Bounded Laplace transforms, primitives and semigroup orbits

Abstract.

Let $f : \mathbb{R}_{+} \rightarrow \mathbb{C}$ be an exponentially bounded, measurable function whose Laplace transform has a bounded holomorphic extension to the open right half-plane. It is known that there is a constant C such that $\mid \int\limits^t_0 f(s) ds \mid\, \leq C (1 + t)$ for all $t \geq 0$. We show that this estimate is sharp. Furthermore, the corresponding estimates for orbits of $C_0$-semigroups are also sharp.