- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:48
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
31
2015
2
Some constructions for the fractional Laplacian on noncompact manifolds
Valeria
Banica
Université d'Evry - Val d'Essonne, EVRY, FRANCE
María
del Mar González
Universitat Politècnica de Catalunya, BARCELONA, SPAIN
Mariel
Sáez
Pontificia Universidad Católica de Chile, SANTIAGO DE CHILE, CHILE
Fractional Laplacian, non-compact manifolds, hyperbolic space, extension problem, Fourier symbol, singular kernel
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.
Real functions
Partial differential equations
Global analysis, analysis on manifolds
681
712
10.4171/RMI/850
http://www.ems-ph.org/doi/10.4171/RMI/850