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| Format: | Preprint |
| Publié: |
2025
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| Accès en ligne: | https://arxiv.org/abs/2510.16494 |
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| _version_ | 1866915562790060032 |
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| author | Petrov, Ljupcho |
| author_facet | Petrov, Ljupcho |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_16494 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Harmonic Extensions on $\mathbb{Z} \times \mathbb{N}$ and a Discrete Hilbert Transform Petrov, Ljupcho Classical Analysis and ODEs 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. |
| title | Harmonic Extensions on $\mathbb{Z} \times \mathbb{N}$ and a Discrete Hilbert Transform |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2510.16494 |