On Fubini Type Property in Lorentz Spaces

Abstract

We study Fubini-type property for Lorentz spaces ${L}^{p,r}({\mathbb{R}}^{2})$. This problem is twofold. First we assume that all linear sections of a function f in directions of coordinate axes belong to ${L}^{p,r}(\mathbb{R})$, and their one-dimensional L ^{p, r}-norms belong to ${L}^{p,r}(\mathbb{R}).$ We show that for p ≠ r it does not imply that $f \in {L}^{p,r}({\mathbb{R}}^{2})$ (this complements one result by Cwikel). Conversely, we assume that $f \in {L}^{p,r}({\mathbb{R}}^{2})$, and we show that then for r < p almost all linear sections of f belong to ${L}^{p,r}(\mathbb{R})$, but for p < r all linear sections may have infinite one-dimensional L ^{p, r}-norms.