Fractional Laplacians and extension problems: The higher rank case

González, María del Mar; Sáez, Mariel

doi:10.1090/tran/7267

Fractional Laplacians and extension problems: The higher rank case
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by María del Mar González and Mariel Sáez PDF

Trans. Amer. Math. Soc. 370 (2018), 8171-8213 Request permission

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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