Solving the Kohn Laplacian on asymptotically flat CR manifolds of dimension 3

Abstract

Let (Xˆ,T1,0Xˆ) be a compact orientable CR embeddable three dimensional strongly pseudoconvex CR manifold, where T1,0Xˆ is a CR structure on Xˆ. Fix a point p∈Xˆ and take a global contact form θˆ so that θˆ is asymptotically flat near p. Then (Xˆ,T1,0Xˆ,θˆ) is a pseudohermitian 3-manifold. Let Gp∈C∞(Xˆ∖{p}), Gp>0, with Gp(x)∼ϑ(x,p)−2 near p, where ϑ(x,y) denotes the natural pseudohermitian distance on Xˆ. Consider the new pseudohermitian 3-manifold with a blow-up of contact form (Xˆ∖{p},T1,0Xˆ,Gp2θˆ) and let □b denote the corresponding Kohn Laplacian on Xˆ∖{p}.In this paper, we prove that the weighted Kohn Laplacian Gp2□b has closed range in L2 with respect to the weighted volume form Gp2θˆ∧dθˆ, and that the associated partial inverse and the Szegö projection enjoy some regularity properties near p. As an application, we prove the existence of some special functions in the kernel of □b that grow at a specific rate at p. The existence of such functions provides an important ingredient for the proof of a positive mass theorem in 3-dimensional CR geometry by Cheng, Malchiodi and Yang [5].