Balanced measures, sparse domination and complexity-dependent weight classes

TL;DR

This paper establishes sparse domination results for Haar shift operators on spaces with atomic filtrations, characterizing weight classes for boundedness that depend on the operator's complexity, with sharp qualitative results.

Contribution

It introduces a complexity-dependent weight class characterization for Haar shift boundedness using sparse domination techniques.

Findings

Sparse domination for Haar shifts is achievable under a weak regularity condition.

The weight class for boundedness depends on the complexity of the Haar shift.

Results are qualitatively sharp and provide a new perspective on weighted inequalities.

Abstract

We study sparse domination for operators defined with respect to an atomic filtration on a space equipped with a general measure $\mu$. In the case of Haar shifts, $L^p$-boundedness is known to require a weak regularity condition, which we prove to be sufficient to have a sparse domination-like theorem. Our result allows us to characterize the class of weights where Haar shifts are bounded. A surprising novelty is that said class depends on the complexity of the Haar shift operator under consideration. Our results are qualitatively sharp.