Laplacian截断函数序列
Sequences of Laplacian cut-off functions
摘要：
本文给出了拉普拉斯截断函数序列概念在黎曼流形上的几个新应用（我们证明存在于具有非负里奇曲率的测地线完备黎曼流形上）：特别地，我们证明了该存在性蕴含着梯度的$\mathsf{L}^q$ -估计，这是索伯列夫空间中光滑紧支撑函数在整个$\mathsf{L}^q$ -尺度上的一个新的密度结果，以及Braverman，Milatovic和Shubin关于$(-Δ+1)f\geq 0$的$\mathsf{L}^2$ -解$f$的非负性的猜想的一个稍弱和稍强的变体.后一个事实是证明了一个新的概念的积极性保持的黎曼流形，这是有关随机完备性。
英文摘要：
We derive several new applications of the concept of sequences of Laplacian cut-off functions on Riemannian manifolds (which we prove to exist on geodesically complete Riemannian manifolds with nonnegative Ricci curvature): In particular, we prove that this existence implies $\mathsf{L}^q$-estimates of the gradient, a new density result of smooth compactly supported functions in Sobolev spaces on the whole $\mathsf{L}^q$-scale, and a slightly weaker and slightly stronger variant of the conjecture of Braverman, Milatovic and Shubin on the nonnegativity of $\mathsf{L}^2$-solutions $f$ of $(-Δ+1)f\geq 0$. The latter fact is proved within a new notion of positivity preservation for Riemannian manifolds which is related to stochastic completeness.
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