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

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.