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

Abstract

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.