  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 $q\leq
s<\infty$.
