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Session S38 - Geometric Potential Analysis

Thursday, July 15, 19:00 ~ 19:30 UTC-3

Fractional Laplacians on the sphere, the Minakshisundaram zeta function and semigroups

Pablo De Nápoli

Universidad de Buenos Aires // IMAS (Conicet), Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

We show novel underlying connections between fractional powers of the Laplacian on the unit sphere and functions from analytic number theory and differential geometry, like the Hurwitz zeta function and the Minakshisundaram zeta function. Inspired by Minakshisundaram's ideas, we find a precise pointwise description of the fractional Laplacian on the sphere in terms of fractional powers of the Dirichlet-to-Neumann map in the unit ball. The Poisson kernel for the unit ball will be essential for this part of the analysis. On the other hand, by using the heat semigroup on the sphere, additional pointwise integro-differential formulas are obtained. Finally, we prove a characterization with a local extension problem and the interior Harnack inequality.

Joint work with Pablo Raúl Stinga (Iowa State University,Argentina).

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