<p>We prove Sobolev inequalities for variable Riesz potentials <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2025_450_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$I_{\alpha (\cdot )}f$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>I</mi> <mrow> <mi>α</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </msub> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation> of functions in central Musielak–Orlicz–Morrey spaces <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2025_450_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">${{\mathcal {B}}}^{\Phi ,\omega }(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="script">B</mi> </mrow> <mrow> <mi mathvariant="normal">Φ</mi> <mo>,</mo> <mi>ω</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> over bounded spaces. As an application and useful example, we give Sobolev inequalities for double phase functionals with variable exponents <Equation ID="Equ11"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2025_450_Article_Equ11.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="183" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \Phi (x,t) = t^{p(x)} + a(x) t^{q(x)}, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi mathvariant="normal">Φ</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>t</mi> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>+</mo> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>t</mi> <mrow> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2025_450_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="86" /> </InlineMediaObject> <EquationSource Format="TEX">$p(x)<q(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2025_450_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$x\in G$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <mi>G</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2025_450_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">$a(\cdot )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is non-negative, bounded and Hölder continuous of order <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2025_450_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">$\theta \in (0,1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

Sobolev inequalities in central Musielak–Orlicz–Morrey spaces and double phase functionals

期刊论文
发表日期：

2025年8月20日

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Takao Ohno,
Tetsu Shimomura

摘要

We prove Sobolev inequalities for variable Riesz potentials $I_{\alpha (\cdot )}f$ $I_{α (\cdot)} f$ of functions in central Musielak–Orlicz–Morrey spaces ${{\mathcal {B}}}^{\Phi ,\omega }(G)$ $B^{Φ, ω} (G)$ over bounded spaces. As an application and useful example, we give Sobolev inequalities for double phase functionals with variable exponents $\begin{aligned} \Phi (x,t) = t^{p(x)} + a(x) t^{q(x)}, \end{aligned}$ $\begin{matrix} Φ (x, t) = t^{p (x)} + a (x) t^{q (x)}, \end{matrix}$ where $p(x)<q(x)$ $p (x) < q (x)$ for $x\in G$ $x \in G$ , $a(\cdot )$ $a (\cdot)$ is non-negative, bounded and Hölder continuous of order $\theta \in (0,1]$ $θ \in (0, 1]$ .