Abstract
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 \(\mathbb{R}^{n}\) plays the key role.
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Translated by A. Muravnik
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Kalachev, G.V., Sadov, S.Y. An Existence Criterion for Maximizers of Convolution Operators in \(\boldsymbol{L}_{\mathbf{1}}\boldsymbol{(\mathbb{R}^{n})}\). Moscow Univ. Math. Bull. 76, 161–167 (2021). https://doi.org/10.3103/S0027132221040033
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DOI: https://doi.org/10.3103/S0027132221040033