Geodesic
An existence criterion for maximizers of convolution operators in $L_1(\mathbb{R}^n)$
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 4 (2021), pp. 17-22

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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. 
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     author = {G. V. Kalachev and S. Yu. Sadov},
     title = {An existence criterion for maximizers of convolution operators in $L_1(\mathbb{R}^n)$},
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     url = {http://geodesic.mathdoc.fr/item/VMUMM_2021_4_a2/}
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G. V. Kalachev; S. Yu. Sadov. An existence criterion for maximizers of convolution operators in $L_1(\mathbb{R}^n)$. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 4 (2021), pp. 17-22. http://geodesic.mathdoc.fr/item/VMUMM_2021_4_a2/