\int^t_0 (t-s)^{-\beta}\int^s_0 (s-r)^{2(\beta-\gamma)}{\mathbb{E}}\|g(r, \cdot)\|^2_{L_2(\ell_2)} drds\leq N\|g\|^2_{{\mathbb{L}}_2(t,\ell_2)}.$$ This proves .

Note that the second equality in just follows from the definition of $\|u\|^2_{{\mathcal{H}}^{\sigma+2}_2(t)}$. Hence to prove (\[eqn 1.27.2\]) it is enough to consider a convolution with $k_{\beta}$ and use , which implies $$(k_{\beta}*k_{1-\beta}*{\mathbb{E}}\|u\|^2_{L_2})(t)=\int^t_0 {\mathbb{E}}\|u\|^2_{L_2}ds=\|u\|^2_{{\mathbb{L}}_2(t)}.$$ Hence the theorem is now proved.

Define $$P_{\beta,\gamma}(t,x)=\begin{cases}I^{\beta-\gamma}p (\cdot,x) (t)\quad &\text{if} \quad\beta\geq \gamma\\
\partial^{\gamma-\beta}_t p (\cdot,x) (t) \quad &\text{if}\quad \beta<\gamma.
\end{cases}$$

\[lemma main estimate\] Let $g\in {\mathbb{H}}^{\infty}_0(T,\ell_2)$ and $u$ be defined by $$u(t,x)=\sum_{k=1}^{\infty}\int^t_0 \int_{{\mathbb{R}}^d}P_{\beta,\gamma}(t-s, x-y)g^k(s,y)dy dW^k_s.$$ Let $\sigma \leq 2\wedge (\frac{1-2\gamma}{\beta} +2)$ if $\gamma \neq 1/2$, and $\sigma<2$ if $\gamma=1/2$. Then it holds that $${\mathbb{E}}\int^T_0 \|\Delta^{\sigma/2}u(t, \cdot)\|^2_{L_2} dt\leq N
 \|g\|^2_{{\mathbb{L}}_2(T, \ell_2)}.$$ In general, for any $\gamma_1 \in {\mathbb{R}}$, $${\mathbb{E}}\int^T_0 \|u(t, \cdot)\|^2_{H^{\gamma_1+\sigma}_2} dt\leq N
 \|g\|^2_{{\mathbb{H}}^{\gamma_1}_2(T, \ell_2)}.$$

Let $a:=\gamma-\beta<1/2$. Recall that $E_{\alpha,\gamma}(t)$ is bounded on $(-\infty,0]$. By the Fourier transform and Lemma \[l:DaE\], we have for any $\sigma\geq 0$ $$\begin{aligned}
&&{\mathbb{E}}\int^T_0\|\Delta^{\sigma/2}u(t, \cdot)\|^2_{L_2} dt\\ &\leq&N \sum_{k=1}^\infty {\mathbb{E}}\int^T_0 \int^t_0\int_{{\mathbb{R}}^d} |\xi|^{2\sigma}(t-s)^{-2a}E^2_{\beta, 1-a}(-|\xi|^2(t-s)^{\beta})|\hat{g}^k(s, \xi)|^2 d\xi ds\,dt\\
&\leq&N \|g\|^2_{{\mathbb{L}}_2(T, \ell_2)}\\
&&+N {\mathbb{E}}\int^T_0 \int^T_s\int_{|\xi|\geq 1} |\xi|^{2\sigma}(t-s)^{-2a}E^2_{\beta, 1-a}(-|\xi|^2(t-s)^{\beta})
|\hat{g}(s, \xi)|^2_{\ell_2} d\xi \,dt\,ds.\end{aligned}$$ By the substitution $r=|\xi|^{2/\beta} (t-s)$, the last term above is bounded by constant times of $${\mathbb{E}}\int_{|\xi|\geq 1}\int^T_0 |\hat{g}(s,\xi)|_{\ell_2}^2\int^{T|\xi|^{\frac{2}{\beta}}}_0 |\xi|^{2(\sigma+(2a-1)/{\beta})} r^{-2a}E^2_{\beta, 1-a}(-r^{\beta}) dr ds\,d\xi.$$ Let $\gamma>1/2$. Then, since $E_{\beta, 1-a}(-r^{\beta})$ is bounded on $[0,\infty)$ and 