Anomaly of the fractional heat propagator in abstract settings

We study the following time-fractional heat equation: \begin{equation*} ^{C}\partial_{t}^αu(t)+\mathscr{L}u(t)=0,\quad u(0)=u_0\in X, \quad t\in[0,T],\quad T>0,\quad 0<α<1, \end{equation*} where ${}^C{\mathrm{\partial }}_t^{\alpha }$ is the Djrbashian-Caputo fractional derivative, $X$ is a complex Banach space and $\mathcal{L}:\mathcal{D}(\mathcal{L})\subset X\to X$ is a closed linear operator. The solution operator of the equation above is given by the strongly continuous operator $E_{\alpha }(-t^{\alpha }\mathcal{L})$ for any $t⩾0$, closely related with the Mittag-Leffler function $E_{\alpha }(-x)$ for $x⩾0.$ There are different ways to present explicitly this operator and one of the most popular is given in terms of the $C_0$-semigroup generated by $-\mathcal{L}$ $(\{e^{-t\mathcal{L}}{\}}_{t⩾0})$ as follows: $$E_{\alpha }(-t^{\alpha }\mathcal{L})={\int }_0^{+\mathrm{\infty }}{M}_{\alpha }(s)e^{-s{t}^{\alpha }\mathcal{L}}\mathrm{d}s,t⩾0,$$ where $M_{\alpha }(s)$ is a Wright-type function. We will see that the latter expression is not always optimal (regarding restrictions: endpoint lost) to estimate different norms. An additional restriction appears while bounding the above integral, which can be avoided by using directly the function itself and its well-known uniform bound $\mathrm{\mid }{E}_{\alpha }(-x)\mathrm{\mid }⩽\frac{C}{1+x},$ $x⩾0.$