A two weight inequality for Calder\'{o}n-Zygmund operators on spaces of homogeneous type with applications

TL;DR

This paper characterizes the boundedness of Calderón-Zygmund operators between weighted L2 spaces on spaces of homogeneous type using an A2 condition and testing conditions, extending classical results to more general metric measure spaces.

Contribution

It provides a new characterization of two-weight inequalities for Calderón-Zygmund operators on spaces of homogeneous type, incorporating a pivotal side condition and advanced decomposition techniques.

Findings

Established necessary and sufficient conditions for boundedness in the two-weight setting.

Extended classical Calderón-Zygmund theory to spaces of homogeneous type.

Utilized stopping cubes and corona decompositions for the proof.

Abstract

Let $(X,d,\mu )$ be a space of homogeneous type in the sense of Coifman and Weiss, i.e. $d$ is a quasi metric on $X$ and $\mu $ is a positive measure satisfying the doubling condition. Suppose that $u$ and $v$ are two locally finite positive Borel measures on $(X,d,\mu )$. Subject to the pair of weights satisfying a side condition, we characterize the boundedness of a Calder\'{o}n--Zygmund operator $T$ from $L^{2}(u)$ to $L^{2}(v)$ in terms of the $A_{2}$ condition and two testing conditions. For every cube $B\subset X$, we have the following testing conditions, with $\mathbf{1}_{B}$ taken as the indicator of $B$ \begin{equation*} \Vert T(u\mathbf{1}_{B})\Vert _{L^{2}(B, v)}\leq \mathcal{T}\Vert 1_{B}\Vert _{L^{2}(u)}, \end{equation*} \begin{equation*} \Vert T^{\ast }(v\mathbf{1}_{B})\Vert _{L^{2}(B, u)}\leq \mathcal{T}\Vert 1_{B}\Vert _{L^{2}(v)}. \end{equation*} The proof uses stopping cubes and corona decompositions originating in work of Nazarov, Treil and Volberg, along with the pivotal side condition.