Bounded convolutions and solutions of inhomogeneous Cauchy problems

Let X be a complex Banach space, T: R+ --> B(X) and f: R+ --> X be bounded functions, and suppose that the singular points of the Laplace transforms o f 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 inhomogeneou s Cauchy problems on R+: u'(t) = Au(t) + f(t) (t greater than or equal to 0 ), where A is the generator of a bounded C-0-semigroup on X. For holomorphi c semigroups, a result of this type has been obtained by Basit. 1