Discrete analogues of second-order Riesz transforms

TL;DR

The paper identifies the exact $\ell^p$ norms of discrete second-order Riesz transforms by constructing probabilistic discrete transforms $\mathcal{R}^{(jk)}$ on $\mathbb{Z}^d$ via space-time martingale transforms. These operators preserve the $L^p$-norms of their continuous counterparts, i.e., $\|\mathcal{R}^{(jk)}\|_{\ell^p\to\ell^p} = \|R^{(jk)}\|_{L^p\to L^p}$, while differing from the standard discrete Riesz transforms by a convolution with an $\ell^1$ kernel. The framework uses the periodic heat kernel, the heat extension, and Doob-type martingale inequalities to derive sharp bounds and a kernel decomposition, enabling discrete Beurling--Ahlfors and Calderón--Zygmund analyses. The work links probabilistic discretization with CZ theory and provides a robust method for transferring sharp $L^p$ bounds from the continuous to the discrete setting, with potential applications to discrete Beurling--Ahlfors and related operators.

Abstract

Discrete analogues of many classical operators in harmonic analysis have been widely studied for many years with interesting connections to other areas in mathematics, including ergodic theory and analytic number theory. This paper concerns the problem of identifying the $\ell^p$-norms of discrete analogues of second-order Riesz transforms. Using probabilistic techniques, a class of second-order discrete Riesz transforms $\mathcal{R}^{(jk)}$ is constructed on the lattice $\mathbb{Z}^d$, $d\geq 2$. It is shown that their $\ell^p(\mathbb{Z}^d)$ norms, $1<p<\infty$, are the same as the norms of the classical second-order Riesz transforms $R^{(jk)}$ in $L^p(\mathbb{R}^d)$. The operators $\mathcal{R}^{(jk)}$ differ from the harmonic analysis analogues of the discrete Riesz transforms, $R^{(jk)}_{\mathrm{dis}}$, by convolution with a function in $\ell^1(\mathbb{Z}^d)$. Applications are given to the discrete Beurling-Ahlfors operator. It is shown that the operators $\mathcal{R}^{(jk)}$ arise by discretization of a class of Calderón-Zygmund singular integrals $T^{(jk)}$ which differ from the classical Riesz transforms $R^{(jk)}$ by convolution with a function in $L^1(\mathbb{R}^d)$.

Figures (1)

• Figure 1: The graphs of $q_0(t)$ (Left) and $q_1(t)$ (Right) for $1/8\le t\le 1/2$

Theorems & Definitions (44)

• Theorem 1.1
• Theorem 1.2
• Conjecture 1.3
• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Remark 2.3
• Lemma 2.4
• proof