Weighted Weak Type estimates for non-integral Square Functions

TL;DR

This paper establishes quantitative weighted weak type bounds for non-integral square functions at the critical exponent p=2, utilizing a decoupling approach and Gehring's lemma, with extensions to other p values.

Contribution

It introduces a novel decoupling method and quantitative analysis for weighted weak type estimates of non-integral square functions at p=2.

Findings

Quantitative weak type estimates at p=2 in terms of A_p and reverse H"older constants.

Decoupling method using a quantitative version of Gehring's lemma.

Extension of results to other p in the boundedness range.

Abstract

We provide quantitative weighted weak type estimates for non-integral square functions in the critical case $p=2$ in terms of the $A_p$ and reverse H\"older constants associated to the weight. The method of proof uses a decoupling of the role of the weights via a quantitative version of Gehring's lemma. The results can be extended to other $p$ in the range of boundedness of the square function at hand.