具有 $${{v\arepsilon }$$ 范围的 Bakry-Émery-Ricci 曲率下限下的一些函数不等式

Pub Date : 2024-04-02 DOI:10.1007/s00229-024-01537-3

Yasuaki Fujitani

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引用次数: 0

摘要

对于 n 维加权黎曼流形，Lu-Minguzzi-Ohta（Anal Geom Metr Spaces 10(1):1-30,2022）提出的具有 ${\varepsilon }$-range 的下 m-Bakry-Émery-Ricci 曲率界值，以权重函数的形式整合了常数下界和某些变量下界。在本文中，我们证明了具有 ${v\arepsilon }$-range 的 m-Bakry-Émery-Ricci 曲率下限下的 Cheng 型不等式和局部 Sobolev 不等式。这些将恒定曲率边界下的（m （n,\infty ））不等式推广到（m （-\infty ,1]\cup \\{infty \}）。

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Some functional inequalities under lower Bakry–Émery–Ricci curvature bounds with $${\varepsilon }$$ -range

For n-dimensional weighted Riemannian manifolds, lower m-Bakry–Émery–Ricci curvature bounds with ${\varepsilon }$-range, introduced by Lu-Minguzzi-Ohta (Anal Geom Metr Spaces 10(1):1–30, 2022), integrate constant lower bounds and certain variable lower bounds in terms of weight functions. In this paper, we prove a Cheng type inequality and a local Sobolev inequality under lower m-Bakry–Émery–Ricci curvature bounds with ${\varepsilon }$-range. These generalize those inequalities under constant curvature bounds for $m \in (n,\infty )$ to $m\in (-\infty ,1]\cup \{\infty \}$.

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