EXISTENCE OF CONVOLUTION MAXIMIZERS IN Lp(Rn) WITH KERNELS FROM LORENTZ SPACES.

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 R n with kernel from some L q , 1 < q < ∞ . 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 ≤ s < ∞ . [ABSTRACT FROM AUTHOR]