The paper studies sharp weighted $L^p$ inequalities for the martingale maximal function. Proofs exploit properties of certain special functions of four variables and self-improving properties of $A_p$ weights.
"Sharp $L^p$-bounds for the martingale maximal function." Tohoku Math. J. (2) 70 (1) 121 - 138, 2018. https://doi.org/10.2748/tmj/1520564421