Convergence at the origin of integrated semigroups

Abstract

We study a classification of the kappa-times integrated semigroups (for kappa>0) by the (uniform) rate of convergence at the origin: $\|S(t)\|=O(t^α)$, $0\leqα\leqκ$. By an improved generation theorem we characterize this behaviour by Hille-Yosida type estimates. Then we consider integrated semigroups with holomorphic extension and characterize the convergence at the origin, as well as the existence of boundary values, by estimates of the associated holomorphic semigroup. Different examples illustrate these results. The particular case $α=κ$, which corresponds to the notions of Riesz means or tempered integrated semigroups, is of special interest : as an application, it leads to an integrated version of Euler's exponential formula.