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

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