Non-homogeneous square functions on general sets: suppression and big pieces methods

TL;DR

This paper extends local $Tb$ theorems to non-homogeneous sets using big pieces and suppression methods, introducing new criteria for boundedness of conical square functions on general sets with metric space techniques.

Contribution

It introduces a novel setting for local $Tb$ theorems involving conical square functions on arbitrary sets with non-homogeneous measures, utilizing metric space methods.

Findings

Established boundedness criteria based on weak type testing on regular balls.

Applied metric space techniques to generalize local $Tb$ theorems.

Demonstrated the effectiveness of suppression and big pieces methods in this context.

Abstract

We aim to showcase the wide applicability and power of the big pieces and suppression methods in the theory of local $Tb$ theorems. The setting is new: we consider conical square functions with cones $\{x \in \mathbb{R}^n \setminus E: |x-y| < 2 \operatorname{dist}(x,E)\}$, $y \in E$, defined on general closed subsets $E \subset \mathbb{R}^n$ supporting a non-homogeneous measure $\mu$. We obtain boundedness criteria in this generality in terms of weak type testing of measures on regular balls $B \subset E$, which are doubling and of small boundary. Due to the general set $E$ we use metric space methods. Therefore, we also demonstrate the recent techniques from the metric space point of view, and show that they yield the most general known local $Tb$ theorems even with assumptions formulated using balls rather than the abstract dyadic metric cubes.