论文标题
高斯估计高阶的热核Schrödinger操作员具有广义Schechter类中的潜力
Gaussian Estimates for Heat Kernels of Higher Order Schrödinger Operators with Potentials in Generalized Schechter Classes
论文作者
论文摘要
令$ m \ in \ mathbb n $，$ p（d）：= \ sum_ {|α| = 2M}（ - 1）^ma_αd^α$是$ 2M $ $级 - 阶椭圆型运算符，具有$ \ MATHBB {r}^n $和$ v $ a $ a $ a $ a $ a $ a $ n $ n of con的均等椭圆型运算符和$ a $ a n $ n n $ n n $ n n $ n con。在本文中，作者介绍了有关$ v $的新的广义Schechter类，并表明高级schrödingeroperator $ \ nathcal {l}：= p（d）+v $具有满足高斯上限的热核，而当$ v $属于这个新班级时，则具有满足高斯上限和Höld的规律性。戴维斯（Davies） - 加夫尼（Gaffney）对相关的半群及其本地版本的估计。这些结果为许多进一步研究$ \ MATHCAL {L} $的进一步研究铺平了道路。
Let $m\in\mathbb N$, $P(D):=\sum_{|α|=2m}(-1)^m a_αD^α$ be a $2m$-order homogeneous elliptic operator with real constant coefficients on $\mathbb{R}^n$, and $V$ a measurable function on $\mathbb{R}^n$. In this article, the authors introduce a new generalized Schechter class concerning $V$ and show that the higher order Schrödinger operator $\mathcal{L}:=P(D)+V$ possesses a heat kernel that satisfies the Gaussian upper bound and the Hölder regularity when $V$ belongs to this new class. The Davies--Gaffney estimates for the associated semigroup and their local versions are also given. These results pave the way for many further studies on the analysis of $\mathcal{L}$.
