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Transactions of the American Mathematical Society

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Fractional Laplacians and extension problems: The higher rank case

Authors: María del Mar González and Mariel Sáez
Journal: Trans. Amer. Math. Soc. 370 (2018), 8171-8213
MSC (2010): Primary 53A30, 35J57; Secondary 53C35, 17B15
Published electronically: July 31, 2018
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Abstract: The aim of this paper is to define conformal operators that arise from an extension problem of codimension two. To this end we interpret and extend results of representation theory from a purely analytic point of view.

The first part of the paper is an interpretation of the fractional Laplacian and the conformal fractional Laplacian in the general framework of representation theory on symmetric spaces. In the flat case, these results are well known from the representation theory perspective but have been much less explored in the context of non-local operators in partial differential equations. This analytic approach will be needed in order to consider the curved case.

In the second part of the paper we construct new boundary operators with good conformal properties that generalize the fractional Laplacian in $ \mathbb{R}^n$ using an extension problem in which the boundary is of codimension two. Then we extend these results to more general manifolds that are not necessarily symmetric spaces.

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Additional Information

María del Mar González
Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, Madrid 28049, Spain

Mariel Sáez
Affiliation: Pontificia Universidad Católica de Chile, Santiago Avda. Vicuña Mackenna 4860, Macul, 6904441 Santiago, Chile

Received by editor(s): December 6, 2016
Received by editor(s) in revised form: April 10, 2017, and April 19, 2017
Published electronically: July 31, 2018
Additional Notes: The first author was supported by Spanish government grant MTM2014-52402-C3-1-P, and the BBVA Foundation grant for Investigadores y Creadores Culturales 2016.
The second author was partially supported by Proyecto Fondecyt Regular 1150014
Both authors would like to acknowledge the support of NSF grant DMS-1440140 while both authors were in residence at the Mathematical Sciences Research Institute in Berkeley, CA, during the Spring 2016 semester.
Article copyright: © Copyright 2018 American Mathematical Society

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