Generalized Ornstein-Uhlenbeck operators perturbed by multipolar inverse square potentials in \(L^2\)-spaces
Consider an Ornstein-Uhlenbeck opeartor \[ A_{\Phi,G}=\Delta-\nabla\Phi\cdot\nabla+G\cdot\nabla \] and assume that \begin{itemize} \item \(\Phi\in C^2(\mathbb{R}^N,\mathbb{R})\) and \(\int_{\mathbb{R}^N}\exp(-\Phi(x))dx<+\infty\) \item \(G\in C^1(\mathbb{R}^N,\mathbb{R}^N)\) and \(|G|\leq const (|\nabla\Phi|+1)\) \item for every \(\tau>0\) there exists \(C_\tau>0\) s.t. \(|D^2\Phi|+|\mathrm{div} G|\leq \tau|\nabla\Phi|^2+C_\tau\) \item there exists \(\beta\geq0\) s.t. \(\mathrm{div} G-G\cdot\nabla\Phi\geq-\beta\) \end{itemize} Consider potential \(V=\sum_{i=1}^n c/|x-a_i|^2\). If \(N\geq3\) and \(c\in(0,(N-2)^2/16)\), then the operator \(-A=A_{\Phi,G}+V\) generates a quasi-contractive, positive and analytic semigroup on weighted space \(L^2_\mu(\mathbb{R}^N)\) with \(d\mu=\exp(-\Phi(x))dx\). Weighthed Hardy-type inequalities are obtained.
scientific article; zbMATH DE number 7394764
scientific article