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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 convolutionoperator 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 anextremizer for the Hardy-Littlewood-Sobolev inequality it is natural to askwhether a convolution maximizer exists for any kernel from weak $L_q$. Theanswer in the negative was given by Lieb in the above citation. In this paperwe prove the existence of maximizers for kernels from a slightly more narrowclass than weak $L_q$, which contains all Lorentz spaces $L_{q,s}$ with $q\leqs<\infty$.

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