Some constructions for the fractional Laplacian on noncompact manifolds

Abstract

We give a definition of the fractional Laplacian on some noncompact manifolds, through an extension problem introduced by Caffarelli-Silvestre. While this definition in the compact case is straightforward, in the noncompact setting one needs to have a precise control of the behavior of the metric at infinity 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.

Accepted in Revista Matematica Iberoamericana

References in corpus (3)

• From the long jump random walk to the fractional Laplacian
• Fractional Laplacian on the torus
• Heat kernels of metric trees and applications

Cited by in corpus (4)

• Hardy-Sobolev-Maz'ya inequalities for higher order derivatives on half spaces
• An extension problem for the CR fractional Laplacian
• Extension problem and fractional operators: semigroups and wave equations
• Sharp Hardy-Adams inequalities for bi-Laplacian on hyperbolic space of dimension four