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arXiv · 1212.3109

Some constructions for the fractional Laplacian on noncompact manifolds

Valeria Banica·Maria del Mar González·Mariel Sáez
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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.

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Valeria Banica, Maria del Mar González, Mariel Sáez. 2014-05-08. Some constructions for the fractional Laplacian on noncompact manifolds. https://arxiv.org/abs/1212.3109

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