Global Regularity for the Tangential Cauchy-Riemann Operator on Weakly Pseudoconvex CR Manifolds

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Let M of real dimension 2n-1 be a compact, orientable, weakly pseudoconvex manifold of dimension at least five, embedded in Cⁿ (n less than or equal to N), of codimension one or more in Cⁿ, and endowed with the induced CR structure. We show the tangential Cauchy-Riemann operator has closed range on such a manifold M, hence we get global existence and regularity results for the ∂b problem. We also show the middle (i.e. corresponding to (p,q) forms for q between 1 and n-2) ∂b cohomology groups of M with respect to L², Sobolev s, and smooth coefficients are finite and isomorphic to each other. The results are obtained by microlocalization using a new type of weight function called strongly CR plurisubharmonic.