Sharp $\ell^p$ inequalities for discrete singular integrals on the lattice $\mathbb{Z}^d$

TL;DR

This paper extends the understanding of discrete singular integrals on lattices, showing they have the same $p$-norms as their continuous counterparts, and introduces new methods and results for higher-dimensional cases.

Contribution

It computes $\ell^p$-norms of discrete operators on $\mathbb{Z}^d$, generalizing known results and developing a discrete rotation method and new Fourier-based proofs.

Findings

Discrete operators have the same $p$-norms as classical Riesz transforms.

A discrete analogue of the method of rotations is established.

New Fourier transform techniques are used to prove key identities.

Abstract

This paper investigates higher dimensional versions of the longstanding conjecture verified in [Ba\~nuelos and Kwa\'snicki, Duke Math. J. (2019)] that the $\ell^p$-norm of the discrete Hilbert transform on the integers is the same as the $L^p$-norm of the Hilbert transform on the real line. It computes the $\ell^p$-norms of a family of discrete operators on the lattice $\mathbb{Z}^{d}$, $d\geq 1.$ They are discretizations of a new class of singular integrals on $\mathbb{R}^d$ that have the same kernels as the classical Riesz transforms near zero and similar behavior at infinity. The discrete operators have the same $p$-norms as the classical Riesz transforms on $\mathbb{R}^d$. They are constructed as conditional expectations of martingale transforms of Doob h-processes conditioned to exit the upper--half space $\mathbb{R}^d\times \mathbb{R}_{+}$ only on the lattice $\mathbb{Z}^d$. The paper also presents a discrete analogue of the classical method of rotations which gives the norm of a different variant of discrete Riesz transforms on $\mathbb{Z}^d$. Along the way a new proof is given based on Fourier transform techniques of the key identity used to identify the norm of the discrete Hilbert transform in [Ba\~nuelos and Kwa\'snicki, Duke Math. J. (2019)]. Open problems are stated.