Inequalities for Hilbert operator and its extensions: The probabilistic approach

We present a probabilistic study of the Hilbert operator Tf(x)=1π∫∞0f(y)dyx+y,x≥0, Tf(x)=1π∫0∞f(y)dyx+y,x≥0, defined on integrable functions ff on the positive halfline. Using appropriate novel estimates for orthogonal martingales satisfying the differential subordination, we establish sharp moment, weak-type and ΦΦ-inequalities for TT. 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.