The Annals of Probability

Inequalities for Hilbert operator and its extensions: The probabilistic approach

Adam Osȩkowski

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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

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

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

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Mathematical Reviews number (MathSciNet)

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

Hilbert operator martingale differential subordination best constants


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.

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