Square Function Estimates in Spaces of Homogeneous Type and on Uniformly Rectifiable Euclidean Sets

TL;DR

This paper develops a local $T(b)$ theorem and an inductive scheme to establish $L^p$ square function estimates for integral operators on Ahlfors-David regular and uniformly rectifiable sets, extending analysis in quasi-metric spaces.

Contribution

It introduces a new inductive scheme based on a local $T(b)$ theorem that ensures stability of $L^2$ square function estimates on complex sets, advancing the understanding of integral operators on irregular sets.

Findings

Established $L^p$ square function estimates for uniformly rectifiable sets.

Proved stability of $L^2$ estimates under big pieces functor.

Extended analysis to Ahlfors-David regular sets in quasi-metric spaces.

Abstract

We announce a local $T(b)$ theorem, an inductive scheme, and $L^p$ extrapolation results for $L^2$ square function estimates related to the analysis of integral operators that act on Ahlfors-David regular sets of arbitrary codimension in ambient quasi-metric spaces. The inductive scheme is a natural application of the local $T(b)$ theorem and it implies the stability of $L^2$ square function estimates under the so-called big pieces functor. In particular, this analysis implies $L^p$ and Hardy space square function estimates for integral operators on uniformly rectifiable subsets of the Euclidean space.