Two-sided Dirichlet heat kernel estimates of symmetric stable processes on horn-shaped regions

Abstract

In this paper, we consider symmetric a-stable processes on (unbounded) horn-shaped regions which are non-uniformly C-1,C-1 near infinity. By using probabilistic approaches extensively, we establish two-sided Dirichlet heat kernel estimates of such processes for all time. The estimates are very sensitive with respect to the reference function corresponding to each horn-shaped region. Our results also cover the case that the associated Dirichlet semigroup is not intrinsically ultracontractive. A striking observation from our estimates is that, even when the associated Dirichlet semigroup is intrinsically ultracontractive, the so-called Varopoulos-type estimates do not hold in general for symmetric stable processes on horn-shaped regions.