EXISTENCE OF CONVOLUTION MAXIMIZERS IN $L_p(\mathbb {R}^n)$ WITH KERNELS FROM LORENTZ SPACES

Abstract

The paper extends an earlier result of G.V. Kalachev et al. (Sb. Math. 210(8):1129–1147, 2019) on the existence of a maximizer of convolution operator acting between two Lebesgue spaces on $\mathbb {R}^n$ with kernel from some $L_q$, $1<q<\infty$. On the other hand, E. Lieb (Ann. of Math. 118:(2):349–374, 1983) proved the existence of a maximizer for the Hardy-Littlewood-Sobolev inequality and remarked that in general a convolution maximizer for a kernel from weak $L_q$ may not exist. In this paper we axiomatize some properties used in the proof of the Kalachev-Sadov 2019 theorem and obtain a more general result. As a consequence, we prove that the convolution maximizer always exists for kernels from a slightly more narrow class than weak $L_q$, which contains all Lorentz spaces $L_{q,s}$ with $q\le s<\infty$.