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 $\ell^{p}$ 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.