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 [Bañuelos and Kwaśnicki, 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 ${\mathbb{Z}}^d$, $d\ge 1.$ 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 [Bañuelos and Kwaśnicki, Duke Math. J. (2019)]. Open problems are stated.

66 pages, 4 figures