发表机构
Institut für Mathematik, Universität Potsdam（波茨坦大学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对图上逐点高斯上界刻画存在顶点度数依赖误差的问题，提出时空平均形式的热半群，结合积分版Davies方法从大尺度Faber-Krahn不等式推导无误差的高斯上界，还证明了反向蕴含关系，给出热核渐近行为的泛函不等式刻画。
AI 中文摘要
在几何可能无界的图上，用局部化泛函不等式刻画逐点高斯上界时，会出现依赖顶点度数的误差。我们引入了一种新的时空平均形式的热半群，基于时间平均的$\boldsymbol{\frac{q}{q-1}}-\boldsymbol{q}$型估计，并从大尺度Faber-Krahn不等式出发推导高斯上界，避免了这类误差。核心分析工具是Davies方法的积分版本，它能给出该平均量的非对角估计，且当时间趋于无穷时，该估计会收敛到逐点高斯界。反之，在大尺度下，体积加倍性质与该平均热半群范数的高斯界，可推出具有指定相对测度的子集的相对Faber-Krahn不等式。我们的证明基于用这些平均量表示的Dirichlet特征值下界。Faber-Krahn维数是尺度依赖的，但随着半径增大，它会收敛到加倍维数。这一结果用Faber-Krahn不等式刻画了高斯热核的渐近行为。
英文摘要
Characterizations of pointwise Gaussian upper bounds on graphs with possibly unbounded geometry in terms of localized functional inequalities contain errors depending on the vertex degree. We introduce a new space-time averaged form of the heat semigroup in terms of time-averaged $\ell^{q/(q-1)}-\ell^q$-estimates and obtain Gaussian upper bounds from large-scale Faber-Krahn inequalities which avoid such errors. A main analytic ingredient is an integrated version of Davies' method which yields off-diagonal estimates for this averaged quantity, and which tends to pointwise Gaussian bounds as time tends to infinity. Conversely, on large scales volume doubling and Gaussian bounds on this averaged heat semigroup norms imply relative Faber-Krahn inequalities for subsets of prescribed relative measure. Our proof is based on a lower bound for Dirichlet eigenvalues in terms of these averages. The Faber-Krahn dimension is scale-dependent, but converges to the doubling dimension for increasing radii. This gives a characterization of asymptotic Gaussian heat kernel behavior in terms of Faber-Krahn inequalities.
Comments24 pages. Comments are welcome!
