Some constructions for the fractional Laplacian on noncompact manifolds

Abstract

We give a de nition of the fractional Laplacian on some noncompact manifolds, through an extension problem introduced by Ca arelli-Silvestre. While this de nition in the compact case is straightforward, in the noncompact setting one needs to have a precise control of the behavior of the metric at in nity and geometry plays a crucial role. First we give explicit calculations in the hyperbolic space, including a formula for the kernel and a trace Sobolev inequality. Then we consider more general noncompact manifolds, where the problem reduces to obtain suitable upper bounds for the heat kernel.