Weighted maximal inequality for differentially subordinate martingales

Abstract

We establish a weighted maximal $L^1$-inequality for differentially subordinate martingales taking values in $\mathbb{R} ^\nu $, $\nu \geq 1$, under the assumption that the weight satisfies Muckenhoupt’s condition $A_1$. An optimal dependence of the constant on the $A_1$ characteristics is identified.