Intrinsic ultracontractivity and conditional gauge for symmetric stable processes (Q1371035)

The theories of Green functions and Poisson kernels are systematically extended to a symmetric stable process with index \(\alpha\), killed at the boundary of a \(C^{1,1}\) domain \(D\). The intrinsic ultracontractivity, i.e. the ultracontractivity of Doob's \(h\)-process with \(h=\varphi\), where \(\varphi\) is the principal eigenfunction of the generator of this killed process, is obtained by using a logarithmic Sobolev inequality. This \(\varphi\)-process and its behavior are the analogue of the killed conditioning process in the diffusion case, see \textit{G. Gong, M. Qian} and \textit{Z. Zhao} [Probab. Theory Relat. Fields 80, No. 1, 151-167 (1988; Zbl 0631.60073)]. Then for the Feynman-Kac semigroup of the killed process with the potential term \(q\in K_{n,\alpha}\) (the Kato class): \(\lim_{r\to 0}\sup_{x\in R^n}\int_{|x-y|\leq r}\frac{|q(y)|}{|x-y|^{n-\alpha}}dy=0\), the following conditional gauge theorem \[ 1/c\leq\inf_{x,y\in D} E_y^x[ e_q (\tau_{D \setminus \{ y\}})]\leq\sup_{x,y \in D} E_y^x[ e_q (\tau_{D \setminus \{ y \}})]\leq c \] with \(e_q(t)=e^{\int _0^t |q(X_s)|ds}\), where \(\tau_{D\setminus{\{ y \}}}\) is the life time of the killed process, is proven.