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 (Calculus on Heisenberg manifolds. Annals of mathematics studies, vol 119. Princeton University Press, Princeton, 1988), 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 et al. (Adv Math 308:276–347, 2017), which they then applied to study the CR Yamabe problem in 3 dimensions.