The ∂¯b equation on weakly pseudo-convex CR manifolds of dimension 3

Let M be a smooth, compact, orientable, weakly pseudoconvex manifold of dimension 3, embedded in CN (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 L2(M) in order to rule out the Rossi example, we push regularity up to show ∂¯b has closed range in Hs(M) for all s>0. We then use the Szego 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.