Bounded Convolutions and Solutions of Inhomogeneous Cauchy Problems
نویسنده
چکیده
Let X be a complex Banach space, 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 0 (t) = Au(t) + f(t) (t 0), 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.
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