The \bar\partial_b Equation on Weakly Pseudoconvex CR Manifolds of   Dimension 3
  Let M be a smooth, compact, orientable, weakly pseudoconvex manifold of dimension 3, embedded in C^N (N greater than or equal to 2), of codimension one or more in C^N, and endowed with the induced CR structure. Assuming that the tangential Cauchy-Riemann operator \bar\partial_b has closed range in L^2 in order to rule out the Rossi example, we push regularity up to show \bar\partial_b has closed range in all Sobolev spaces s for s greater than zero. We then use the Szeg\"{o} projection to show there is a smooth solution to the \bar\partial_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. 
Joseph J. Kohn and Andreea Nicoara