<p> We investigate the sparse bound for maximal oscillatory singular integrals given by<Equation ID="Equa"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_95_Article_Equa.gif" Format="GIF" Height="45" Rendition="HTML" Resolution="72" Type="Linedraw" Width="342" /> </MediaObject> <EquationSource Format="TEX">$T_{P,K}^f(x)=\sup_{\epsilon>0} \bigg| \int_{|x-y|>\epsilon}e^{iP(x,y)}K(x,y)f(y) \, dy \bigg| ,$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msubsup> <mi>T</mi> <mrow> <mi>P</mi> <mo>,</mo> <mi>K</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>></mo> <mn>0</mn> </mrow> </munder> <mrow> <mo maxsize="2.047em" minsize="2.047em" 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>></mo> <mi>ϵ</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow> <mi>i</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mi>K</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="0.166667em" /> <mi>d</mi> <mi>y</mi> <mrow> <mo maxsize="2.047em" minsize="2.047em" stretchy="true">|</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_95_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">$P(x,y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a real-valued polynomial on <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_95_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb{R}^n\times \mathbb{R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo>×</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_95_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$K$</EquationSource> <EquationSource Format="MATHML"><math> <mi>K</mi> </math></EquationSource> </InlineEquation> is a Calderón–Zygmund non-convolutional type kernel. We show that <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_95_Article_IEq5.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">$T_{P,K}^$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>T</mi> <mrow> <mi>P</mi> <mo>,</mo> <mi>K</mi> </mrow> <mo>∗</mo> </msubsup> </math></EquationSource> </InlineEquation> satisfies an <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_95_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">$(r,r)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-sparse bound for <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_95_Article_IEq7.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$1<r<2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>r</mi> <mo><</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, which implies the weighted <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_95_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="111" /> </InlineMediaObject> <EquationSource Format="TEX">$L^p(1<p<\infty)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> estimate for the operator <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_95_Article_IEq5.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">$T_{P,K}^*$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>T</mi> <mrow> <mi>P</mi> <mo>,</mo> <mi>K</mi> </mrow> <mo>∗</mo> </msubsup> </math></EquationSource> </InlineEquation>.</p>

Sparse bounds for maximally truncated oscillatory singular integrals with non-convolutional Hölder class kernels

期刊论文
发表日期：

2025年6月19日

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W. Sun,
S. Wang

摘要

We investigate the sparse bound for maximal oscillatory singular integrals given by $T_{P,K}^*f(x)=\sup_{\epsilon>0} \bigg| \int_{|x-y|>\epsilon}e^{iP(x,y)}K(x,y)f(y) \, dy \bigg| ,$ $T_{P, K}^{*} f (x) = sup_{ϵ > 0} | \int_{| x - y | > ϵ} e^{i P (x, y)} K (x, y) f (y) d y |,$ where $P(x,y)$ $P (x, y)$ is a real-valued polynomial on $\mathbb{R}^n\times \mathbb{R}^n$ $R^{n} \times R^{n}$ and $K$ $K$ is a Calderón–Zygmund non-convolutional type kernel. We show that $T_{P,K}^*$ $T_{P, K}^{*}$ satisfies an $(r,r)$ $(r, r)$ -sparse bound for $1<r<2$ $1 < r < 2$ , which implies the weighted $L^p(1<p<\infty)$ $L^{p} (1 < p < \infty)$ estimate for the operator $T_{P,K}^*$ $T_{P, K}^{*}$ .