Weighted Davis Inequalities for Martingale Square Functions

Abstract

For a Hilbert space-valued martingale (f_{n}) and an adapted sequence of positive random variables (w_{n}), we show the weighted Davis-type inequality E|f0|w0+14∑n=1N|dfn|2fn∗wn≤E(fN∗wN∗).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\mathbb {E}}\\left( {|}f_{0}{|} w_{0} + \\frac{1}{4} \\sum _{n=1}^{N} \\frac{{|}df_{n}{|}^{2}}{f^{*}_{n}} w_{n} \\right) \\le {\\mathbb {E}}( f^{*}_{N} w^{*}_{N}). \\end{aligned}$$\\end{document}More generally, for a martingale (f_{n}) with values in a (q,delta )-uniformly convex Banach space, we show that E|f0|w0+δ∑n=1∞|dfn|q(fn∗)q-1wn≤CqE(f∗w∗).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\mathbb {E}}\\left( {|}f_{0}{|} w_{0} + \\delta \\sum _{n=1}^{\\infty } \\frac{{|}df_{n}{|}^{q}}{(f^{*}_{n})^{q-1}} w_{n} \\right) \\le C_{q} {\\mathbb {E}}( f^{*} w^{*}). \\end{aligned}$$\\end{document}These inequalities unify several results about the martingale square function.