The $\ell^p$ norm of the Riesz--Titchmarsh transform for even integer $p$

TL;DR

This paper confirms a long-standing conjecture relating the $ ext{ell}^p$ norm of the discrete Riesz--Titchmarsh transform to the classical Hilbert transform's $L^p$ norm for specific even integer values of p, using algebraic methods.

Contribution

It proves the conjecture for p = 2n or p/(p-1) = 2n, expanding understanding of the operator's norms at these special points.

Findings

Confirmed the conjecture for p = 2n and p/(p-1) = 2n.

Relied on a sharp estimate for a variant of the operator for all p.

Used algebraic techniques based on recent norm estimates.

Abstract

The long-standing conjecture that for $p \in (1, \infty)$ the $\ell^p(\mathbb Z)$ norm of the Riesz--Titchmarsh discrete Hilbert transform is the same as the $L^p(\mathbb R)$ norm of the classical Hilbert transform, is verified when $p = 2 n$ or $\frac{p}{p - 1} = 2 n$, for $n \in \mathbb N$. The proof, which is algebraic in nature, depends in a crucial way on the sharp estimate for the $\ell^p(\mathbb Z)$ norm of a different variant of this operator for the full range of $p$. The latter result was recently proved by the authors in [Ba\~nuelos, Kwa\'snicki, On the $\ell^p$-norm of the discrete Hilbert transform, Duke Math. J. 168(3) (2019): 471-504].