|x|^{-2}))^{-1}}^{t^\alpha }|\phi (\varDelta )^\gamma D^m_x p(r,x)|t^{-\beta } dr \\&=:I_1 + I_2. \end{aligned}$$

By Lemma A.3 (recall that $t^{\alpha }\phi (|x|^{-2})\ge 1$),

$$\begin{aligned} I_1&\le C t^{-\beta } \int _0^{(\phi (|x|^{-2}))^{-1}} \frac{\phi (|x|^{-2})^{\gamma }}{|x|^{d+m}}dr\\&= C t^{-\beta } \frac{\phi (|x|^{-2})^{\gamma -1}}{|x|^{d+m}} = Ct^{-\beta } \int _{(\phi (|x|^{-2}))^{-1}}^{2(\phi (|x|^{-2}))^{-1}} \frac{\phi (|x|^{-2})^{\gamma }}{|x|^{d+m}} dr \end{aligned}$$

Note that if $r \le 2(\phi (|x|^{-2}))^{-1}$, then by (3.4)

$$\begin{aligned} |x|^{-2} \le \phi ^{-1}(2r^{-1}) \le \left( \frac{2}{k_0} \right) ^{1/\delta _{0}} \phi ^{-1}(r^{-1}). \end{aligned}$$

(A.10)

Thus, using $r\le 2(\phi (|x|^{-2}))^{-1}$, we get

$$\begin{aligned} I_1&\le C t^{-\beta }\int _{(\phi (|x|^{-2}))^{-1}}^{2(\phi (|x|^{-2}))^{-1}} (\phi ^{-1}(r^{-1}))^{(d+m)/2} r^{-\gamma } dr \nonumber \\&\le Ct^{-\beta } \int _{(\phi (|x|^{-2}))^{-1}}^{2t^{\alpha }} (\phi ^{-1}(r^{-1}))^{(d+m)/2} r^{-\gamma } dr. \end{aligned}$$

We also get, by Lemma A.3,

$$\begin{aligned} I_2 \le C t^{-\beta } \int _{(\phi (|x|^{-2}))^{-1}}^{2t^{\alpha }} (\phi ^{-1}(r^{-1}))^{(d+m)/2} r^{-\gamma }dr. \end{aligned}$$

Thus, I is handled. Next we estimate II. By (3.4), we find that

$$\begin{aligned} \phi ^{-1}(r^{-1})\le t^\alpha r^{-1} \phi ^{-1}(t^{-\alpha }), \quad t^\alpha \le r. \end{aligned}$$

Therefore, by Lemma A.3 and the change of variables $rt^{-\alpha }\rightarrow r$,

$$\begin{aligned} II\le & {} Ct^{-\beta } \int _{t^\alpha }^\infty r^{-\gamma }(\phi ^{-1}(r^{-1}))^{(d+m)/2} e^{-c(rt^{-\alpha })^{1/(1-\alpha )}} dr \nonumber \\\le & {} Ct^{-\beta } \int _{t^\alpha }^\infty r^{-\gamma }(t^\alpha r^{-1} \phi ^{-1}(t^{-\alpha }))^{(d+m)/2} e^{-c(rt^{-\alpha })^{1/(1-\alpha )}} dr \nonumber \\= & {} C t^{(1-\gamma )\alpha -\beta } (\phi ^{-1}(t^{-\alpha }))^{(d+m)/2} \int _1^\infty r^{-\gamma -(d+m)/2}e^{-cr^{1/(1-\alpha )}} dr \nonumber \\\le & {} C t^{(1-\gamma )\alpha -\beta } (\phi ^{-1}(t^{-\alpha }))^{(d+m)/2}. \end{aligned}$$

(A.11)

As (A.10), if $r \le 2t^{\alpha }$, then by (3.4)

$$\begin{aligned} \phi ^{-1}(t^{-\alpha })\le \phi ^{-1}(2r^{-1}) \le \left( \frac{2}{\kappa _0} \right) ^{1/\delta _{0}} \phi ^{-1}(r^{-1}). \end{aligned}$$

Therefore,

$$\begin{aligned} t^{(1-\gamma )\alpha -\beta } (\phi ^{-1}(t^{-\alpha }))^{(d+m)/2}&\le C \int _{t^\alpha }^{2t^{\alpha }} (\phi ^{-1}(r^{-1}))^{(d+m)/2} r^{-\gamma }t^{-\beta } dr \\&\le C \int _{(\phi (|x|^{-2}))^{-1}}^{2t^{\alpha }} (\phi ^{-1}(r^{-1}))^{(d+m)/2} r^{-\gamma }t^{-\beta } dr. \end{aligned}$$

This and (A.11) take care of II, and consequently (3.9) is proved.

(iv) See [27, Corollary 3.9] for (3.10). We prove (3.11). By (3.8), (3.9), Fubini’s theorem, and (A.1),

$$\begin{aligned} \int _{{\mathbb {R}}^d} |q^\gamma _{\alpha ,\beta }(t,x)|dx= & {} \int _{|x|\ge (\phi ^{-1}(t^{-\alpha }))^{-\frac{1}{2}}} |q^\gamma _{\alpha ,\beta }(t,x)|dx \\&+\int _{|x|< (\phi ^{-1}(t^{-\alpha }))^{-\frac{1}{2}}} |q^\gamma _{\alpha ,\beta }(t,x)|dx \\\le & {} C \int _{|x|\ge (\phi ^{-1}(t^{-\alpha }))^{-\frac{1}{2}}} t^{\alpha -\beta }\frac{\phi (|x|^{-2})^\gamma }{|x|^d} dx \\&+ C \int _{|x|< (\phi ^{-1}(t^{-\alpha }))^{-\frac{1}{2}}} \int _{(\phi (|x|^{-2}))^{-1}}^{2t^{\alpha }} (\phi ^{-1}(r^{-1}))^{d/2} r^{-\gamma }t^{-\beta } dr dx \\\le & {} C \int _{r\ge \left( \phi ^{-1}(t^{-\alpha }) \right) ^{-\frac{1}{2}}} t^{\alpha -\beta } \frac{\phi (r^{-2})^\gamma }{r}dr \\&+ C \int _{0}^{2t^\alpha } \int _{(\phi (|x|^{-2}))^{-1}\le r} (\phi ^{-1}(r^{-1}))^{d/2} r^{-\gamma }t^{-\beta } dx dr \\\le & {} C t^{(1-\gamma )\alpha -\beta } + C \int _0^{2t^\alpha } r^{-\gamma }t^{-\beta } dr \le C t^{(1-\gamma )\alpha -\beta }. \end{aligned}$$

(v) By (3.24) in [27] (or see (34) of [18]),

$$\begin{aligned} \int _0^\infty e^{-sr}\varphi _{\alpha ,\beta }(t,r)dr = t^{\alpha -\beta } E_{\alpha ,1-\beta +\alpha }(-st^\alpha ). \end{aligned}$$

Hence, by Fubini’s theorem and (A.5) for $\gamma \in (0,1)$,

$$\begin{aligned} {\mathcal {F}}_d(q^\gamma _{\alpha ,\beta })(t,\xi )&= \int _0^\infty \varphi _{\alpha ,\beta }(t,r)\left[ \int _{{\mathbb {R}}^d}e^{-ix\cdot \xi } \phi (\varDelta )^\gamma p(r,x)dx \right] dr \\&=-\int _0^\infty \varphi _{\alpha ,\beta }(t,r) \phi (|\xi |^2)^\gamma e^{-r\phi (|\xi |^2)} dr \\&=-t^{\alpha -\beta } \phi (|\xi |)^\gamma E_{\alpha ,1-\beta +\alpha }(-t^\alpha \phi (|\xi |^2))). \end{aligned}$$

Similarly, we get

$$\begin{aligned} {\mathcal {F}}_d(q_{\alpha ,\beta })(t,\xi )&= \int _0^\infty \varphi _{\alpha ,\beta }(t,r)\left[ \int _{{\mathbb {R}}^d}e^{-ix\cdot \xi } p(r,x)dx \right] dr \\&=t^{\alpha -\beta } E_{\alpha ,1-\beta +\alpha }(-t^\alpha \phi (|\xi |^2)). \end{aligned}$$

Thus (v) is also proved. $\square$