Let f be a martingale with values in a uniformly p-smooth Banach space and w any positive weight. We show that , where is the martingale maximal operator and is the sum of martingale increments.
I thank Mark Veraar for making me aware of the recent developments concerning vector-valued extensions of Burkholder–Davis–Gundy inequalities. I thank the anonymous referee for helpful improvement suggestions.
" Fefferman–Stein inequality for maximal functions of martingales in uniformly smooth spaces." Electron. J. Probab. 26 1 - 18, 2021. https://doi.org/10.1214/21-EJP680