This is the abstract of the paper "Bounded Convolutions and Solutions of Inhomogeneous Cauchy Problems" by C.J.K. Batty and R. Chill. For the whole paper send the command get evolve-l 98-00037 to: "listserv@rz.uni-karlsruhe.de". Instead you may get the paper at "http://ma1serv.mathematik.uni-karlsruhe.de/evolve-l/index.html" too. Abstract: Let $X$ be a complex Banach space, $T : R_+ \to B(X)$ and $f : R_+ \to X$ be bounded functions, and suppose that the singular points of the Laplace transforms of $T$ and $f$ do not coincide. Under various supplementary assumptions, we show that the convolution $T*f$ is bounded. When $T(t)=I$, this is a classical result of Ingham. Our results are applied to mild solutions of inhomogeneous Cauchy problems on $R_+$: $u'(t) = Au(t) + f(t) \; (t\ge0)$, where $A$ is the generator of a bounded $C_0$-semigroup on $X$. For holomorphic semigroups, a result of this type has been obtained by Basit.