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

Rodrigo Bañuelos, Mateusz Kwaśnicki

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 [Bañuelos, Kwaśnicki, On the $\ell^p$-norm of the discrete Hilbert transform, Duke Math. J. 168(3) (2019): 471-504].

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

Abstract

The long-standing conjecture that for the norm of the Riesz--Titchmarsh discrete Hilbert transform is the same as the norm of the classical Hilbert transform, is verified when or , for . The proof, which is algebraic in nature, depends in a crucial way on the sharp estimate for the norm of a different variant of this operator for the full range of . The latter result was recently proved by the authors in [Bañuelos, Kwaśnicki, On the -norm of the discrete Hilbert transform, Duke Math. J. 168(3) (2019): 471-504].
Paper Structure (7 sections, 14 theorems, 108 equations)

This paper contains 7 sections, 14 theorems, 108 equations.

Table of Contents

  1. Introduction and statement of main result
  2. Preliminaries
  3. Historical remarks
  4. Idea of the proof
  5. Skeletons, frames and buildings
  6. Norms and constants
  7. Cotangents

Key Result

Theorem 1.1

For $p = 2 n$ or $p = \frac{2 n}{2 n - 1}$, $n \in \mathds{N}$, we have for every sequence $(a_n)$ in $\ell^p$. The constant is best possible. In particular, the operator norm of $\mathscr{R}$ on $\ell^p$ is equal to the operator norm of $H$ on $L^p(\mathds{R})$ for such $p$.

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2: BanKwa, Theorem 1.1
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 24 more