Sparse domination on non-homogeneous spaces with an application to $A_p$ weights

TL;DR

This paper extends sparse domination techniques for Calderón--Zygmund operators to non-homogeneous metric spaces, providing sharp weighted estimates involving $A_p$ weights, which broadens the applicability of such methods.

Contribution

It introduces a new sparse domination theorem for upper doubling metric spaces, differing from recent results, and achieves sharp weighted bounds with respect to $A_p$ weights.

Findings

Established sparse domination in non-homogeneous spaces.

Derived sharp weighted $A_p$ bounds for Calderón--Zygmund operators.

Extended the scope of sparse domination techniques beyond doubling measures.

Abstract

We extend Lerner's recent approach to sparse domination of Calder\'on--Zygmund operators to upper doubling (but not necessarily doubling), geometrically doubling metric measure spaces. Our domination theorem is different from the one obtained recently by Conde-Alonso and Parcet and yields a weighted estimate with the sharp power $\max(1,1/(p-1))$ of the $A_p$ characteristic of the weight.