This paper is devoted to studying the boundedness of commutators <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1037_Article_IEq1.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{H}_{\Omega ,\beta }^b$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mtext>H</mtext> <mrow> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi>β</mi> </mrow> <mi>b</mi> </msubsup> </math></EquationSource> </InlineEquation> generated by the rough fractional Hardy operators <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1037_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{H}_{\Omega ,\beta }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>H</mtext> <mrow> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi>β</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> with the symbol b on the mixed radial-angular spaces. When b is a mixed radial-angular central bounded mean oscillation function and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1037_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="98" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega \in L^s(S^{n-1})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>∈</mo> <msup> <mi>L</mi> <mi>s</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>S</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for some <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1037_Article_IEq4.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$s>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, the boundedness of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1037_Article_IEq5.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{H}_{\Omega ,\beta }^b$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mtext>H</mtext> <mrow> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi>β</mi> </mrow> <mi>b</mi> </msubsup> </math></EquationSource> </InlineEquation> on the mixed radial-angular homogeneous Herz spaces is established. Meanwhile, the boundedness for <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1037_Article_IEq6.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{H}_{\Omega ,\beta }^b$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mtext>H</mtext> <mrow> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi>β</mi> </mrow> <mi>b</mi> </msubsup> </math></EquationSource> </InlineEquation> on the mixed radial-angular homogeneous <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1037_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-central Morrey spaces is also obtained, provided that b belongs to the mixed radial-angular homogeneous <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1037_Article_IEq8.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-central bounded mean oscillation spaces and <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1037_Article_IEq9.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="98" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega \in L^s(S^{n-1})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>∈</mo> <msup> <mi>L</mi> <mi>s</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>S</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for some <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1037_Article_IEq10.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$s>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.

Mixed radial-angular integrabilities for commutators of fractional Hardy operators with rough kernels

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发表日期：

2025年2月27日

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Ronghui Liu,
Shuangping Tao,
Huoxiong Wu

摘要

This paper is devoted to studying the boundedness of commutators $\textrm{H}_{\Omega ,\beta }^b$ $H_{Ω, β}^{b}$ generated by the rough fractional Hardy operators $\textrm{H}_{\Omega ,\beta }$ $H_{Ω, β}$ with the symbol b on the mixed radial-angular spaces. When b is a mixed radial-angular central bounded mean oscillation function and $\Omega \in L^s(S^{n-1})$ $Ω \in L^{s} (S^{n - 1})$ for some $s>1$ $s > 1$ , the boundedness of $\textrm{H}_{\Omega ,\beta }^b$ $H_{Ω, β}^{b}$ on the mixed radial-angular homogeneous Herz spaces is established. Meanwhile, the boundedness for $\textrm{H}_{\Omega ,\beta }^b$ $H_{Ω, β}^{b}$ on the mixed radial-angular homogeneous $\lambda $ $λ$ -central Morrey spaces is also obtained, provided that b belongs to the mixed radial-angular homogeneous $\lambda $ $λ$ -central bounded mean oscillation spaces and $\Omega \in L^s(S^{n-1})$ $Ω \in L^{s} (S^{n - 1})$ for some $s>1$ $s > 1$ .