Translator Disclaimer
January 2017 Inequalities for Hilbert operator and its extensions: The probabilistic approach
Adam Osȩkowski
Ann. Probab. 45(1): 535-563 (January 2017). DOI: 10.1214/15-AOP1026

Abstract

We present a probabilistic study of the Hilbert operator

\[Tf(x)=\frac{1}{\pi}\int_{0}^{\infty}\frac{f(y)\,\mathrm{d}y}{x+y},\qquad x\geq0,\] 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.

Citation

Download Citation

Adam Osȩkowski. "Inequalities for Hilbert operator and its extensions: The probabilistic approach." Ann. Probab. 45 (1) 535 - 563, January 2017. https://doi.org/10.1214/15-AOP1026

Information

Received: 1 June 2014; Revised: 1 March 2015; Published: January 2017
First available in Project Euclid: 26 January 2017

MathSciNet: MR3601656
Digital Object Identifier: 10.1214/15-AOP1026

Subjects:
Primary: 60G44
Secondary: 31B05

Rights: Copyright © 2017 Institute of Mathematical Statistics

JOURNAL ARTICLE
29 PAGES


SHARE
Vol.45 • No. 1 • January 2017
Back to Top