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.
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Valeria Banica, María del Mar González, Mariel Sáez, Some constructions for the fractional Laplacian on noncompact manifolds. Rev. Mat. Iberoam. 31 (2015), no. 2, pp. 681–712DOI 10.4171/RMI/850