Harmonic Extensions on $\mathbb{Z}\times \mathbb{N}$ and a Discrete Hilbert Transform

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 ${\mathrm{\ell }}^p$ 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.