On the $l^p$-norm of the discrete Hilbert transform

Abstract

Using a representation of the discrete Hilbert transform in terms of martingales arising from Doob $h$-processes, we prove that its $l^p$-norm, $1<p<\infty$, is bounded above by the $L^p$-norm of the continuous Hilbert transform. Together with the already known lower bound, this resolves the long-standing conjecture that the norms of these operators are equal.

References in corpus (3)

• Harmonic Analysis associated with a discrete Laplacian
• Sharp $L^p$ estimates for discrete second order {R}iesz transforms
• One-parameter groups of operators and discrete Hilbert transforms

Cited by in corpus (2)

• The Ionescu--Wainger multiplier theorem and the adeles
• Sharp inequalities for linear combinations of orthogonal martingales