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

An <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript normal infinity"> <mml:semantics> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">A_{\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -weight on a Lipschitz curve <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda"> <mml:semantics> <mml:mi mathvariant="normal"> Λ </mml:mi> <mml:annotation encoding="application/x-tex">\Lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the plane can be extended analytically to the graph Lipschitz domain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"> <mml:semantics> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:annotation encoding="application/x-tex">\Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> above it. This problem was studied by C. Kenig [Amer. J. Math. 102 (1980), pp. 129–163], who introduced the class <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A upper E"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>E</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">AE</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">H^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -integrability condition. As a consequence, one of the conditions in the definition of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A upper E"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>E</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">AE</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"> <mml:semantics> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:annotation encoding="application/x-tex">\Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with boundary data in weighted spaces.