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

DE NÁPOLI, PABLO LUIS; STINGA. PABLO RAÚL

New Developments in the Analysis of Nonlocal Operators (in the series: Contemporary Mathematics)

American Mathematical Society

Año: 2019; p. 167 - 189

In this paper we show novel underlying connections between fractional powersof the Laplacian on the unit sphere and functions from analytic number theoryand differential geometry, like the Hurwitz zeta function and theMinakshisundaram zeta function. Inspired by inakshisundaram´s ideas, we find a precise pointwise description of the fractional powers of the Laplace-Beltrami operator in terms of fractional powers of the Dirichlet-to-Neumann map on the sphere. The Poissonkernel for the unit ball will be essential for this part of the analysis. Onthe other hand, by using the heat semigroup on the sphere, additional pointwiseintegro-differential formulas are obtained. Finally, we prove acharacterization with a local extension problem and the interior Harnackinequality.