This paper deals with the fractional Sobolev spaces <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$W^{s, p}(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>W</mi> <mrow> <mi>s</mi> <mo>,</mo> <mi>p</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$s\in (0, 1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$p\in [1,+\infty ]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mo>+</mo> <mi>∞</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>. Here, we use the interpolation results in [<CitationRef CitationID="CR4">4</CitationRef>] to provide suitable conditions on the exponents s and p so that the spaces <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$W^{s, p}(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>W</mi> <mrow> <mi>s</mi> <mo>,</mo> <mi>p</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> realize a continuous embedding when either <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Omega =\mathbb {R}^N$</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> or <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is any open and bounded domain with Lipschitz boundary. Our results enhance the classical continuous embedding and, when <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is any open bounded domain with Lipschitz boundary, we also improve the classical compact embeddings. All the results stated here are proved to be optimal. Also, our strategy does not require the use of Besov or other interpolation spaces.

Optimal embedding results for fractional Sobolev spaces

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2026年6月15日

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Serena Dipierro,
Edoardo Proietti Lippi,
Caterina Sportelli,
Enrico Valdinoci

摘要

This paper deals with the fractional Sobolev spaces $W^{s, p}(\Omega )$ $W^{s, p} (Ω)$ , with $s\in (0, 1]$ $s \in (0, 1]$ and $p\in [1,+\infty ]$ $p \in [1, + \infty]$ . Here, we use the interpolation results in [4] to provide suitable conditions on the exponents s and p so that the spaces $W^{s, p}(\Omega )$ $W^{s, p} (Ω)$ realize a continuous embedding when either $\Omega =\mathbb {R}^N$ $Ω = R^{N}$ or $\Omega $ $Ω$ is any open and bounded domain with Lipschitz boundary. Our results enhance the classical continuous embedding and, when $\Omega $ $Ω$ is any open bounded domain with Lipschitz boundary, we also improve the classical compact embeddings. All the results stated here are proved to be optimal. Also, our strategy does not require the use of Besov or other interpolation spaces.