Sharp weighted norm estimates for martingale square functions

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.