Boundedness of Convolution Operators and Input–Output Maps between Weighted Spaces

Abstract

We ask when convolution operators with scalar- or operator-valued kernel functions map between weighted L2 spaces of Hilbert space-valued functions. For a certain class of decreasing weights, including negative powers (t + a)−m for example, we solve the one-weight problem completely by using Laplace transforms and Bergman-type spaces of vector-valued analytic functions. For a much more general class of decreasing weights, we solve the one-weight problem for all positive real kernels (also for Lp(w) with p > 1), by results on Steklov operators which generalise the weighted Hardy inequality. When the kernel function is a strongly continuous semigroup of bounded linear Hilbert space operators, which arises from input–output maps of certain linear systems, then the most obvious sufficient condition for boundedness, obtained by taking norm signs inside the integrals, is also necessary in many cases, but not in general.