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

TL;DR

This paper proves the existence of convolution maximizers in Lorentz spaces, extending previous results from Lebesgue spaces and addressing a question about kernels from weak $L_q$ spaces.

Contribution

It establishes the existence of convolution maximizers for kernels in Lorentz spaces $L_{q,s}$, broadening the class of kernels beyond weak $L_q$ spaces.

Findings

Maximizers exist for kernels in Lorentz spaces $L_{q,s}$ with $q \\leq s<\\infty$.

Extends previous results from Lebesgue spaces to Lorentz spaces.

Addresses the question of extremizers for kernels from weak $L_q$ 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 $q\leq s<\infty$.