Weak factorization of Hardy spaces in the Bessel setting

TL;DR

This paper establishes a weak factorization for Hardy spaces in the Bessel setting and characterizes the boundedness of certain commutators, extending previous work through generalizations and adaptations.

Contribution

It provides the first weak factorization of Hardy spaces in the Bessel setting and characterizes commutator boundedness for Lipschitz functions, generalizing prior results.

Findings

Weak factorization of Hardy spaces in the Bessel setting established.

Characterization of commutator boundedness for Lipschitz functions obtained.

Results extend and modify previous work by Duong, Li, Yang, and Uchiyama.

Abstract

We provide the weak factorization of the Hardy spaces $H^{p}(\mathbb{R}_+, dm_{\lambda})$ in the Bessel setting, for $p\in \left(\frac{2\lambda + 1}{2\lambda + 2}, 1\right]$. As a corollary we obtain a characterization of the boundedness of the commutator $[b, R_{\Delta_{\lambda}}]$ from $L^{q}(\mathbb{R}_+, dm_{\lambda})$ to $L^{r}(\mathbb{R}_+, dm_{\lambda})$ when $b\in \textrm{Lip}_{\alpha}(\mathbb{R}_+, dm_{\lambda})$ provided that $\alpha = \frac{1}{q} - \frac{1}{r}$. The results are a slight generalization and modification of the work of Duong, Li, Yang, and the second named author, which in turn are based on modifications and adaptations of work by Uchiyama.