$E_{\beta, 1-a}(-r^{\beta}) \le N r^{-\beta}$ if $r \geq 1$ (see (\[mittag\])), we have $$\begin{aligned}
&&\int^{T|\xi|^{\frac{2}{\beta}}}_0 |\xi|^{2(\sigma+(2a-1)/{\beta})} r^{-2a}E^2_{\beta, 1-a}(-r^{\beta}) dr\\
&\leq& \int^{\infty}_0 r^{-2a} E^2_{\beta, 1-a}(-r^{\beta}) dr\\
&\le & N \left(\int_0^1 r^{-2a} dr+ \int_1^\infty r^{-2 \gamma} dr \right)<\infty.\end{aligned}$$ If $\gamma=1/2$, then since $|\xi|\geq 1$ and $\sigma<2$, $$\begin{aligned}
&&\int^{T|\xi|^{\frac{2}{\beta}}}_0|\xi|^{2(\sigma+(2a-1)/{\beta})} r^{-2a}E^2_{\beta, 1-a}(-r^{\beta})  dr\\
&=&\int^{T|\xi|^{\frac{2}{\beta}}}_0|\xi|^{-2(2-\sigma)} r^{-2a}E^2_{\beta, 1-a}(-r^{\beta})  dr\\
&\leq& \int^1_0r^{-2a} E^2_{\beta, 1-a}(-r^{\beta})dr+N
|\xi|^{-2(2-\sigma)}  \int^{T|\xi|^{\frac{2}{\beta}}}_1 r^{-1} dr \\
 &\leq&N \int_0^1 r^{-2a} dr+N|\xi|^{-2(2-\sigma)} \ln|\xi|\leq N<\infty.\end{aligned}$$ The case $\gamma<1/2$ is treated similarly using $\sigma\leq 2$. Indeed, $$\begin{aligned}
&&\int^{T|\xi|^{\frac{2}{\beta}}}_0|\xi|^{2(\sigma+(2a-1)/{\beta})} r^{-2a}E^2_{\beta, 1-a}(-r^{\beta})  dr\\
&\leq& \int^1_0 r^{-2a}E^2_{\beta, 1-a}(-r^{\beta})dr+N
|\xi|^{2(\sigma+(2a-1)/{\beta})}  \int^{T|\xi|^{\frac{2}{\beta}}}_1 r^{-2\beta} dr\\
 &\leq&N\int_0^1 r^{-2a} dr+N |\xi|^{2\sigma-4}\leq N<\infty.\end{aligned}$$ Therefore the lemma is proved.

Lemma \[lemma main estimate\] says that $u$ (which is a solution of (\[time-space\]) below) is smoother than $g$ by order $2\wedge ((1-2\gamma)\beta^{-1}+2)$ if $\gamma\neq 1/2$ and $2-\varepsilon$ if $\gamma=1/2$, where $\varepsilon>0$. Thus, for example, to estimate the second derivative of solution $u$ we need to assume $$\|g\|_{{\mathbb{H}}^{\gamma_0}_2(T,\ell_2)}<\infty \quad \text{if}\,\, \gamma\neq 1/2,  \quad \text{and}\quad  \|g\|_{{\mathbb{H}}^{\varepsilon}_2(T,\ell_2)}<\infty \quad \text{if}\,\, \gamma=1/2.$$ Recall $\gamma_0=(2\gamma-1)_+/{\beta}<2$, which is defined in (\[gamma0\]).

We first consider the equation $$\label{time-space}
    \partial^{\beta}_tu (t,x)=\Delta u(t,x)+f(t,x)
    +\sum_{k=1}^{\infty} \partial^{\gamma}_t \int^t_0 g^k(s,x)\, dW^k_s.$$ Note that by letting $\beta\to 1$ and $\gamma \to 1$ we get the classical stochastic partial differentia equations.

\[lemma formula\] Let 