1983/07/01 by Patrick Ahern, William S. Cohn, William Cohn
Mathematics · #Holomorphic and Operator Theory #Meromorphic and Entire Functions #Analytic and geometric function theory
paper · pdf · doi:10.1090/s0002-9939-1983-0699413-2
A connected open set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O subset-of-or-equal-to bold upper C"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:mo> ⊆ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">C</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal O ⊆ \mathbf C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is called an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N Superscript plus"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">N^ + </mml:annotation> </mml:semantics> </mml:math> </inline-formula> domain if every holomorphic function defined in the unit disc and taking values in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal O</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is necessarily in the Smirnov class <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N Superscript plus"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">N^ + </mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We show that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal O</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N Superscript plus"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">N^ + </mml:annotation> </mml:semantics> </mml:math> </inline-formula> domain if and only if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal infinity"> <mml:semantics> <mml:mi mathvariant="normal"> ∞ </mml:mi> <mml:annotation encoding="application/x-tex">∞</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a regular point for the solution of the Dirichlet problem for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal O</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We get a similar characterization when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N Superscript plus"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">N^ + </mml:annotation> </mml:semantics> </mml:math> </inline-formula> is replaced by the class of outer functions.