Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2111_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="78" /> </InlineMediaObject> <EquationSource Format="TEX">$0<\alpha <n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo><</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2111_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">$M_{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation> be the fractional maximal function. For a locally integrable function b, we denote by <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2111_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">$M_{\alpha ,b}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2111_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$[b,M_{\alpha }]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mi>b</mi> <mo>,</mo> <msub> <mi>M</mi> <mi>α</mi> </msub> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> the maximal commutator and the commutator of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2111_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">$M_{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation> with b. In this paper, we consider Bloom-type estimates for <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2111_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">$M_{\alpha ,b}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2111_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$[b,M_{\alpha }]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mi>b</mi> <mo>,</mo> <msub> <mi>M</mi> <mi>α</mi> </msub> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>. Some necessary and sufficient conditions to characterize the Bloom-type two-weight norm inequalities are given.

On Bloom-type estimates for commutators of the fractional maximal function

期刊论文
发表日期：

2025年3月12日

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Jie Sun,
Jianglong Wu,
Pu Zhang

摘要

Let $0<\alpha <n$ $0 < α < n$ and $M_{\alpha }$ $M_{α}$ be the fractional maximal function. For a locally integrable function b, we denote by $M_{\alpha ,b}$ $M_{α, b}$ and $[b,M_{\alpha }]$ $[b, M_{α}]$ the maximal commutator and the commutator of $M_{\alpha }$ $M_{α}$ with b. In this paper, we consider Bloom-type estimates for $M_{\alpha ,b}$ $M_{α, b}$ and $[b,M_{\alpha }]$ $[b, M_{α}]$ . Some necessary and sufficient conditions to characterize the Bloom-type two-weight norm inequalities are given.