Fractional Laplacians and extension problems: the higher rank case

TL;DR

This paper explores fractional Laplacians within symmetric spaces using representation theory and introduces new boundary operators with conformal properties via an extension problem involving codimension-two boundaries.

Contribution

It generalizes fractional Laplacians through a representation-theoretic framework and constructs novel boundary operators with conformal invariance in higher codimension settings.

Findings

Representation theory provides a unifying framework for fractional Laplacians.

New boundary operators with conformal properties are constructed for codimension-two boundaries.

Extension problems are used to define and analyze these generalized operators.

Abstract

The aim of this paper is two-fold: first, we look at the fractional Laplacian and the conformal fractional Laplacian from the general framework of representation theory on symmetric spaces and, second, we construct new boundary operators with good conformal properties that generalize the fractional Laplacian using an extension problem in which the boundary is of codimension two.