The Hardy Space $H^1$ on Non-homogeneous Metric Spaces

TL;DR

This paper develops the theory of atomic Hardy spaces on non-homogeneous metric measure spaces, establishing duality with RBMO and criteria for operator boundedness, including Calderón–Zygmund operators.

Contribution

It introduces the atomic Hardy space $H^1(rac{ ext{mu}})$ on non-homogeneous spaces and proves its duality with RBMO, extending harmonic analysis tools.

Findings

Duality between $H^1(rac{ ext{mu}})$ and RBMO established.

Criterion for boundedness of linear operators from $H^1(rac{ ext{mu}})$ to Banach spaces.

Boundedness of Calderón–Zygmund operators from $H^1(rac{ ext{mu}})$ to $L^1(rac{ ext{mu}})$ proved.

Abstract

Let $({\mathcal X}, d, \mu)$ be a metric measure space and satisfy the so-called upper doubling condition and the geometrical doubling condition. In this paper, we introduce the atomic Hardy space $H^1(\mu)$ and prove that its dual space is the known space ${\rm RBMO}(\mu)$ in this context. Using this duality, we establish a criterion for the boundedness of linear operators from $H^1(\mu)$ to any Banach space. As an application of this criterion, we obtain the boundedness of Calder\'on--Zygmund operators from $H^1(\mu)$ to $L^1(\mu)$.