Boundedness of Calder\'on-Zygmund Operators on Non-homogeneous Metric Measure Spaces

TL;DR

This paper establishes the equivalence of boundedness conditions for Calderón-Zygmund operators on non-homogeneous metric measure spaces with specific geometric and measure-theoretic properties, extending classical results.

Contribution

It generalizes the boundedness criteria for Calderón-Zygmund operators to non-homogeneous spaces satisfying upper doubling and geometrical doubling conditions.

Findings

Boundedness on L^2(μ) is equivalent to boundedness on L^p(μ) for some p in (1, ∞).

Calderón-Zygmund operators bounded on L^2(μ) have maximal operators bounded on all L^p(μ).

Results extend classical theorems to more general metric measure spaces.

Abstract

Let $({\mathcal X}, d, \mu)$ be a separable metric measure space satisfying the known upper doubling condition, the geometrical doubling condition and the non-atomic condition that $\mu(\{x\})=0$ for all $x\in{\mathcal X}$. In this paper, we show that the boundedness of a Calder\'on-Zygmund operator $T$ on $L^2(\mu)$ is equivalent to that of $T$ on $L^p(\mu)$ for some $p\in (1, \infty)$, and that of $T$ from $L^1(\mu)$ to $L^{1,\,\infty}(\mu).$ As an application, we prove that if $T$ is a Calder\'on-Zygmund operator bounded on $L^2(\mu)$, then its maximal operator is bounded on $L^p(\mu)$ for all $p\in (1, \infty)$ and from the space of all complex-valued Borel measures on ${\mathcal X}$ to $L^{1,\,\infty}(\mu)$. All these results generalize the corresponding results of Nazarov et al. on metric spaces with measures satisfying the so-called polynomial growth condition.