Lipschitz conditions on bounded harmonic functions on the upper half-space
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913666613379072 |
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| author | Markovic, Marijan |
| author_facet | Markovic, Marijan |
| contents | This work is devoted to Lipschitz conditions on bounded harmonic functions on the upper half-space in $\mathbb {R}^n$. Among other results we prove the following one. Let $U(x',x_n)$ be a real-valued bounded harmonic function on the upper half-space $\mathbb {R}^n_+ = \{(x',x_n):x'\in \mathbb{R}^{n-1}, x_n\in (0,\infty)\}$, which is continuous on the closure of this domain. Assume that for $α\in (0,1)$ there exists a constant $C$ such that for every $x'\in \mathbb{R}^{n-1}$ we have $| |U|(x',x_n) - |U|(x',0)|\le Cx_n^α,\, x_n\in (0,\infty)$. Then there exists a constant $\tilde {C}$ such that $|U(x) - U (y)| \le \tilde{C} |x-y|^α,\, x,y\in \mathbb{R}^{n}_+$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_15315 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lipschitz conditions on bounded harmonic functions on the upper half-space Markovic, Marijan Complex Variables This work is devoted to Lipschitz conditions on bounded harmonic functions on the upper half-space in $\mathbb {R}^n$. Among other results we prove the following one. Let $U(x',x_n)$ be a real-valued bounded harmonic function on the upper half-space $\mathbb {R}^n_+ = \{(x',x_n):x'\in \mathbb{R}^{n-1}, x_n\in (0,\infty)\}$, which is continuous on the closure of this domain. Assume that for $α\in (0,1)$ there exists a constant $C$ such that for every $x'\in \mathbb{R}^{n-1}$ we have $| |U|(x',x_n) - |U|(x',0)|\le Cx_n^α,\, x_n\in (0,\infty)$. Then there exists a constant $\tilde {C}$ such that $|U(x) - U (y)| \le \tilde{C} |x-y|^α,\, x,y\in \mathbb{R}^{n}_+$. |
| title | Lipschitz conditions on bounded harmonic functions on the upper half-space |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2501.15315 |