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

TL;DR

This paper explores the properties of the Hardy space $H^{ ext{log}}$, offering an alternative dyadic approach to its relation with $H^1$ and $BMO$, and extends classical harmonic analysis results to this setting.

Contribution

It provides a new dyadic framework for understanding $H^{ ext{log}}$ and extends classical theorems to this space and related Musielak-Orlicz spaces.

Findings

Dyadic paraproducts characterize $H^{ ext{log}}$ functions.

Classical harmonic analysis results are extended to $H^{ ext{log}}$.

Alternative approach simplifies the relation between $H^1$, $BMO$, and $H^{ ext{log}}$.

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.