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

Let (X,T^1,0X) be a compact orientable CR embeddable three dimensional strongly pseudoconvex CR manifold, where T^1,0X is a CR structure on X. Fix a point p∈X and take a global contact form θ so that θ is asymptotically flat near p. Then (X,T^1,0X,θ) is a pseudohermitian 3-manifold. Let G_p ∈ C^∞(X\{p}), G_p > 0, with G_p(x)~ θ{symbol}(x, p)^-2 near p, where θ{symbol} (x, y) denotes the natural pseudohermitian distance on X. Consider the new pseudohermitian 3-manifold with a blow-up of contact form (X\{p}, T^1,0X, G_p²θ) and let □_b denote the corresponding Kohn Laplacian on X\{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 Gp2θ∧dθ, 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].