Toward a geometric characterization of intrinsic ultracontractivity for Dirichlet Laplacians
Denote the transition density of the killed Brownian motion on a \(\mathbb{R}^2\) domain D by \(p_t^D(x,y)\). A domain D is called intrinsically ultracontractive (IU), if the \(L^2(D,\varphi_0^2 dx)\) kernel \[ e^{\lambda t}p_t^D(x,y)/\varphi_0(x)\varphi_0(y) \] is ultracontractive, where \(\varphi_0\) is the positive normalized eigenfunction of the main eigenvalue of \(\frac 12\Delta\) on D. It is then an interesting problem to describe the geometric characterization for IU for a rather wide class of domains. A simply connected domain D is called having the wide access property (WA), if there is \(o\in D\), such that for any \(x\in D\) there exists a path \(\Gamma\) from \(o\) to \(x\) with \(\inf_{z\in\Gamma}d(z)\geq C_{\Gamma} d(x)\), where d(y) denotes the distance from y to the boundary of D. In the present paper, a conjecture posed by Benelos-Davis on the necessary and sufficient condition of IU property of WA domains is partly and mostly solved. And when D is a tube along a geodesic, the necessary and sufficient condition is also found in this paper, which extends a result of Benelos.
scientific article; zbMATH DE number 1548340