$L\log \log L$ versions of Stein's and Zygmund's theorems for the Hardy space $H^{\log }({\mathbb{R}}^d)$

Abstract

We obtain versions of some classical results of Zygmund and Stein for functions belonging to the Hardy space $H^{\log }({\mathbb{R}}^d)$ introduced by Bonami, Grellier, and Ky. We present further applications in the context of more general Orlicz spaces.