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Weak-L∞ inequality for non-symmetric martingale transforms and Haar system
Statistics & Probability Letters ( IF 0.8 ) Pub Date : 2020-08-01 , DOI: 10.1016/j.spl.2020.108778
Meryem Akboudj , Yong Jiao , Adam Osękowski

Abstract Let b B be two real numbers. Suppose that f = ( f n ) n ≥ 0 and g = ( g n ) n ≥ 0 are two Hilbert-space-valued martingales satisfying | d g n − B + b 2 d f n | ≤ | B − b 2 d f n | , n = 0 , 1 , 2 , … . The paper contains the proof of the sharp weak-type inequality ‖ g ‖ W ( Ω ) ≤ 2 max ( − b , B ) ‖ f ‖ L ∞ , where W is the weak- L ∞ space introduced by Bennett, DeVore and Sharpley. As applications, we obtain related estimates for the Haar system and harmonic functions on Euclidean domains.

中文翻译:

非对称鞅变换和 Haar 系统的弱 L∞ 不等式

摘要 设 b B 是两个实数。假设 f = ( fn ) n ≥ 0 和 g = ( gn ) n ≥ 0 是两个希尔伯特空间值鞅满足 | dgn − B + b 2 dfn | ≤ | B − b 2 dfn | , n = 0 , 1 , 2 , ... 。该论文包含了尖锐弱型不等式 ‖ g ‖ W ( Ω ) ≤ 2 max ( − b , B ) ‖ f ‖ L ∞ 的证明,其中 W 是 Bennett、DeVore 和 Sharpley 引入的弱 L ∞ 空间. 作为应用,我们获得了欧几里德域上 Haar 系统和谐波函数的相关估计。
更新日期:2020-08-01
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