Intrinsic Ultracontractivity for Domains in Negatively Curved Manifolds
Intrinsic Ultracontractivity for Domains in Negatively Curved Manifolds
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负曲流形域的本征超收缩性
DOI：
10.1007/s40315-021-00402-8
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发表时间：
2021
期刊：
影响因子：
--
通讯作者：
J.
中科院分区：
文献类型：
--
作者：
Aikawa;H.;van den Berg;M. & Masamune;J.
设M是一个完备的、非紧的、连通的黎曼流形，其Ricci曲率由一个负常数从下有界。得到了Dirichlet热半群对应的开集和连通集DinM本质超压缩的一个充分条件。这一条件是用电容宽度表示的。结果表明，当Dirichlet拉普拉斯函数足够小时，Dirichlet拉普拉斯函数谱底的倒数和扭转函数的上确界与Df的电容宽度的平方是相当的。技术关键是Dirichlet热核在有限尺度下的体积倍增性质、Poincaré不等式和Li-Yau Gauss估计。
LetMbe a complete, non-compact, connected Riemannian manifold with Ricci curvature bounded from below by a negative constant. A sufficient condition is obtained for open and connected setsDinMfor which the corresponding Dirichlet heat semigroup is intrinsically ultracontractive. That condition is formulated in terms of capacitary width. It is shown that both the reciprocal of the bottom of the spectrum of the Dirichlet Laplacian acting in, and the supremum of the torsion function forDare comparable with the square of the capacitary width forDif the latter is sufficiently small. The technical key ingredients are the volume doubling property, the Poincaré inequality and the Li-Yau Gaussian estimate for the Dirichlet heat kernel at finite scale.