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Abstract. We prove hypoellipticity with loss of k−1 m derivatives in two ways using different a priori estimates, together with analytic hypoellipticity, for, the loss is shown to be optimal.

The local real analytic regularity of solutions to O b, the complex boundary Laplacian, and related operators is proved for (p,q) forms on a nondegenerate, abstract, real analytic Cauchy-Riemann (C-R) manifold of dimension 2n -1 satisfying J. J. Kohn's condition Y(q). The problem is reduced to the study of general, "variable coefficient" operators, satisfying the same a priori estimates, on the Heisenberg group. L2 methods only are used.Let M denote a real analytic, Cauchy-Riemann (C-R) (or "partially complex") manifold of real dimension 2n -1 with nondegenerate Levi form. This means that the complexified the matrix c1k has either min(q + 1,n -q) pairs of Y(q) eigenvalues of opposite sign or max(q + 1,n -q) eigenvalues of the same sign and guarantees that, with P =°b ,

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No abstract