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arXiv · 2505.15278

Analytic extensions of $A_{\infty}$-weights on Lipschitz curves and their use in weighted Hardy spaces

Fernando Ballesta-Yagüe
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Abstract

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.

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Fernando Ballesta-Yagüe. 2025-06-27. Analytic extensions of $A_{\infty}$-weights on Lipschitz curves and their use in weighted Hardy spaces. https://arxiv.org/abs/2505.15278

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