INEQUALITIES FOR HILBERT OPERATOR AND ITS EXTENSIONS: THE PROBABILISTIC APPROACH

• 成果类型: Article
• 署名作者: Osekowski, Adam
• 署名单位: University of Warsaw
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/15-AOP1026
• 发表日期: 2017
• 页码: 535-563
• 关键词: logarithmic inequalities; orthogonal martingales; sharp inequalities; elementary proof; riesz transforms; SUBORDINATION; constants; ahlfors; zeros

摘要

We present a probabilistic study of the Hilbert operator, T f (x) = 1/pi integral(infinity)(0) f(y) dy/x+y, x >= 0, defined on integrable functions f on the positive halfline. Using appropriate novel estimates for orthogonal martingales satisfying the differential subordination, we establish sharp moment, weak-type and phi-inequalities for T. We also show similar estimates for higher dimensional analogues of the Hilbert operator, and by the further careful modification of martingale methods, we obtain related sharp localized inequalities for Hilbert and Riesz transforms.