## The Annals of Probability

### Inequalities for Hilbert operator and its extensions: The probabilistic approach

Adam Osȩkowski

#### 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.

#### Article information

Source
Ann. Probab., Volume 45, Number 1 (2017), 535-563.

Dates
Received: June 2014
Revised: March 2015
First available in Project Euclid: 26 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1485421339

Digital Object Identifier
doi:10.1214/15-AOP1026

Mathematical Reviews number (MathSciNet)
MR3601656

Subjects
Primary: 60G44: Martingales with continuous parameter
Secondary: 31B05: Harmonic, subharmonic, superharmonic functions

#### Citation

Osȩkowski, Adam. Inequalities for Hilbert operator and its extensions: The probabilistic approach. Ann. Probab. 45 (2017), no. 1, 535--563. doi:10.1214/15-AOP1026. https://projecteuclid.org/euclid.aop/1485421339

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