Dirichlet Laplace算子迹的泛界
Universal Bounds for Traces of the Dirichlet Laplace Operator
摘要：
本文给出了Dirichlet Laplace算子的热核$Z(t)$在开集$Ω\subset \R^d$，$d \geq 2$上的迹的上界.在有限体积区域上，所得结果改进了Kac的一个不等式.利用同样的方法，我们给出了无限体积区域上$Z(t)$的界.对于有限体积的区域，当$t$趋于无穷大时，$Z(t)$上的界指数衰减，它包含反映$Z(t)$短时渐近性质的尖锐首项和修正项.为了证明这个结果，我们采用了特征值平均的精化Berezin-Li-Yau不等式。
英文摘要：
We derive upper bounds for the trace of the heat kernel $Z(t)$ of the Dirichlet Laplace operator in an open set $Ω\subset \R^d$, $d \geq 2$. In domains of finite volume the result improves an inequality of Kac. Using the same methods we give bounds on $Z(t)$ in domains of infinite volume. For domains of finite volume the bound on $Z(t)$ decays exponentially as $t$ tends to infinity and it contains the sharp first term and a correction term reflecting the properties of the short time asymptotics of $Z(t)$. To prove the result we employ refined Berezin-Li-Yau inequalities for eigenvalue means.
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