(t)\leq 2\int^t_0 (\frac{\partial}{ \partial s}(\kappa*w)(s, \cdot),w(s, \cdot))_{L_2}ds.$$ We take (see [@Za]) a sequence of such functions $\kappa_n\in H^1_1([0,T])$ so that $\kappa_n \to k_{1-\beta}$ in $L_1([0,T]$ and $\frac{\partial}{ \partial t}(\kappa_n*w)\to \frac{\partial}{ \partial t}(k_{1-\beta}*w)$ in $L_2([0,T],L_2)$. Finally for (\[eqn 8.27.1\]) it is enough to note that $$\kappa_n *\|w\|^2_{L_2} \to k_{1-\beta}*\|w\|^2_{L_2} \quad \text{in} \quad L_1([0,T])$$ and both sides of (\[eqn 8.27.1\]) are continuous in $t$. The lemma is proved.

Recall that for any $\sigma$, $$\|u\|^2_{{\mathbb{H}}^{\sigma}_2(t)}:={\mathbb{E}}\int^t_0\|u(s)\|^2_{H^{\sigma}_2}ds.$$

\[lemm 8.28.6\] Let $u\in {\mathcal{H}}^{\sigma+2}_2(T)$. Then for any $t\leq T$, $$\label{eqn 1.27.1}
(k_{1-\beta}*{\mathbb{E}}\|u\|^2_{H^{\sigma}_2})(t)\leq N ({\mathbb{E}}\|u(0)\|^2_{H^{\sigma}_2}+\|{\mathbb{D}}u\|^2_{{\mathbb{H}}^{\sigma}_2(t)}+\|{\mathbb{S}}u\|^2_{{\mathbb{H}}^{\sigma}_2(t,\ell_2)}+\|u\|^2_{{\mathbb{H}}^{\sigma}_2(t)})
\leq N \|u\|^2_{{\mathcal{H}}^{\sigma+2}_2(t)},$$ where $N$ depends only on $T, \beta$ and $\gamma$. In particular, for any $t\leq T$, $$\label{eqn 1.27.2}
\|u\|^2_{{\mathbb{H}}^{\sigma}_2(t)}\leq N \int^t_0 (t-s)^{-1+\beta}\|u\|^2_{{\mathcal{H}}^{\sigma+2}_2(s)}\, ds.$$

We only consider the case $\sigma=0$. In general, one can consider $\Delta^{\sigma/2}u$ in place of $u$. Denote $v$ as in Lemma \[Banach\]. Then by (\[eqn 8.27.1\]), $$(k_{1-\beta}*{\mathbb{E}}\|u\|^2_{L_2})(t)\leq 2 (k_{1-\beta}*{\mathbb{E}}\|u(0)+v\|^2_{L_2})(t)+2{\mathbb{E}}\int^t_0(f(s, \cdot),u(s, \cdot)-u(0, \cdot)-v(s, \cdot))_{L_2} ds.$$ Note that by Fubini’s theorem and Davis’s inequality $$\begin{aligned}
                              \nonumber
&&{\mathbb{E}}\int^t_0\|v(s)\|^2_{L_2}ds = \int^{t}_0 {\mathbb{E}}\|v(s)\|^2_{L_2}ds\\
&\leq& N {\mathbb{E}}\int^t_0 \int^s_0 (s-r)^{2(\beta-\gamma)}
\|g(r, \cdot)\|^2_{L_2(\ell_2)} drds  \leq N\|g\|^2_{{\mathbb{L}}_2(t,\ell_2)}.   \label{eqn 8.28.5}\end{aligned}$$ For the last inequality we use the fact $2(\beta-\gamma)>-1$. Therefore, by young’s inequality $${\mathbb{E}}\int^t_0(f(s, \cdot),u(s, \cdot)-u(0, \cdot)-v(s, \cdot))_{L_2} ds\leq N({\mathbb{E}}\|u(0)\|^2_{L_2}+\|f\|^2_{{\mathbb{L}}_2(t)}+\|g\|^2_{{\mathbb{L}}_2(t,\ell_2)}+\|u\|^2_{{\mathbb{L}}_2(t)}).$$ Also, $$(k_{1-\beta}*{\mathbb{E}}\|u(0)+v\|^2_{L_2})(t)\leq N {\mathbb{E}}\|u(0)\|^2_{L_2}+ N k_{1-\beta}*{\mathbb{E}}\|v\|^2_{L_2}(t),$$ $$k_{1-\beta}*{\mathbb{E}}\|v\|^2_{L_2}(t)\leq N 