Archiv der Mathematik ( IF 0.6 ) Pub Date : 2021-06-05 , DOI: 10.1007/s00013-021-01625-w Imen Metoui
It is shown that the Ornstein–Uhlenbeck operator perturbed by a multipolar inverse square potential
$$\begin{aligned} A_{\Phi ,G}+V=\Delta -\nabla \Phi \cdot \nabla +G\cdot \nabla +\sum \limits _{i=1}^{n}\frac{c}{|x-a_{i}|^{2}} \end{aligned}$$with suitable domain generates a quasi-contractive and positive analytic \(C_{0}\)-semigroup on the weighted space \(L^{2}(\mathbb {R}^{N},d\mu )\), where \(d\mu =\exp (-\Phi (x))dx\), \(\Phi \in C^{2}(\mathbb {R}^{N}, \mathbb {R})\), \(G \in C^{1}(\mathbb {R}^{N},\mathbb {R}^{N})\), \(c>0\), and \(a_{1},\ldots , a_{n}\in \mathbb {R}^{N}\). The proofs are based on an \(L^{2}\)-weighted Hardy inequality and bilinear form techniques.
中文翻译：
广义 Ornstein-Uhlenbeck 算子受 $$L^{2}$$ L 2 空间中多极反平方势扰动
结果表明，Ornstein-Uhlenbeck 算子受多极平方反比势扰动
$$\begin{aligned} A_{\Phi ,G}+V=\Delta -\nabla \Phi \cdot \nabla +G\cdot \nabla +\sum \limits _{i=1}^{n}\压裂{c}{|x-a_{i}|^{2}} \end{aligned}$$具有合适的域在加权空间\(L^{2}(\mathbb {R}^{N},d\mu )\) 上生成准收缩和正解析\(C_{0}\) -半群，其中\(d\mu =\exp (-\Phi (x))dx\) , \(\Phi \in C^{2}(\mathbb {R}^{N}, \mathbb {R})\ )、\(G \in C^{1}(\mathbb {R}^{N},\mathbb {R}^{N})\)、\(c>0\)和\(a_{1 },\ldots , a_{n}\in \mathbb {R}^{N}\)。证明基于\(L^{2}\)加权哈代不等式和双线性形式技术。
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