A weighted maximal inequality for differentially subordinate martingales

• Published in: Proceedings of the American Mathematical Society 2018-05, Vol.146 (5), p.2263-2275
• Main Authors: Rodrigo Bañuelos, Adam Osękowski
• Format: Article
• Language: English
• Subjects: F. STATISTICS AND PROBABILITY
• ISSN: 0002-9939; 1088-6826
• DOI: 10.1090/proc/13912

The paper contains the proof of a weighted Fefferman-Stein inequality in a probabilistic setting. Suppose that f=(f_n)_{n\geq 0}, g=(g_n)_{n\geq 0} are martingales such that g is differentially subordinate to f, and let w=(w_n)_{n\geq 0} be a weight, i.e., a nonnegative, uniformly integrable martingale. Denoting by Mf=\sup _{n\geq 0}\vert f_n\vert, Mw=\sup _{n\geq 0}w_n the maximal functions of f and w, we prove the weighted inequality \displaystyle \vert\vert g\vert\vert _{L^1(w)}\leq C\vert\vert Mf\vert\vert _{L^1(Mw)}, where 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.