Inequalities for Hilbert operator and its extensions: The probabilistic approach

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.