<p>Let <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2169_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$M_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2169_Article_IEq4.gif" Format="GIF" Height="9" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$\cdots $</EquationSource> <EquationSource Format="MATHML"><math> <mo>⋯</mo> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2169_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">$M_\ell $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mi>ℓ</mi> </msub> </math></EquationSource> </InlineEquation> be complete, connected and non-collapsed manifolds of the same dimension, where <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2169_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">$2\le \ell \in \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>≤</mo> <mi>ℓ</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>, and suppose that each <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2169_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$M_i$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation> satisfies a doubling condition and a Gaussian upper bound for the heat kernel. If each manifold <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2169_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$M_i$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation> has volume growth either bigger than two or equal to two, then we show that the Riesz transform <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2169_Article_IEq9.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">$\nabla {\mathscr {L}}^{-1/2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">∇</mi> <msup> <mrow> <mi mathvariant="script">L</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> is bounded on <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2169_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$L^p(M)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for each <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2169_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$1<p<2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> on the gluing manifold <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2169_Article_IEq12.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="180" /> </InlineMediaObject> <EquationSource Format="TEX">$M=M_1\#M_2\#\cdots \# M_\ell .$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mo>=</mo> <msub> <mi>M</mi> <mn>1</mn> </msub> <mo>#</mo> <msub> <mi>M</mi> <mn>2</mn> </msub> <mo>#</mo> <mo>⋯</mo> <mo>#</mo> <msub> <mi>M</mi> <mi>ℓ</mi> </msub> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>

Riesz Transform on Manifolds with Ends of Different Volume Growth for \(1

期刊论文
发表日期：

2025年9月6日

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Ren-Jin Jiang,
Hong-Quan Li,
Hai-Bo Lin

摘要

Let $M_1$ $M_{1}$ , $\cdots $ $\dots$ , $M_\ell $ $M_{ℓ}$ be complete, connected and non-collapsed manifolds of the same dimension, where $2\le \ell \in \mathbb {N}$ $2 \leq ℓ \in N$ , and suppose that each $M_i$ $M_{i}$ satisfies a doubling condition and a Gaussian upper bound for the heat kernel. If each manifold $M_i$ $M_{i}$ has volume growth either bigger than two or equal to two, then we show that the Riesz transform $\nabla {\mathscr {L}}^{-1/2}$ $\nabla L^{- 1 / 2}$ is bounded on $L^p(M)$ $L^{p} (M)$ for each $1<p<2$ $1 < p < 2$ on the gluing manifold $M=M_1\#M_2\#\cdots \# M_\ell .$ $M = M_{1} # M_{2} # \dots # M_{ℓ} .$