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• W1556430892 abstract "For an arbitrary open set <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Omega subset-of double-struck upper R Superscript n> <mml:semantics> <mml:mrow> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>Omega subset mathbb {R}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we characterize all functions <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=application/x-tex>G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on the real line such that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G ring u element-of upper W Superscript 1 comma p Baseline left-parenthesis normal upper Omega right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>∘<!-- ∘ --></mml:mo> <mml:mi>u</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>Gcirc uin W^{1,p}(Omega )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=u element-of upper W Superscript 1 comma p Baseline left-parenthesis normal upper Omega right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>uin W^{1,p}(Omega )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. New element in the proof is based on Maz’ya’s capacitary criterion for the imbedding <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper W Superscript 1 comma p Baseline left-parenthesis normal upper Omega right-parenthesis right-arrow with hook upper L Superscript normal infinity Baseline left-parenthesis normal upper Omega right-parenthesis> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>↪<!-- ↪ --></mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{W^{1,p}(Omega )hookrightarrow L^infty (Omega )}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
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• W1556430892 date "2000-05-11" @default.
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• W1556430892 title "Superposition operator in Sobolev spaces on domains" @default.
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