<p>We prove Hardy-Sobolev inequalities in weighted Herz-Morrey-Orlicz spaces on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\mathbb R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>. As an application, we discuss Hardy-Sobolev inequalities for generalized Riesz potentials <Equation ID="Equ15"> <EquationSource Format="TEX">$\begin{aligned} & R_{\alpha ,m}f(x) = \int _{{\mathbb R}^n} R_{\alpha ,m} (x,y) f(y) dy, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd /> <mtd columnalign="left"> <mrow> <msub> <mi>R</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </msub> <msub> <mi>R</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> <mi>y</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$R_{\alpha }(x)=|x|^{\alpha -n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mi>α</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>α</mi> <mo>-</mo> <mi>n</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> and <Equation ID="Equ16"> <EquationSource Format="TEX">$ \displaystyle {R_{\alpha ,m}(x,y) = R_{\alpha }(x-y) - \sum _{|\ell | \le m-1} \frac{x^{\ell }}{\ell !}(D^{\ell }R_{\alpha })(-y)}. $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>R</mi> <mi>α</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>-</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <munder> <mo>∑</mo> <mrow> <mo stretchy="false">|</mo> <mi>ℓ</mi> <mo stretchy="false">|</mo> <mo>≤</mo> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </munder> <mfrac> <msup> <mi>x</mi> <mi>ℓ</mi> </msup> <mrow> <mi>ℓ</mi> <mo>!</mo> </mrow> </mfrac> <mrow> <mo stretchy="false">(</mo> <msup> <mi>D</mi> <mi>ℓ</mi> </msup> <msub> <mi>R</mi> <mi>α</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mo>.</mo> </mrow> </mstyle> </math></EquationSource> </Equation></p>

Hardy type inequalities in weighted Herz-Morrey-Orlicz spaces

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2026年6月20日

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Yoshihiro Mizuta,
Tetsu Shimomura

摘要

We prove Hardy-Sobolev inequalities in weighted Herz-Morrey-Orlicz spaces on ${\mathbb R}^n$ $R^{n}$ . As an application, we discuss Hardy-Sobolev inequalities for generalized Riesz potentials $\begin{aligned} & R_{\alpha ,m}f(x) = \int _{{\mathbb R}^n} R_{\alpha ,m} (x,y) f(y) dy, \end{aligned}$ $\begin{matrix} R_{α, m} f (x) = \int_{R^{n}} R_{α, m} (x, y) f (y) d y, \end{matrix}$ where $R_{\alpha }(x)=|x|^{\alpha -n}$ $R_{α} (x) = {| x |}^{α - n}$ and $ \displaystyle {R_{\alpha ,m}(x,y) = R_{\alpha }(x-y) - \sum _{|\ell | \le m-1} \frac{x^{\ell }}{\ell !}(D^{\ell }R_{\alpha })(-y)}. $ $R_{α, m} (x, y) = R_{α} (x - y) - \sum_{| ℓ | \leq m - 1} \frac{x^{ℓ}}{ℓ!} (D^{ℓ} R_{α}) (- y) .$