a,b>0.$$ The lemma is proved.

Define $$p(t,x)={\mathcal{F}}^{-1} (E_{\beta}(-|\xi|^{2}t^{\beta})), \quad q(t,x)=D^{1-\beta}_t p.$$ Actually (\[mittag\]) shows that if $d>1$ then $E_{\beta}(-|\xi|^2t^{\beta})\not\in L_1({\mathbb{R}}^d)$ for fixed $t>0$. Thus we understand $p(t,x)$ as the inverse transform of a radial function in the sense of improper integral, or we can define $p(t,x)$ first as in [@EIK Section 5.2.2] so that $p(t,\cdot)\in L_1({\mathbb{R}}^d)$ and ${\mathcal{F}}(p(t,x))(\xi)=E_{\beta}(-|\xi|^2t^{\beta})$.

Since, for $x\neq 0$, $p(t,x)\to 0$ as $t\downarrow 0$ (see [@EIK]), the Riemann-Liouville derivative of $p(\cdot, x)$ coincides with the Caputo derivative of $p(\cdot , x)$ for every $x\not= 0$.

Here is a list of other useful properties of $p$ and $q$.

\[lemma list\] (i) For $(t,x)\in (0,T]\times {\mathbb{R}}^d \setminus \{0\}$, we have $\partial^{\beta}_t p=D^{\beta}_t p=\Delta p$.

\(ii) For each $x\neq 0$ and $m\leq 3$, $$\label{eqn 9.21.1}
\lim_{t\to 0+} D^m_x p(t,x)=\lim_{t\to 0+}D^m_x q(t,x)=0.$$

\(iii) $D^m_x p(t,\cdot)$ is integrable for each $t>0$ if $m\leq 1$.

\(iv) For each $t>0$ and $m\leq 3$, $D^m_x q(t,\cdot)$ is integrable in ${\mathbb{R}}^d$ uniformly on $[\varepsilon,T]$ for any $\varepsilon>0$.

\(v) For each $x\neq 0$, $p(\cdot,x)$ is absolutely continuous and $\frac{
\partial}{ \partial t}p(t,x)\to 0$ as $t\downarrow 0$. Moreover, $\frac{\partial}{ \partial t}p(t,\cdot)$ is integrable in ${\mathbb{R}}^d$ uniformly on $[\varepsilon,T]$ for any $\varepsilon>0$.

\(vi) For any compact set $K\subset {\mathbb{R}}^d\setminus 