Harmonic Extensions on $\mathbb{Z} \times \mathbb{N}$ and a Discrete Hilbert Transform
Ljupcho Petrov
TL;DR
The paper develops a fully discrete harmonic-extension framework on $\mathbb{Z}\times\mathbb{N}$ that mimics Poisson and conjugate–Poisson integrals, yielding a discrete Hilbert transform $H_d$ arising from a lattice Cauchy–Riemann system. It constructs $U$ and $V$ via discrete Poisson kernels $P_k$ and $Q_k$, establishes discrete harmonicity with respect to the five-point Laplacian, and identifies the boundary data as $(a,H_d a)$ through a Fourier multiplier $m_d(\theta)$. It proves weak-type $(1,1)$ and $\ell^p$ bounds for $H_d$, shows $H_d=H^++D*$ with an $\ell^1$-small perturbation, and provides asymptotics $K_d(n)=\frac{1}{\pi(n+1/2)}+O(n^{-2})$; it further extends the construction to $\mathbb{Z}^s\times\mathbb{N}$, yielding discrete Riesz transforms with analogous $\ell^p(w)$ bounds and weak-type estimates. The work offers a robust discrete analytic model parallel to the classical theory and supports higher-dimensional generalizations with weighted inequality frameworks.
Abstract
For a given boundary sequence $a=(a_n)_{n\in\mathbb{Z}}$, we construct harmonic extensions $U,V:\mathbb{Z}\times\ \mathbb{N}\to \mathbb{R}$ that serve as discrete analogs of the Poisson and conjugate-Poisson integrals. The construction is characterized by: (i) discrete harmonicity with respect to a two-dimensional Laplacian, (ii) a Cauchy-Riemann system, and (iii) boundary values involving a discrete Hilbert transform: $U(n,0)=a_n,\;V(n,0)=(H_{\mathrm d}a)_n$. We compare $H_{\mathrm d}$ to the Riesz-Titchmarsh transform and prove weak-type $(1,1)$ and $\ell^{p}$ bounds for $p>1$. We also extend the constructions to harmonic extensions on $\mathbb{Z}^s \times \mathbb{N}$. These results provide a discrete harmonic-analytic model analogous to the classical theory.