Regularity of semigroups for exponentially tempered stable processes with drift

In this paper, we study the regularity of semigroups for the exponentially tempered 2 ss-stable processes. First, we use the Fourier analysis technique to prove that for 1 <= p <8, the semigroup for the tempered stable process without drift is analytic in L-p(R-n). Next, it is shown that the semigroup for the tempered stable process with drift is an analytic semigroup if 1/2 <= ss< 1and the semigroup is a Gevrey semigroup of order dwith delta> 1/(2 ss) if 0 < ss< 1/2. Finally, we find that the result on Gevrey regularity is optimal in some sense. (c) 2023 Elsevier Inc. All rights reserved.