The [formula omitted] equation on weakly pseudo-convex CR manifolds of dimension 3

Abstract

Let M be a smooth, compact, orientable, weakly pseudoconvex manifold of dimension 3, embedded in C N ( N ⩾ 2 ) , of codimension one or more, and endowed with the induced CR structure. Assuming that the tangential Cauchy-Riemann operator ∂ ¯ b has closed range in L 2 ( M ) in order to rule out the Rossi example, we push regularity up to show ∂ ¯ b has closed range in H s ( M ) for all s > 0 . We then use the Szegö projection to show there is a smooth solution for the ∂ ¯ b problem given smooth data. The results are obtained via microlocalization by piecing together estimates for functions and (0,1) forms that hold on different microlocal regions.