An extension problem for the CR fractional Laplacian

Abstract

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.

A nonlinear Liouville theorem for fractional equations in the Heisenberg group

“…Since it will be of utmost importance in the sequel, we recall here the extension result proven in [16]. Theorem 1.1 (see Theorem 1.1 in [16]).…”

“…The operators considered in [15] are different in nature from the ones in [16], they correspond to the pure fractional powers of the sub-Laplacian and do not enjoy the CR covariance property.…”

“…In [16] Frank, Gonzalez, Monticelli and one of the authors, study CR covariant operators of fractional orders on orientable and strictly pseudoconvex CR manifolds. In particular, they focuse on the construction of such operators as the Dirichlet-toNeumann map associated to a degenarate elliptic equation in the spirit of Caffarelli and Silvestre [9].…”

“…In [16], in order to construct fractional CR covariant operators in the specific case of the Heisenberg group, H n is identified with the boundary of the Siegel domain in R 2n+2 (see Section 2 for the precise definition) and it is crucial to use its underlying complex hyperbolic structure.…”

“…We establish a Liouville-type theorem for a subcritical nonlinear problem, involving a fractional power of the subLaplacian in the Heisenberg group. To prove our result we will use the local realization of fractional CR covariant operators, which can be constructed as the Dirichlet-to-Neumann operator of a degenerate elliptic equation in the spirit of Caffarelli and Silvestre [9], as established in [16]. The main tools in our proof are the CR inversion and the moving plane method, applied to the solution of the lifted problem in the half-space H n × R + .…”