Regularity of algebraically tempered stable semigroups

In the present letter, we deal with the nonlocal operators which are the generators of the tempered stable processes. It is shown that the semigroups generated by the nonlocal operators are analytic in \(L^p(\mathbb {R}^d)\) $L^p({\mathbb{R}}^d)$ for any \(p \in [1,\infty )\) $p\in [1,\infty )$ . In particular, the result holds for not only exponentially tempered stable processes but also algebraically tempered ones.