The ∂b Equation on Weakly Pseudoconvex CR Manifolds of Dimension 3
Joseph J. Kohn, Andreea Nicoara
Abstract
Let M be a smooth, compact, orientable, weakly pseudoconvex manifold of dimension 3, embedded in CN (N greater than or equal to 2), of codimension one or more in CN, and endowed with the induced CR structure. Assuming that the tangential Cauchy-Riemann operator ∂b has closed range in L2 in order to rule out the Rossi example, we push regularity up to show ∂b has closed range in all Sobolev spaces s for s greater than zero. We then use the Szegö projection to show there is a smooth solution to 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.
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