An Existence Criterion for Maximizers of Convolution Operators in L1(Rn)

The convolution operator with a complex-valued integrable kernel in the space of integrable functions is considered; a necessary and sufficient condition for the existence of a maximizer, i.e., a norm-one function that maximizes the norm of the convolution, is given. The analysis of measurable solutions of Pexider's functional equation defined on subsets of positive measure in R-n plays the key role.