On Weakly Pseudoconvex Cr Manifolds of Dimension 3

Let M be a compact, COO CR manifold of dimension 3 over R. Associated to the CR structure is a first-order differential operator, Db' on M. We study the regularity properties, in terms of L P Sobolev and Holder norms, of the equation Db u = f. M is said to be CR if there is given a COO sub-bundle, denoted T I .o M , of the complexified tangent bundle TM, such that each fiber T~'o M is of dimension lover C, and T I .0 M n T I .0 M = {O} for all x EM. Define x x T o.1 M = T I .0 M and let Bo. 1 M denote its dual bundle. Then for any COO function u on M , Db U is the smooth section of Bo. 1 M obtained by restricting du to To. 1 M. In other words if Z E T~·I M then (Dbu)(Z) = (Zu)(x). The boundary M of any smoothly bounded relatively compact open set Q C C2 carries a natural CR structure: T ~ .0 M is the subspace of the tangent space of C2 at x consisting of all holomorphic tangent vectors, that is, linear combinations of ,};\ ' ;~2 ' which are tangent to M. A prime motivation for the study of Db is its connection with complex analysis on Q, in the case M = a Q. In order to construct holomorphic functions in Q one often needs to solve D u = 0' in Q, where 0' is a given (0, 1) form satisfying the necessary condition DO' = O. It is desirable to have as much control over the regularity of some solution u as possible. Currently much more is known in terms of L2 and L2 Sobolev norms than L P , L P Sobolev and Holder norms. In particular the algebra of functions holomorphic on Q and continuous on Q is of interest, so one seeks conditions on 0' which guarantee the existence of a continuous solution u. J. J. Kohn has pointed out [K3] that if it were proved that