Solution of the tangential Kohn Laplacian on a class of non-compact CR manifolds

Abstract

We solve ${\square }_b$ on a class of non-compact 3-dimensional strongly pseudoconvex CR manifolds via a certain conformal equivalence. The idea is to make use of a related ${\square }_b$ operator on a compact 3-dimensional strongly pseudoconvex CR manifold, which we solve using a pseudodifferential calculus. The way we solve ${\square }_b$ works whenever ${\overline{\partial }}_b$ on the compact CR manifold has closed range in $L^2$; in particular, as in Beals and Greiner, it does not require the CR manifold to be the boundary of a strongly pseudoconvex domain in ${\mathbb{C}}^2$. Our result provides in turn a key step in the proof of a positive mass theorem in 3-dimensional CR geometry, by Cheng, Malchiodi and Yang, which they then applied to study the CR Yamabe problem in 3 dimensions.

to appear in Calc. Var. Partial Differential Equations