A weighted maximal inequality for differentially subordinate martingales

Abstract

The paper contains the proof of a weighted Fefferman-Stein inequality in a probabilistic setting. Suppose that f = ( f n ) n ≥ 0 f=(f_n)_{n\geq 0} , g = ( g n ) n ≥ 0 g=(g_n)_{n\geq 0} are martingales such that g g is differentially subordinate to f f , and let w = ( w n ) n ≥ 0 w=(w_n)_{n\geq 0} be a weight, i.e., a nonnegative, uniformly integrable martingale. Denoting by M f = sup n ≥ 0 | f n | Mf=\sup _{n\geq 0}|f_n| , M w = sup n ≥ 0 w n Mw=\sup _{n\geq 0}w_n the maximal functions of f f and w w , we prove the weighted inequality | | g | | L 1 ( w ) ≤ C | | M f | | L 1 ( M w ) , \begin{equation*} ||g||_{L^1(w)}\leq C||Mf||_{L^1(Mw)}, \end{equation*} where C = 3 + 2 + 4 ln ⁡ 2 = 7.186802 … C=3+\sqrt {2}+4\ln 2=7.186802\ldots . The proof rests on the existence of a special function enjoying appropriate majorization and concavity.