CR-invariants and the scattering operator for complex manifolds with CR-boundary

An extension problem for the CR fractional Laplacian

“…Fractional CR covariant operators of order 2γ, γ ∈ R, may be defined from scattering theory on a Kähler-Einstein manifold X [21,45,40,33], they are pseudodifferential operators whose principal symbol agrees with the pure fractional powers of the CR sub-Laplacian (−∆ b ) γ on the boundary M = ∂X . In the particular case of the Heisenberg group H n , they are simply the intertwining operators on the CR sphere calculated in [8,51,9] using classical representation theory tools.…”

“…Under certain conditions (see [15,55,45]), Fefferman's local approximate solution of the Monge-Ampère equation can be globalized to an approximate solution of the Monge-Ampère equation near the boundary of a complex manifold X with strictly pseudoconvex boundary M. It follows that X carries an approximate Kähler-Einstein metric in the sense of (2.17). This ϕ is called a globally defined approximate solution of the Monge-Ampère equation in X .…”

“…Most of this section is taken from the well written introduction in [45], but we recall it here for convenience of the reader. Two good survey references on Heisenberg groups and CR manifolds are [5], [20].…”

“…The Poisson operator is constructed as follows (see [45]). Let s ∈ C such that ℜ(s) ≥ m 2 , s ∈ Z and 2s − m ∈ Z.…”

A nonlinear Liouville theorem for fractional equations in the Heisenberg group

“…They can be defined using scattering theory, as done in [12,22,21,20]. In the particular case of the Heisenberg group, they are the intertwining operators on the CR sphere, which can be calculated using representation theory techniques (see [8]).…”