<p>In this paper, the authors consider the endpoint boundedness properties for the rough maximal Calderón commutator defined by <Equation ID="Equ43"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3152_Article_Equ43.gif" Format="GIF" Height="45" Rendition="HTML" Resolution="72" Type="Linedraw" Width="406" /> </MediaObject> <EquationSource Format="TEX">\(\begin{aligned} T_{\Omega ,\,a}^*f(x)=\sup _{\epsilon&gt;0}\Big |\int _{|x-y|&gt;\epsilon }\frac{\Omega (x-y)}{|x-y|^{d+1}} \big (a(x)-a(y)\big )f(y)dy\Big |, \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msubsup> <mi>T</mi> <mrow> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mspace width="0.166667em" /> <mi>a</mi> </mrow> <mo>∗</mo> </msubsup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <munder> <mo movablelimits="true">sup</mo> <mrow> <mi>ϵ</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </munder> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">|</mo> </mrow> <msub> <mo>∫</mo> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo>-</mo> <mi>y</mi> <mo stretchy="false">|</mo> <mo>&gt;</mo> <mi>ϵ</mi> </mrow> </msub> <mfrac> <mrow> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>-</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo>-</mo> <mi>y</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>d</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mfrac> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> <mi>y</mi> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">|</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3152_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Omega \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is homogeneous of degree zero, integrable on <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3152_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">\(S^{d-1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>S</mi> <mrow> <mi>d</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> and has vanishing moment of order one, <i>a</i> is a function on <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3152_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb {R}}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3152_Article_IEq4.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="104" /> </InlineMediaObject> <EquationSource Format="TEX">\(\nabla a\in L^{\infty }({\mathbb {R}}^d).\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">∇</mi> <mi>a</mi> <mo>∈</mo> <msup> <mi>L</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> The authors give an elemental method for establishing the weak type endpoint estimate of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3152_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">\(L\log \log L\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo>log</mo> <mo>log</mo> <mi>L</mi> </mrow> </math></EquationSource> </InlineEquation> type for the maximal commutator <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3152_Article_IEq6.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">\(T^*_{\Omega ,\,a},\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>T</mi> <mrow> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mspace width="0.166667em" /> <mi>a</mi> </mrow> <mo>∗</mo> </msubsup> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> if <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3152_Article_IEq7.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="134" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Omega \in L\log L(S^{d-1}).\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>∈</mo> <mi>L</mi> <mo>log</mo> <mi>L</mi> <mo stretchy="false">(</mo> <msup> <mi>S</mi> <mrow> <mi>d</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>

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An endpoint estimate for the maximal Calderón commutator with rough kernel

  • Guoen Hu,
  • Xudong Lai,
  • Xiangxing Tao,
  • Qingying Xue

摘要

In this paper, the authors consider the endpoint boundedness properties for the rough maximal Calderón commutator defined by \(\begin{aligned} T_{\Omega ,\,a}^*f(x)=\sup _{\epsilon>0}\Big |\int _{|x-y|>\epsilon }\frac{\Omega (x-y)}{|x-y|^{d+1}} \big (a(x)-a(y)\big )f(y)dy\Big |, \end{aligned}\) T Ω , a f ( x ) = sup ϵ > 0 | | x - y | > ϵ Ω ( x - y ) | x - y | d + 1 ( a ( x ) - a ( y ) ) f ( y ) d y | , where \(\Omega \) Ω is homogeneous of degree zero, integrable on \(S^{d-1}\) S d - 1 and has vanishing moment of order one, a is a function on \({\mathbb {R}}^d\) R d such that \(\nabla a\in L^{\infty }({\mathbb {R}}^d).\) a L ( R d ) . The authors give an elemental method for establishing the weak type endpoint estimate of \(L\log \log L\) L log log L type for the maximal commutator \(T^*_{\Omega ,\,a},\) T Ω , a , if \(\Omega \in L\log L(S^{d-1}).\) Ω L log L ( S d - 1 ) .