This is the abstract of the paper "Bounded Laplace transforms, primitives and semigroup orbits" by Charles J.K. Batty. For the whole paper send the command get evolve-l 01-00005 to: "listserv@uni-karlsruhe.de". Instead you may get the paper at "http://ma1serv.mathematik.uni-karlsruhe.de/evolve-l/index.html" too. Abstract: Let $f : \R_+ \to \C$ 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 $|\int_0^t f(s)\,ds| \le C(1+t)$ for all $t\ge0$. We show that this estimate is sharp. Furthermore, the corresponding estimates for orbits of $C_0$-semigroups are also sharp. ------------------ http://www.uni-karlsruhe.de/~listserv/ -------------------