| In Chapter 2, we study a conjecture concerning a geometrical characterization in terms of the areas of Whitney cubes along quasihyperbolic geodesics, for Intrinsic Ultracontractivity (IU) of the semigroup associated to the Dirichlet Laplacian in domains which have the "wide access" property. In particular, we prove that any such domain which is IU satisfies this geometric condition. We also prove that this condition characterizes IU for tubes along geodesics, and that it implies one-half intrinsic ultracontractivity for domains with the "wide access" property.; In Chapter 3, we study the moments of integrability of tq,a , the first exit time of an n-dimensional symmetric alpha-stable process, alpha ∈ (0, 2), from a circular cone of angle q,0<q<p . We show that there exists a constant pq,a,n such that for all x in the cone, Extpq,a <infinity , if p<pq,a,n and ExTpq=infinity , if p<pq,a,n . We characterize pq,a,n in terms of the principle eigenvalue of a degenerate differential operator. We also give some explicit upper and lower bounds for pq,a,n . |
