On the convolution operators arising in the study of abstract initial boundary value problems

We consider convolution operators arising in the study of abstract initial boundary value problems. These operators are of the form g-->Gamma g(t)=(zI-A(0)) integral(0)(t) S(t-s)K(z)g(s) ds, t greater than or equal to 0, where {S(t)}(t greater than or equal to 0) is a C-0-semigroup in a Banach space X, with infinitesimal generator A(0):D(A(0)) subset of X-->X, and K(z): Y-->X is a linear, continuous mapping defined in another Banach space Y. We study the continuity of Gamma between the spaces L(p)[(0), + infinity), Y) and L(q)([0, + infinity), X), 1 less than or equal to p, q less than or equal to + infinity. We give several examples of the applicability of the results to some familiar initial boundary value problems, including both parabolic and hyperbolic cases.