An Interpolation Theorem for Sublinear Operators on Non-homogeneous Metric Measure Spaces

TL;DR

This paper proves an interpolation theorem for sublinear operators on non-homogeneous metric measure spaces, extending boundedness results from Hardy and BMO spaces to all L^p spaces.

Contribution

It establishes a new interpolation result for sublinear operators on non-homogeneous metric measure spaces, improving previous bounds.

Findings

Sublinear operators bounded from Hardy space to weak L^1 and from L^0 to RBMO are also bounded on all L^p spaces.

The result extends classical interpolation theorems to non-homogeneous metric measure spaces.

The theorem applies under upper doubling and geometrically doubling conditions.

Abstract

Let $({\mathcal X}, d, \mu)$ be a metric measure space and satisfy the so-called upper doubling condition and the geometrically doubling condition. In this paper, the authors establish an interpolation result that a sublinear operator which is bounded from the Hardy space $H^1(\mu)$ to $L^{1,\,\infty}(\mu)$ and from $L^\infty(\mu)$ to the BMO-type space ${\mathop\mathrm{RBMO}}(\mu)$ is also bounded on $L^p(\mu)$ for all $p\in(1,\,\infty)$. This extension is not completely straightforward and improves the existing result.