Endpoint estimates for Haar shift operators with balanced measures

TL;DR

This paper establishes endpoint inequalities for Haar shift operators with non-homogeneous measures, providing streamlined proofs of existing $L^p$ results and analyzing regularity properties in martingale Lipschitz spaces.

Contribution

It introduces endpoint estimates for Haar shifts with non-homogeneous measures and offers a simplified proof of known $L^p$ bounds, also exploring regularity and measure sharpness.

Findings

Proved $ ext{H}^1$ and $ ext{BMO}$ endpoint inequalities for Haar shifts.

Provided a streamlined proof of existing $L^p$ results.

Analyzed regularity properties and measure sharpness for Haar shift operators.

Abstract

We prove $\mathrm{H}^1$ and $\mathrm{BMO}$ endpoint inequalities for generic cancellative Haar shifts defined with respect to a possibly non-homogeneous Borel measure $\mu$ satisfying a weak regularity condition. This immediately yields a new, highly streamlined proof of the $L^p$-results for the same operators due to L\'opez-Sanchez, Martell, and Parcet. We also prove regularity properties for the Haar shift operators on the natural martingale Lipschitz spaces defined with respect to the underlying dyadic system, and show that the class of measures that we consider is sharp.