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

TL;DR

This paper simplifies the conditions for analytic extensions of $A_{}$-weights on Lipschitz curves and applies these results to analyze weighted Hardy spaces, aiding the study of boundary value problems.

Contribution

It shows the equivalence of a Smirnov-type condition to an $H^1$-integrability condition, allowing for a simplified framework to study weighted Hardy spaces.

Findings

Smirnov-type condition is equivalent to an $H^1$-integrability condition.

One condition in the class $AE$ can be dropped due to this equivalence.

Application of simplified theory to weighted Hardy spaces for boundary value problems.

Abstract

An $A_{\infty}$-weight on a Lipschitz curve $\Lambda$ in the plane can be extended analytically to the graph Lipschitz domain $\Omega$ 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 $\Omega$ with boundary data in weighted spaces.