<p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1905_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="97" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega \in L^1(\mathbb {S}^{n-1}) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>∈</mo> <msup> <mi>L</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and consider the maximal function <Equation ID="Equ38"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1905_Article_Equ38.gif" Format="GIF" Height="44" Rendition="HTML" Resolution="72" Type="Linedraw" Width="398" /> </MediaObject> <EquationSource Format="TEX">$M_\Omega f(x)=\sup _{r>0}\frac{1}{r^n}\int _{|y|<r}| \Omega (y/|y|)||f(x-y)|dy,\ \ x\in \mathbb {R}^n.$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi>M</mi> <mi mathvariant="normal">Ω</mi> </msub> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <munder> <mo movablelimits="true">sup</mo> <mrow> <mi>r</mi> <mo>></mo> <mn>0</mn> </mrow> </munder> <mfrac> <mn>1</mn> <msup> <mi>r</mi> <mi>n</mi> </msup> </mfrac> <msub> <mo>∫</mo> <mrow> <mo stretchy="false">|</mo> <mi>y</mi> <mo stretchy="false">|</mo> <mo><</mo> <mi>r</mi> </mrow> </msub> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>y</mi> </mrow> <mo stretchy="false">/</mo> <mrow> <mo stretchy="false">|</mo> <mi>y</mi> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> <mo stretchy="false">|</mo> <mo stretchy="false">|</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>-</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> <mi>d</mi> <mi>y</mi> <mo>,</mo> <mspace width="4pt" /> <mspace width="4pt" /> <mi>x</mi> </mrow> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo>.</mo> </mrow> </math></EquationSource> </Equation>We prove that <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1905_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">$M_\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mi mathvariant="normal">Ω</mi> </msub> </math></EquationSource> </InlineEquation> is of weak type (1, 1) if the rough kernel function <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1905_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> belongs to the block space <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1905_Article_IEq4.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">$B_2^{0,0}(\mathbb {S}^{n-1})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>B</mi> <mn>2</mn> <mrow> <mn>0</mn> <mo>,</mo> <mn>0</mn> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. The main result substantially extends a classical result of Christ and Rubio de Francia (Invent Math 93:225–237, 1988).</p>

Weak Type (1, 1) Bounds for Maximal Functions with Rough Kernels

期刊论文
发表日期：

2025年2月6日

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Yanping Chen,
Feng Liu,
Huoxiong Wu

摘要

Let $\Omega \in L^1(\mathbb {S}^{n-1}) $ $Ω \in L^{1} (S^{n - 1})$ and consider the maximal function $M_\Omega f(x)=\sup _{r>0}\frac{1}{r^n}\int _{|y|<r}| \Omega (y/|y|)||f(x-y)|dy,\ \ x\in \mathbb {R}^n.$ $M_{Ω} f (x) = sup_{r > 0} \frac{1}{r^{n}} \int_{| y | < r} | Ω (y / | y |) | | f (x - y) | d y, x \in R^{n} .$ We prove that $M_\Omega $ $M_{Ω}$ is of weak type (1, 1) if the rough kernel function $\Omega $ $Ω$ belongs to the block space $B_2^{0,0}(\mathbb {S}^{n-1})$ $B_{2}^{0, 0} (S^{n - 1})$ . The main result substantially extends a classical result of Christ and Rubio de Francia (Invent Math 93:225–237, 1988).