This article is cited in 2 papers

On maximizers of a convolution operator in $L_p$-spaces

G. V. Kalachev^a, S. Yu. Sadov<sup>b

a</sup> Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
^b Moscow, Russia

Abstract: The paper is concerned with convolution operators in $\mathbb R^d$, whose kernels are in $L_q$, which act from $L_p$ into $L_s$, where $1/p+1/q=1+1/s$. It is shown that for $1<q,p,s<\infty$ there exists a maximizer (a function with $L_p$-norm $1$) at which the supremum of the $s$-norm of the convolution is attained. A special analysis is carried out for the cases in which one of the exponents $q,p$, or $s$ is $1$ or $\infty$.
Bibliography: 12 titles.

Keywords: convolution, Young inequality, existence of an extremal function, tight sequence, concentration compactness.

UDC: 517.44+517.972.4

MSC: 44A35, 46E30, 49J99

Received: 18.03.2018 and 16.01.2019

DOI: 10.4213/sm9099

 English version: Sbornik: Mathematics, 2019, 210:8, 1129–1147

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