Revista Matemática Iberoamericana

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Volume 31, Issue 2, 2015, pp. 681–712
DOI: 10.4171/RMI/850

Published online: 2015-07-16

Some constructions for the fractional Laplacian on noncompact manifolds

Valeria Banica[1], María del Mar González[2] and Mariel Sáez[3]

(1) Université d'Evry - Val d'Essonne, France
(2) Universitat Politècnica de Catalunya, Barcelona, Spain
(3) Pontificia Universidad Católica de Chile, Santiago de Chile, Chile

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.

Keywords: Fractional Laplacian, non-compact manifolds, hyperbolic space, extension problem, Fourier symbol, singular kernel

Banica Valeria, González María del Mar, Sáez Mariel: Some constructions for the fractional Laplacian on noncompact manifolds. Rev. Mat. Iberoam. 31 (2015), 681-712. doi: 10.4171/RMI/850