In this paper, we study the relationship between Sobolev extension domains and homogeneous Sobolev extension domains. Precisely, we obtain the following results,<UnorderedList Mark="Bullet"> <ItemContent> Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$1\leq q\leq p\leq \infty$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>q</mi> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. Then a bounded <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(L^{1, p}, L^{1, q})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>p</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>L</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>q</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-extension domain is also a <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(W^{1, p}, W^{1, q})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mi>W</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>p</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>W</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>q</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-extension domain. </ItemContent> <ItemContent> Let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$1\leq q\leq p<q ^{\star} \leq \infty$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>q</mi> <mo>≤</mo> <mi>p</mi> <mo><</mo> <msup> <mi>q</mi> <mo>⋆</mo> </msup> <mo>≤</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$n< q \leq p\leq \infty$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo><</mo> <mi>q</mi> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. Then a bounded domain is a <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(W^{1, p}, W^{1, q})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mi>W</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>p</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>W</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>q</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-extension domain if and only if it is an <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$(L^{1, p}, L^{1, q})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>p</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>L</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>q</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-extension domain. </ItemContent> <ItemContent> For <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$1\leq q<n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>q</mi> <mo><</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$q<nq ^{\star} <p\leq \infty$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo><</mo> <mi>n</mi> <msup> <mi>q</mi> <mo>⋆</mo> </msup> <mo><</mo> <mi>p</mi> <mo>≤</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, there exists a bounded domain <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\Omega\subset\mathbb{R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> which is a <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$(W^{1, p}, W^{1, q})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mi>W</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>p</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>W</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>q</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-extension domain but not an <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$(L^{1, p}, L^{1, q})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>p</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>L</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>q</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-extension domain for <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$1 \leq q <p\leq n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>q</mi> <mo><</mo> <mi>p</mi> <mo>≤</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>. </ItemContent> </UnorderedList>

Sobolev versus homogeneous Sobolev extension

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2025年10月31日

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P. Koskela,
R. Mishra,
Z. Zhu

摘要

In this paper, we study the relationship between Sobolev extension domains and homogeneous Sobolev extension domains. Precisely, we obtain the following results,

Let $1\leq q\leq p\leq \infty$ $1 \leq q \leq p \leq \infty$ . Then a bounded $(L^{1, p}, L^{1, q})$ $(L^{1, p}, L^{1, q})$ -extension domain is also a $(W^{1, p}, W^{1, q})$ $(W^{1, p}, W^{1, q})$ -extension domain.

Let $1\leq q\leq p<q ^{\star} \leq \infty$ $1 \leq q \leq p < q^{⋆} \leq \infty$ or $n< q \leq p\leq \infty$ $n < q \leq p \leq \infty$ . Then a bounded domain is a $(W^{1, p}, W^{1, q})$ $(W^{1, p}, W^{1, q})$ -extension domain if and only if it is an $(L^{1, p}, L^{1, q})$ $(L^{1, p}, L^{1, q})$ -extension domain.

For $1\leq q<n$ $1 \leq q < n$ and $q<nq ^{\star} <p\leq \infty$ $q < n q^{⋆} < p \leq \infty$ , there exists a bounded domain $\Omega\subset\mathbb{R}^n$ $Ω \subset R^{n}$ which is a $(W^{1, p}, W^{1, q})$ $(W^{1, p}, W^{1, q})$ -extension domain but not an $(L^{1, p}, L^{1, q})$ $(L^{1, p}, L^{1, q})$ -extension domain for $1 \leq q <p\leq n$ $1 \leq q < p \leq n$ .