Type: Article
Publication Date: 1992-01-01
Citations: 16
DOI: https://doi.org/10.1090/s0002-9939-1992-1075947-1
We prove that if a domain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D subset-of bold upper R Superscript n"> <mml:semantics> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">D \subset {{\mathbf {R}}^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is quasiconformally equivalent to a uniform domain, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an extension domain for the Sobolev class <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Subscript n Superscript 1"> <mml:semantics> <mml:msubsup> <mml:mi>W</mml:mi> <mml:mi>n</mml:mi> <mml:mn>1</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">W_n^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if and only if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is locally uniform. We provide examples which suggest that this result is best possible. We exhibit a list of equivalent conditions for domains quasiconformally equivalent to uniform domains, one of which characterizes the quasiconformal homeomorphisms between uniform and locally uniform domains.