Completeness of Length-Weighted Sobolev Metrics on the Space of Curves

Gradient estimates for weighted $p$-Laplacian equations on Riemannian manifolds with a Sobolev inequality and integral Ricci bounds

In this paper, we consider the non-linear general $p$-Laplacian equation $Δ_{p,f}u+F(u)=0$ for a smooth function $F$ on smooth metric measure spaces. Assume that a Sobolev inequality holds true on $M$ and an integral Ricci curvature is small, we first prove a local gradient estimate for the equation. Then, as its applications, we prove several Liouville type results on manifolds with lower bounds of Ricci curvature. We also derive new local gradient estimates provided that the integral Ricci curvature is small enough.

加权周围空间中子流形的Sobolev不等式与等周不等式

Sobolev and isoperimetric inequalities for submanifolds in weighted ambient spaces

本文证明了加权流形中子流形的一个Sobolev不等式和等周不等式。我们的结果推广了Hoffman-Spruck不等式。

从SuperPoincaré到加权Log-Sobolev不等式和熵代价不等式

From Super Poincaré to Weighted Log-Sobolev and Entropy-Cost Inequalities

从一类超Poincaré不等式出发，得到了加权log-Sobolev不等式.作为应用，得到了距离较大的Talagrand不等式.特别地，在完备连通黎曼流形上，证明了具有$\dd\in (1,2)$的$\log^\dd$-Sobolev不等式对某个常数$C>0$蕴涵$L^{2/(2-\dd)}$ -运输费用不等式$$W^\rr_{2/(2-\dd)}(fμ,μ)^{2/(2-\dd)}\le Cμ(f\log f), μ(f)=1, f\ge 0$$，并且当相应生成元的曲率有下界时，它们是等价的.加权log-Sobolev和熵成本不等式也推导出一个大类的概率测度$\R^d$。

孤立锥奇点的去奇异化：加权索伯列夫空间的一致估计

Desingularizing isolated conical singularities: Uniform estimates via weighted Sobolev spaces

我们定义了一个非常一般的“参数连通和”构造，它可以用来消除黎曼流形的孤立锥奇点。然后，我们表明，各种重要的分析和椭圆估计，制定加权索伯列夫空间，可以得到独立的参数中使用的建设。具体地说，我们证明了与（i）Sobolev嵌入定理，（ii）拉普拉斯算子的可逆性和（iii）Poincare'和Gagliardo-Nirenberg-Sobolev型不等式有关的一致估计。我们的主要工具是著名的理论加权索伯列夫空间和椭圆算子的“锥”。我们提供了一个概述，连同前一般黎曼流形的延伸。对于我们的结果的几何应用，我们请读者参考我们的论文“特殊拉格朗日锥，II：胶合结构在C^m”。

Geodesic completeness of the $H^{3/2}$ metric on $\mathrm{Diff}(S^{1})$

Of concern is the study of the long-time existence of solutions to the Euler--Arnold equation of the right-invariant $H^{3/2}$-metric on the diffeomorphism group of the circle. In previous work by Escher and Kolev it has been shown that this equation admits long-time solutions if the order $s$ of the metric is greater than $3/2$, the behaviour for the critical Sobolev index $s=3/2$ has been left open. In this article we fill this gap by proving the analogous result also for the boundary case. The behaviour of the $H^{3/2}$-metric is, however, still different from its higher order counter parts…

具有非均匀各向异性介质的不规则外区域上加权索伯列夫空间的Hodge-Helmholtz分解

Hodge-Helmholtz Decompositions of Weighted Sobolev Spaces in Irregular Exterior Domains with Inhomogeneous and Anisotropic Media

我们详细研究了充满非均匀和各向异性介质的非光滑外部区域中的Hodge-Helmholtz分解。我们展示了属于加权索伯列夫空间的交替微分形式分解成无旋和螺线管形式。这些分解是重要的工具，例如，在电磁理论的外部域。在附录中，我们将我们的结果转化为向量分析的经典框架。

尖锐的加权CR迹Sobolev不等式

Sharp weighted CR trace Sobolev inequalities

本文在Siegel域上建立了涉及加权范数的尖锐Sobolev迹不等式，并将其与Heisenberg群上次Laplace算子分数幂的广义Dirichlet-to-Neumann实现联系起来，推广了已有结果。

加权图的流分解与随机完备性

Flow decomposition and stochastic completeness of weighted graphs

本文通过流分解方法，利用有限域估计和相对容量，给出了加权图随机完备性的Grigor'yan型积分准则的新证明，并恢复了体积准则。

Sobolev spaces and $\nabla$-differential operators on manifolds I: basic properties and weighted spaces

We study {\em $\nabla$-Sobolev spaces} and {\em $\nabla$-differential operators} with coefficients in general Hermitian vector bundles on Riemannian manifolds, stressing a coordinate free approach that uses connections (which are typically denoted $\nabla$). These concepts arise naturally from Partial Differential Equations, including some that are formulated on plain Euclidean domains, such as the weighted Sobolev spaces used to study PDEs on singular domains. We prove several basic properties of the $\nabla$-Sobolev spaces and of the $\nabla$-differential operators on general manifolds. For …