Some constructions for the fractional Laplacian on noncompact manifolds

By Valeria Banica, Mar'ia de Mar Gonz'alez, Mariel S'aez. 2012

74 citations recorded by the source corpus.

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

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/270034832a015f11c22e133401097f88d0ea1850

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Paneitz operators on hyperbolic spaces and high order Hardy-Sobolev-Maz'ya inequalities on half spaces (2019, background): “For more information about fractional Laplacian on hyperbolic space, we refer to [2, 4].”
An extension problem for the CR fractional Laplacian (2013, unknown): “t contains an energy equality for the fractional norm S˙γ(Hn), using the extension problem from Theorem 1.1. This idea of using the extension has been successfully employed in several other settings ([27, 28, 4], for instance). Proposition 6.1. Let 0 <γ<1 and f∈ S˙γ(Hn). Then Eγf∈ H˙1,γ(Ωn+1) ”
Extension problem and fractional operators: semigroups and wave equations (2012, background): “of the extension problem, found applications in [15], where Harnack’s inequality for fractional subLaplacians in Carnot groups are derived, and in [4], where the fractional Laplacians on the hyperbolic space and on some noncompact manifolds are studied.”