Existence of convolution maximizers in $L_p(R^n)$ for kernels from Lorentz spaces

Abstract

The paper extends an earlier result of G.V.~Kalachev and the author (Sb. Math. 2019 or arXiv:1712.08836) on the existence of a maximizer of convolution operator acting between two Lebesgue spaces on $R^n$ with kernel from some $L_q$, $1<q<\infty$. In view of Lieb's result of 1983 about the existence of an extremizer for the Hardy-Littlewood-Sobolev inequality it is natural to ask whether a convolution maximizer exists for any kernel from weak $L_q$. The answer in the negative was given by Lieb in the above citation. In this paper we prove the existence of maximizers for kernels from a slightly more narrow class than weak $L_q$, which contains all Lorentz spaces $L_{q,s}$ with $q\leq s<\infty$.