Convergence at the origin of integrated semigroups
Volume 187 / 2008
Abstract
We study a classification of $\kappa$-times integrated semigroups (for $\kappa>0$) by their (uniform) rate of convergence at the origin: $\|S(t)\|={\cal O}(t^\alpha)$ as $t\rightarrow 0$ (${0\leq\alpha\leq\kappa}$). By an improved generation theorem we characterize this behaviour by Hille–Yosida type estimates. Then we consider integrated semigroups with holomorphic extension and characterize their convergence at the origin, as well as the existence of boundary values, by estimates of the associated holomorphic semigroup. Various examples illustrate these results. The particular case $\alpha=\kappa$, 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.
