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

An A ∞ A_{\infty } -weight on a Lipschitz curve Λ \Lambda in the plane can be extended analytically to the graph Lipschitz domain Ω \Omega above it. This problem was studied by C. Kenig [Amer. J. Math. 102 (1980), pp. 129–163], who introduced the class A E AE of well-behaved analytic extensions. Later, he and D. Jerison [Math. Scand. 50 (1982), pp. 221–247] 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 H^1 -integrability condition. As a consequence, one of the conditions in the definition of A E 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 \Omega with boundary data in weighted spaces.