Hardy spaces $H^p$ over non-homogeneous metric measure spaces and their applications

TL;DR

This paper develops Hardy spaces over non-homogeneous metric measure spaces, introduces atomic and molecular variants, and studies their properties, boundedness of operators, and duality relations, extending classical harmonic analysis to more general settings.

Contribution

It introduces new Hardy space frameworks on non-homogeneous spaces, proves boundedness of Calderón-Zygmund operators, and establishes duality and independence results for these spaces.

Findings

Boundedness of Calderón-Zygmund operators on the new Hardy spaces.

Introduction of the $ ho$-weakly doubling condition for measures.

Identification of the Hardy space as the predual of a Campanato space.

Abstract

Let $({\mathcal X},d,\mu)$ be a metric measure space satisfying both the geometrically doubling and the upper doubling conditions. Let $\rho\in (1,\infty)$, $0<p\le1\le q\le\infty$, $p\neq q$, $\gamma\in[1,\infty)$ and $\epsilon\in(0,\infty)$. In this article, the authors introduce the atomic Hardy space ${\widetilde H_{\mathrm{atb},\,\rho}^{p,\,q,\,\gamma}(\mu)}$ and the molecular Hardy space ${\widetilde H_{\rm{mb},\,\rho}^{p,\,q,\,\gamma,\,\epsilon}(\mu)}$ via the discrete coefficient $\widetilde{K}^{(\rho),\,p}_{B,\,S}$, and prove that the Calder\'on-Zygmund operator is bounded from ${\widetilde H_{\rm{mb},\,\rho}^{p,\,q,\,\gamma,\,\delta}(\mu)}$ (or ${\widetilde H_{\rm{atb},\,\rho}^{p,\,q,\,\gamma}(\mu)}$) into $L^p(\mu)$, and from ${\widetilde H_{\rm{atb},\,\rho(\rho+1)}^{p,\,q,\,\gamma+1}(\mu)}$ into ${\widetilde H_{\rm{mb},\,\rho}^{p,\,q,\,\gamma,\,\frac12(\delta -\frac{\nu}{p}+\nu)}(\mu)}$ whose genealized fractional versions are also obtained. The authors also introduce the $\rho$-weakly doubling condition, with $\rho\in (1,\infty)$, of the measure $\mu$ and construct a non-doubling measure $\mu$ satisfying this condition. If $\mu$ is $\rho$-weakly doubling, the authors further introduce the Campanato space ${\mathcal E}^{\alpha,\,q}_{\rho,\,\eta,\,\gamma}(\mu)$ and show that ${\mathcal E}^{\alpha,\,q}_{\rho,\,\eta,\,\gamma}(\mu)$ is independent of the choices of $\rho$, $\eta$, $\gamma$ and $q$; the authors then introduce the atomic Hardy space $\widehat H_{\rm{atb},\,\rho}^{p,\,q,\,\gamma}(\mu)$ and the molecular Hardy space $\widehat H_{\rm{mb},\,\rho}^{p,\,q,\,\gamma,\,\epsilon}(\mu)$, which coincide with each other; the authors finally prove that $\widehat{H}_{\rm{atb},\,\rho}^{p,\,q,\,\gamma}(\mu)$ is the predual of ${\mathcal E}^{1/p-1,\,1}_{\rho,\,\rho,\,1}(\mu)$. Moreover, relations of these Hardy spaces are also discussed.