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 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 re...