Analytic extensions of $A_{\infty}$-weights on Lipschitz curves and their use in weighted Hardy spaces
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909661371826176 |
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| author | Ballesta-Yagüe, Fernando |
| author_facet | Ballesta-Yagüe, Fernando |
| contents | An $A_{\infty}$-weight on a Lipschitz curve $Λ$ in the plane can be extended analytically to the graph Lipschitz domain $Ω$ above it. This problem was studied by C. Kenig [Ken80], who introduced the class $AE$ of well-behaved analytic extensions. Later, he and D. Jerison [JK82] added a Smirnov-type condition to the definition of this class.
In this note, we show that this Smirnov-type condition is equivalent to an $H^1$-integrability condition. As a consequence, one of the conditions in the definition of $AE$ can be dropped. We use this simplification to apply C. Kenig's theory to prove results about weighted Hardy spaces. These are useful to study the Neumann problem in $Ω$ with boundary data in weighted spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_15278 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Analytic extensions of $A_{\infty}$-weights on Lipschitz curves and their use in weighted Hardy spaces Ballesta-Yagüe, Fernando Classical Analysis and ODEs Analysis of PDEs Complex Variables An $A_{\infty}$-weight on a Lipschitz curve $Λ$ in the plane can be extended analytically to the graph Lipschitz domain $Ω$ above it. This problem was studied by C. Kenig [Ken80], who introduced the class $AE$ of well-behaved analytic extensions. Later, he and D. Jerison [JK82] added a Smirnov-type condition to the definition of this class. In this note, we show that this Smirnov-type condition is equivalent to an $H^1$-integrability condition. As a consequence, one of the conditions in the definition of $AE$ can be dropped. We use this simplification to apply C. Kenig's theory to prove results about weighted Hardy spaces. These are useful to study the Neumann problem in $Ω$ with boundary data in weighted spaces. |
| title | Analytic extensions of $A_{\infty}$-weights on Lipschitz curves and their use in weighted Hardy spaces |
| topic | Classical Analysis and ODEs Analysis of PDEs Complex Variables |
| url | https://arxiv.org/abs/2505.15278 |