<p>Our aim in the present paper is to establish the Sobolev-Trudinger inequality for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation>-potentials <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$I_\rho f$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>I</mi> <mi>ρ</mi> </msub> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation> in weighted Morrey-Orlicz spaces, with the aid of Hedberg’s method by use of maximal functions. As an application, we prove the Sobolev-Trudinger integrabilities for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation>-potentials of double phase <Equation ID="Equ16"> <EquationSource Format="TEX">$ \varphi (x,r) = \varphi _1(r) + \varphi _2(b(x)r) \quad \text {for} x\in {\mathbb {R}}^n \text { and } r \ge 0 , $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>φ</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>φ</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <msub> <mi>φ</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="1em" /> <mtext>for</mtext> <mi>x</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mspace width="0.333333em" /> <mtext>and</mtext> <mspace width="0.333333em" /> <mi>r</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\varphi _1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>φ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\varphi _2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>φ</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> are convex functions on <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$[0,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <i>b</i> is a nonnegative function satisfying the Hölder type condition.</p>

Integrability for fractional potentials in weighted Morrey-Orlicz spaces

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2026年5月21日

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Yoshihiro Mizuta,
Tetsu Shimomura

摘要

Our aim in the present paper is to establish the Sobolev-Trudinger inequality for $\rho $ $ρ$ -potentials $I_\rho f$ $I_{ρ} f$ in weighted Morrey-Orlicz spaces, with the aid of Hedberg’s method by use of maximal functions. As an application, we prove the Sobolev-Trudinger integrabilities for $\rho $ $ρ$ -potentials of double phase $ \varphi (x,r) = \varphi _1(r) + \varphi _2(b(x)r) \quad \text {for} x\in {\mathbb {R}}^n \text { and } r \ge 0 , $ $φ (x, r) = φ_{1} (r) + φ_{2} (b (x) r) for x \in R^{n} and r \geq 0,$ where $\varphi _1$ $φ_{1}$ and $\varphi _2$ $φ_{2}$ are convex functions on $[0,\infty )$ $[0, \infty)$ and b is a nonnegative function satisfying the Hölder type condition.