Analytic extensions of A_∞-weights on Lipschitz curves and their use in weighted Hardy spaces

2025/05/21 by Fernando Ballesta-Yagüe, Ballesta-Yagüe, Fernando
Mathematics · #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2505.15278

openalex publication_date 2025/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

An A_∞-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 H¹-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.

Citations

• Weighted norm inequalities for the conjugate function and Hilbert transform

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