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) 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 < ∞. 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 ≤ s < ∞.