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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.14511 |
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| _version_ | 1866908456808611840 |
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| author | Markovic, Marijan |
| author_facet | Markovic, Marijan |
| contents | Our main result concerns the behavior of bounded harmonic functions on a domain in $\mathbb{R}^N$ which may be represented as a strict epigraph of a Lipschitz function on $\mathbb{R}^{N-1}$. Generally speaking, the result says that the Hölder continuity of a harmonic function on such a domain is equivalent to the uniform Hölder continuity along the straight lines determined by the vector $\mathbf{e}_N$, where $\mathbf{e}_1,\mathbf{e}_2,\dots,\mathbf {e}_N$ is the base of standard vectors in $\mathbb{R}^N$.
More precisely, let $Ψ$ be a Lipschitz function on $\mathbb {R}^{N-1}$, and $U$ be a real-valued bounded harmonic function on $E_Ψ=\{(x',x_N): x'\in\mathbb{R}^{N-1}, x_N>Ψ(x')\}$. We show that for $α\in(0,1)$ the following two conditions on $U$ are equivalent:
(a) There exists a constant $C$ such that \begin{equation*} | U(x',x_N) - U(x',y_N)|\le C |x_N - y_N|^α,\quad x'\in \mathbb {R}^{N-1}, x_N, y_N > Ψ(x'); \end{equation*}
(b) There exists a constant $\tilde {C}$ such that \begin{equation*} |U(x) - U (y)|\le \tilde{C} |x-y|^α,\quad x, y\in E_Ψ. \end{equation*} Moreover, the constant $\tilde {C}$ depends linearly on $C$.
The result holds as well for vector-valued harmonic functions and, therefore, for analytic mappings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_14511 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A condition equivalent to the Hölder continuity of harmonic functions on unbounded Lipschitz domains Markovic, Marijan Complex Variables Our main result concerns the behavior of bounded harmonic functions on a domain in $\mathbb{R}^N$ which may be represented as a strict epigraph of a Lipschitz function on $\mathbb{R}^{N-1}$. Generally speaking, the result says that the Hölder continuity of a harmonic function on such a domain is equivalent to the uniform Hölder continuity along the straight lines determined by the vector $\mathbf{e}_N$, where $\mathbf{e}_1,\mathbf{e}_2,\dots,\mathbf {e}_N$ is the base of standard vectors in $\mathbb{R}^N$. More precisely, let $Ψ$ be a Lipschitz function on $\mathbb {R}^{N-1}$, and $U$ be a real-valued bounded harmonic function on $E_Ψ=\{(x',x_N): x'\in\mathbb{R}^{N-1}, x_N>Ψ(x')\}$. We show that for $α\in(0,1)$ the following two conditions on $U$ are equivalent: (a) There exists a constant $C$ such that \begin{equation*} | U(x',x_N) - U(x',y_N)|\le C |x_N - y_N|^α,\quad x'\in \mathbb {R}^{N-1}, x_N, y_N > Ψ(x'); \end{equation*} (b) There exists a constant $\tilde {C}$ such that \begin{equation*} |U(x) - U (y)|\le \tilde{C} |x-y|^α,\quad x, y\in E_Ψ. \end{equation*} Moreover, the constant $\tilde {C}$ depends linearly on $C$. The result holds as well for vector-valued harmonic functions and, therefore, for analytic mappings. |
| title | A condition equivalent to the Hölder continuity of harmonic functions on unbounded Lipschitz domains |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2507.14511 |