An extension problem for the CR fractional Laplacian

TL;DR

This paper establishes a new extension problem for the CR fractional Laplacian on the Heisenberg group, linking it to the scattering operator on the Siegel upper halfspace, distinct from previous models.

Contribution

It introduces a novel extension problem for the CR fractional Laplacian that differs from existing approaches, connecting it with scattering theory on the Siegel upper halfspace.

Findings

CR fractional powers are characterized via scattering operators

The extension problem is fundamentally different from Caffarelli and Silvestre's model

Provides a new perspective on conformally invariant operators

Abstract

We show that the conformally invariant fractional powers of the sub-Laplacian on the Heisenberg group are given in terms of the scattering operator for an extension problem to the Siegel upper halfspace. Remarkably, this extension problem is different from the one studied, among others, by Caffarelli and Silvestre.