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

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>0Gp>0, with Gp(x)∼ϑ(x,p)−2Gp(x)∼ϑ(x,p)−2 near p  , where ϑ(x,y)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□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 L2L2 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□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].