This article is cited in 1 paper

Mathematics

An existence criterion for maximizers of convolution operators in $L_1(\mathbb{R}^n)$

G. V. Kalachev^a, S. Yu. Sadov

^a Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The operator of convolution 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 convolution, is given. Analysis of measurable solutions of Pexider's functional equation defined on subsets of positive measure in $\mathbb{R}^n$ plays the key role.

Key words: convolution operator, $L_1$ space, maximizer, Pexider's equation, Cauchy's functional equation, measurable solution.

UDC: 517.51, 517.965

Received: 13.12.2019

 English version: Moscow University Mathematics Bulletin, Moscow University Mеchanics Bulletin, 2021, 76:4, 161–167

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