Let $(\hat{ X}, T^{ 1,0} \hat{ X})$ be a compact orientable CR embeddable three dimensional strongly pseudoconvex CR manifold, where $T^{ 1,0} \hat{ X}$ is a CR structure on $\hat{ X}$. Fix a point $p \in \hat{ X}$ and take a global contact form $\hat{ \theta}$ so that $\hat{ \theta}$ is asymptotically flat near $p$. Then $(\hat{ X}, T^{ 1,0} \hat{ X}, \hat{ \theta})$ is a pseudohermitian $3$-manifold. Let $G_p \in C^{ \infty} (\hat{ X} \setminus \{ p\})$, $G_p > 0$, with $G_p(x) \sim \vartheta(x,p)^{ -2}$ near $p$, where $\vartheta(x,y)$ denotes the natural pseudohermitian distance on $\hat{ X}$. Consider the new pseudohermitian $3$-manifold with a blow-up of contact form $(\hat{ X} \setminus \{ p\}, T^{ 1,0} \hat{ X}, G^2_p \hat{ \theta})$ and let $\Box_{ b}$ denote the corresponding Kohn Laplacian on $\hat{ X} \setminus \{ p\}$. In this paper, we prove that the weighted Kohn Laplacian $G^2_p \Box_b$ has closed range in $L^2$ with respect to the weighted volume form $G^2_p \hat{ \theta} \wedge d\hat{ \theta}$, and that the associated partial inverse and the Szeg\"{ o} projection enjoy some regularity properties near $p$. As an application, we prove the existence of some special functions in the kernel of $\Box_{ b}$ 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-Yang.
Categories: math.CV math.AP math.DG