Notes on $H^{\log }$: structural properties, dyadic variants, and bilinear $H^1$-$BMO$ mappings

Abstract

This article is devoted to a study of the Hardy space $H^{\log }({\mathbb{R}}^d)$ introduced by Bonami, Grellier, and Ky. We present an alternative approach to their result relating the product of a function in the real Hardy space $H^1$ and a function in $BMO$ to distributions that belong to $H^{\log }$ based on dyadic paraproducts. We also point out analogues of classical results of Hardy-Littlewood, Zygmund, and Stein for $H^{\log }$ and related Musielak-Orlicz spaces.

This article supersedes and extends a previous article arXiv:1905.13477