Weak Factorizations of the Hardy space $H^1(\mathbb{R}^n)$ in terms of Multilinear Riesz Transforms

TL;DR

This paper presents a constructive proof for weak factorizations of the Hardy space $H^1(R^n)$ using multilinear Riesz transforms, leading to a new characterization of BMO via commutators.

Contribution

It provides a constructive proof of weak factorizations of $H^1(R^n)$ and introduces a new approach to characterize BMO through multilinear Riesz transform commutators.

Findings

Constructive proof of weak factorizations of $H^1(R^n)$.

New proof of BMO characterization via commutators.

Enhanced understanding of multilinear Riesz transforms in harmonic analysis.

Abstract

This paper provides a constructive proof of the weak factorizations of the classical Hardy space $H^1(\mathbb{R}^n)$ in terms of multilinear Riesz transforms. As a direct application, we obtain a new proof of the characterization of ${\rm BMO}(\mathbb{R}^n)$ (the dual of $H^1(\mathbb{R}^n)$) via commutators of the multilinear Riesz transforms.