Factorization for Hardy spaces and characterization for BMO spaces via commutators in the Bessel setting

Abstract

Fix $\lambda >0$. Consider the Hardy space $H^1({\mathbb{R}}_{+},d{m}_{\lambda })$ in the sense of Coifman and Weiss, where ${\mathbb{R}}_{+}:=(0,\infty )$ and $d{m}_{\lambda }:=x^{2\lambda }dx$ with $dx$ the Lebesgue measure. Also consider the Bessel operators ${\Delta }_{\lambda }:=-\frac{d^2}{d{x}^2}-\frac{2\lambda }{x}\frac{d}{dx}$, and $S_{\lambda }:=-\frac{d^2}{d{x}^2}+\frac{{\lambda }^2-\lambda }{x^2}$ on ${\mathbb{R}}_{+}$. The Hardy spaces $H_{{\Delta }_{\lambda }}^1$ and $H_{S_{\lambda }}^1$ associated with ${\Delta }_{\lambda }$ and $S_{\lambda }$ are defined via the Riesz transforms $R_{{\Delta }_{\lambda }}:={\partial }_x({\Delta }_{\lambda }{)}^{-1/2}$ and $R_{S_{\lambda }}:=x^{\lambda }{\partial }_x{x}^{-\lambda }(S_{\lambda }{)}^{-1/2}$, respectively. It is known that $H_{{\Delta }_{\lambda }}^1$ and $H^1({\mathbb{R}}_{+},d{m}_{\lambda })$ coincide but they are different from $H_{S_{\lambda }}^1$. In this article, we prove the following: (a) a weak factorization of $H^1({\mathbb{R}}_{+},d{m}_{\lambda })$ by using a bilinear form of the Riesz transform $R_{{\Delta }_{\lambda }}$, which implies the characterization of the BMO space associated to ${\Delta }_{\lambda }$ via the commutators related to $R_{{\Delta }_{\lambda }}$; (b) the BMO space associated to $S_{\lambda }$ can not be characterized by commutators related to $R_{S_{\lambda }}$, which implies that $H_{S_{\lambda }}^1$ does not have a weak factorization via a bilinear form of the Riesz transform $R_{S_{\lambda }}$.

v2: 21 pages

References in corpus (1)

• Characterizations of $H_{{\Delta }_N}^1({\mathbb{R}}^n)$ and $\mathrm{BM}{\mathrm{O}}_{{\mathrm{\Delta }}_{\mathrm{N}}}({\mathbb{R}}^{\mathrm{n}})$ via Weak Factorizations and Commutators

Cited by in corpus (1)

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