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

TL;DR

This paper proves that the $l^p$-norm of the discrete Hilbert transform equals that of the continuous Hilbert transform, resolving a long-standing conjecture by relating their norms through martingale representations.

Contribution

It establishes the equality of the $l^p$-norms of discrete and continuous Hilbert transforms using martingale techniques, a significant advancement in harmonic analysis.

Findings

The $l^p$-norm of the discrete Hilbert transform is equal to that of the continuous Hilbert transform.

The paper provides an upper bound for the $l^p$-norm of the discrete Hilbert transform.

It confirms the conjecture that these operator norms are identical for all $1<p< $.

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.