A simple proof of the sharp weighted estimate for Calderon-Zygmund operators on homogeneous spaces

TL;DR

This paper provides a simplified proof of the $A_2$ conjecture for Calderon-Zygmund operators on homogeneous spaces using Lerner’s local mean oscillation method, extending known results beyond Euclidean spaces.

Contribution

It introduces a straightforward proof technique for the $A_2$ conjecture applicable to spaces of homogeneous type, generalizing previous Euclidean and geometrically doubling space results.

Findings

Proves the $A_2$ conjecture for homogeneous spaces

Demonstrates linear dependence of bounds on the $A_2$ norm

Simplifies existing proofs using Lerner's method

Abstract

Here we show that Lerner's method of local mean oscillation gives a simple proof of the $A_2$ conjecture for spaces of homogeneous type: that is, the linear dependence on the $A_2$ norm for weighted $L^2$ Calderon-Zygmund operator estimates. In the Euclidean case, the result is due to Hyt\"{o}nen, and for geometrically doubling spaces, Nazarov, Rezinikov, and Volberg obtained the linear bound.