Global regularity for [formula omitted] on weakly pseudoconvex CR manifolds

Abstract

Let M 2 n - 1 be a compact, orientable, weakly pseudoconvex manifold of dimension at least five, embedded in C N ( n ⩽ N ) , of codimension one or more, and endowed with the induced CR structure. We show the tangential Cauchy–Riemann operator ∂ ¯ b has closed range on such a manifold M , hence we get global existence and regularity results for the ∂ ¯ b problem. We also show the middle (i.e. 1 ⩽ q ⩽ n - 2 ) ∂ ¯ b cohomology groups of M , H 0 p , q ( M , ∂ ¯ b ) , H s p , q ( M , ∂ ¯ b ) , and H ∞ p , q ( M , ∂ ¯ b ) with respect to L 2 , Sobolev s norm, and C ∞ coefficients respectively are finite and isomorphic to each other. The results are obtained by microlocalization using a new type of weight function called strongly CR plurisubharmonic.