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

## TL;DR

This paper extends classical harmonic analysis results of Zygmund and Stein to functions in the Hardy space $H^{ ext{log}}(R^d)$, introducing new bounds and applications within Orlicz spaces.

## Contribution

It provides $L ext{log} ext{log} L$ versions of classical theorems for the Hardy space $H^{ ext{log}}(R^d)$, expanding their applicability.

## Key findings

- Derived new bounds for functions in $H^{ ext{log}}(R^d)$
- Extended classical theorems to Orlicz space contexts
- Presented applications in generalized function spaces

## 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.