<p>We study the perturbed Sobolev spaces <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42985_2025_336_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">${H^{s,p}_\alpha (\mathbb {R}^d)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>H</mi> <mi>α</mi> <mrow> <mi>s</mi> <mo>,</mo> <mi>p</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, associated with singular perturbation <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42985_2025_336_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$\Delta _\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation> of Laplace operator in Euclidean space of dimensions 2 and 3. We extend the <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42985_2025_336_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> theory of perturbed Sobolev space to the <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42985_2025_336_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> case, finding an analogue description in terms of standard Sobolev spaces. This enables us to extend the Strichartz estimates to the energy space and to treat the local well-posedness of the Nonlinear Schrödinger equation associated with this singular perturbation, with the contraction method.</p>

Fractional Sobolev spaces for the singular-perturbed Laplace operator in the $L^p$ setting

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发表日期：

2025年6月12日

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Vladimir Georgiev,
Mario Rastrelli

摘要

We study the perturbed Sobolev spaces ${H^{s,p}_\alpha (\mathbb {R}^d)}$ $H_{α}^{s, p} (R^{d})$ , associated with singular perturbation $\Delta _\alpha $ $Δ_{α}$ of Laplace operator in Euclidean space of dimensions 2 and 3. We extend the $L^2$ $L^{2}$ theory of perturbed Sobolev space to the $L^p$ $L^{p}$ case, finding an analogue description in terms of standard Sobolev spaces. This enables us to extend the Strichartz estimates to the energy space and to treat the local well-posedness of the Nonlinear Schrödinger equation associated with this singular perturbation, with the contraction method.

Fractional Sobolev spaces for the singular-perturbed Laplace operator in the \(L^p\) setting

摘要