This paper investigates gradient estimates on graphs satisfying the <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2133_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="100" /> </InlineMediaObject> <EquationSource Format="TEX">$CD\psi (n,-K)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mi>D</mi> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mo>-</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> condition with positive constants n, K, and concave <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2133_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$C^{1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> functions <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2133_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="123" /> </InlineMediaObject> <EquationSource Format="TEX">$\psi :(0,+\infty )\rightarrow \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ψ</mi> <mo>:</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mi>∞</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>. Our study focuses on gradient estimates for positive solutions of the heat equation <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2133_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">$\partial _{t}u=\Delta u$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>∂</mi> <mi>t</mi> </msub> <mi>u</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> </mrow> </math></EquationSource> </InlineEquation>. Additionally, the estimate is extended to a heat-type equation <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2133_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="115" /> </InlineMediaObject> <EquationSource Format="TEX">$\partial _{t}u=\Delta u+cu^{\sigma }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>∂</mi> <mi>t</mi> </msub> <mi>u</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <mi>c</mi> <msup> <mi>u</mi> <mi>σ</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2133_Article_IEq8.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> is a constant and c is a continuous function defined on <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2133_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$[0,+\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, we utilize these estimates to derive heat kernel bounds and Harnack inequalities.

Gradient Estimates on Graphs with the $CD\psi (n,-K)$ Condition

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发表日期：

2025年8月22日

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Yi Li,
Qianwei Zhang

摘要

This paper investigates gradient estimates on graphs satisfying the $CD\psi (n,-K)$ $C D ψ (n, - K)$ condition with positive constants n, K, and concave $C^{1}$ $C^{1}$ functions $\psi :(0,+\infty )\rightarrow \mathbb {R}$ $ψ : (0, + \infty) \to R$ . Our study focuses on gradient estimates for positive solutions of the heat equation $\partial _{t}u=\Delta u$ $\partial_{t} u = Δ u$ . Additionally, the estimate is extended to a heat-type equation $\partial _{t}u=\Delta u+cu^{\sigma }$ $\partial_{t} u = Δ u + c u^{σ}$ , where $\sigma $ $σ$ is a constant and c is a continuous function defined on $[0,+\infty )$ $[0, + \infty)$ . Furthermore, we utilize these estimates to derive heat kernel bounds and Harnack inequalities.

Gradient Estimates on Graphs with the \(CD\psi (n,-K)\) Condition

摘要