We are concerned with weighted Hardy-Sobolev type integrabilities for generalized Riesz potentials <Equation ID="Equa"> <EquationSource Format="TEX">$R_{\alpha, m}f(x)=\int_{\mathbb{R}^{n}}R_{\alpha,m}(x,y)f(y)\mathrm{d}y$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mi>R</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msup> </mrow> </msub> <msub> <mi>R</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mrow> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> </math></EquationSource> </Equation> of functions f in weighted Morrey-Orlicz spaces, where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$R_{\alpha}(x)=\mid x \mid^{\alpha-n}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mi>R</mi> <mrow> <mi>α</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=∣</mo> <mi>x</mi> <msup> <mo stretchy="false">∣</mo> <mrow> <mi>α</mi> <mo>−</mo> <mi>n</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> and <Equation ID="Equb"> <EquationSource Format="TEX">${R_{\alpha ,m}}(x,y) = {R_\alpha }(x - y) - \sum\limits_{\left| \ell \right| \leqslant m - 1} {{{{y^\ell}} \over {\ell!}}} ({D^\ell}{R_\alpha })( - x).$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <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> <mo>=</mo> <mrow> <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> <mo>−</mo> <munder> <mo movablelimits="false">∑</mo> <mrow> <mrow> <mo>|</mo> <mi>ℓ</mi> <mo>|</mo> </mrow> <mo>⩽</mo> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </munder> <mrow> <mrow> <mfrac> <mrow> <mrow> <msup> <mi>y</mi> <mi>ℓ</mi> </msup> </mrow> </mrow> <mrow> <mi>ℓ</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mi>D</mi> <mi>ℓ</mi> </msup> </mrow> <mrow> <msub> <mi>R</mi> <mi>α</mi> </msub> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </math></EquationSource> </Equation>Those potentials are used to give a representation of C1-functions on the punctured space ℝn {0}. As an application, we obtain Hardy-Sobolev type integrabilities for general double phase functionals given by <Equation ID="Equc"> <EquationSource Format="TEX">$\varphi(x,t)=\varphi_{1}(t)+\varphi_{2}(b(x)t).$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>φ</mi> <mrow> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>φ</mi> <mrow> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </math></EquationSource> </Equation>

Weighted Hardy-Sobolev type integrability for generalized Riesz potentials in weighted Morrey-Orlicz spaces

期刊论文
发表日期：

2025年10月1日

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Yoshihiro Mizuta,
Tetsu Shimomura

摘要

We are concerned with weighted Hardy-Sobolev type integrabilities for generalized Riesz potentials $R_{\alpha, m}f(x)=\int_{\mathbb{R}^{n}}R_{\alpha,m}(x,y)f(y)\mathrm{d}y$ $R_{α, m} f (x) = \int_{R^{n}} R_{α, m} (x, y) f (y) d y$ of functions f in weighted Morrey-Orlicz spaces, where $R_{\alpha}(x)=\mid x \mid^{\alpha-n}$ $R_{α} (x) =∣ x ∣^{α - n}$ and ${R_{\alpha ,m}}(x,y) = {R_\alpha }(x - y) - \sum\limits_{\left| \ell \right| \leqslant m - 1} {{{{y^\ell}} \over {\ell!}}} ({D^\ell}{R_\alpha })( - x).$ $R_{α, m} (x, y) = R_{α} (x - y) - \sum_{| ℓ | ⩽ m - 1} \frac{y^{ℓ}}{ℓ!} (D^{ℓ} R_{α}) (- x) .$

Those potentials are used to give a representation of C¹-functions on the punctured space ℝⁿ {0}. As an application, we obtain Hardy-Sobolev type integrabilities for general double phase functionals given by $\varphi(x,t)=\varphi_{1}(t)+\varphi_{2}(b(x)t).$ $φ (x, t) = φ_{1} (t) + φ_{2} (b (x) t) .$