<p>Let <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2045_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="62" /> </InlineMediaObject> <EquationSource Format="TEX">$(X,d,\mu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> denote a space of homogeneous type and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2045_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="95" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {T} :=\{T_\varepsilon \}_{\varepsilon >0}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">T</mi> <mo>:</mo> <mo>=</mo> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>T</mi> <mi>ε</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> be a family of truncated <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2045_Article_IEq6.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation>-Calderón–Zygmund singular integral operators with the kernel <i>K</i> satisfying that for any <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2045_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="120" /> </InlineMediaObject> <EquationSource Format="TEX">$0<\varepsilon \le N< \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>ε</mi> <mo>≤</mo> <mi>N</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, <Equation ID="Equ41"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2045_Article_Equ41.gif" Format="GIF" Height="44" Rendition="HTML" Resolution="72" Type="Linedraw" Width="404" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \int _{\varepsilon \le d(x,y) \le N}K(x,y)d\mu (y)=\int _{\varepsilon \le d(x,y) \le N}K(x,y)d\mu (x)=0. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mo>∫</mo> <mrow> <mi>ε</mi> <mo>≤</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>N</mi> </mrow> </msub> <mi>K</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> <mi>μ</mi> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow> <mi>ε</mi> <mo>≤</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>N</mi> </mrow> </msub> <mi>K</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> <mi>μ</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>The authors obtain the <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2045_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-boundedness of jump operators and variations for the family <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2045_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {T}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">T</mi> </math></EquationSource> </InlineEquation> on spaces of homogeneous type with an additional layer decay property.</p>

The $L^2$-Boundedness of Jump and Variational Operators on Spaces of Homogeneous Type

期刊论文
发表日期：

2025年6月16日

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Tuonan Chen,
Dongyong Yang,
Feng Zhang

摘要

Let $(X,d,\mu )$ $(X, d, μ)$ denote a space of homogeneous type and $\mathcal {T} :=\{T_\varepsilon \}_{\varepsilon >0}$ $T : = {T_{ε}}_{ε > 0}$ be a family of truncated $\omega $ $ω$ -Calderón–Zygmund singular integral operators with the kernel K satisfying that for any $0<\varepsilon \le N< \infty $ $0 < ε \leq N < \infty$ , $\begin{aligned} \int _{\varepsilon \le d(x,y) \le N}K(x,y)d\mu (y)=\int _{\varepsilon \le d(x,y) \le N}K(x,y)d\mu (x)=0. \end{aligned}$ $\begin{matrix} \int_{ε \leq d (x, y) \leq N} K (x, y) d μ (y) = \int_{ε \leq d (x, y) \leq N} K (x, y) d μ (x) = 0 . \end{matrix}$ The authors obtain the $L^2$ $L^{2}$ -boundedness of jump operators and variations for the family $\mathcal {T}$ $T$ on spaces of homogeneous type with an additional layer decay property.

The \(L^2\)-Boundedness of Jump and Variational Operators on Spaces of Homogeneous Type

摘要