Sharp weighted norm estimates for martingale square functions

TL;DR

This paper establishes sharp weighted norm estimates for martingale square functions in scalar and matrix-weighted contexts, utilizing sparse domination techniques and matrix $A_p$ conditions.

Contribution

It introduces a new characterization of $L_p$ estimates for martingale square functions using matrix weights and achieves sharp bounds in the matrix-weighted setting.

Findings

Sharp $L_p$ estimates for matrix-weighted martingale square functions for $1<p extless 2$.

Optimal exponent bounds for scalar-weighted martingale square functions for all $1<p<\infty$.

Use of sparse domination to derive explicit characteristic bounds.

Abstract

This paper is devoted to the study of quantitative weighted norm estimates for martingale square functions in both scalar-weighted and matrix-weighted settings. In particular, we introduce the martingale square functions $S_W$ via matrix weights $W$, and then use the matrix $A_p$ condition, introduced in our previous work \cite{ChenQuanJiaoWu}, to characterize the $L_p$ estimate for $S_W$. Our proof mainly relies on the idea of sparse dominations, which leads to the explicit information on the characteristic of the matrix weight involved. For the range $1<p\leq 2$, our result is sharp in terms of the characteristic of the matrix weight. With some modification on the arguments, we can further improve the result in scalar settings by obtaining the optimal exponent of the characteristic of the weight involved for all indices $1<p<\infty$, addressing a fundamental problem from the classical martingale theory.