Global regularity for ∂¯b on weakly pseudoconvex CR manifolds

AbstractLet M2n-1 be a compact, orientable, weakly pseudoconvex manifold of dimension at least five, embedded in CN(n⩽N), of codimension one or more, and endowed with the induced CR structure. We show the tangential Cauchy–Riemann operator ∂¯b 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. 1⩽q⩽n-2) ∂¯b cohomology groups of M, H0p,q(M,∂¯b), Hsp,q(M,∂¯b), and H∞p,q(M,∂¯b) with respect to L2, Sobolev s norm, and C∞ coefficients respectively are finite and isomorphic to each other. The results are obtained by microlocalization using a new type of weight function called strongly CR plurisubharmonic