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Abstract.Let A/, and M2 be two bounded pseudo-convex domains in C" with smooth boundaries such that Mt c M2. We consider the Cauchy-Riemann operators 3 on the annulus M = M2\Mt. The main result of this paper is the following: Given a 3-closed (p, q) form a, 0 < q < n, which is C°° on M and which is cohomologous to zero on M, there exists a (p, q -1) form u which is C°° on M such that 3 u = a. 0. Introduction. Let Mx and M2 be two bounded pseudo-convex domains in C" with smooth boundaries such that Mx c M2. We consider the annulus M between Mx and M2, i.e., M = M2\MX. The Cauchy-Riemann equation 3 on M is a system of overdetermined first-order differential equations. We ask the following question: Given a(p,q) form a, where 0 < q < n, when can one solve the equationand if a is smooth up to the boundary of M, does there exist a solution u of (0.1) which is also smooth up to the boundary? A necessary condition for a to be solvable is that a must satisfy the compatibility condition (0.2) 3« = 0 since 32 = 0. In this paper, we shall prove that if aj > 3, a has L2 coefficients and satisfies (0.2), and a is orthogonal to a finite-dimensional space (i.e., the harmonic (p, q) forms), then there exists a (p, q -1) form u such that (0.1) holds. Furthermore, if a is smooth up to the boundary of M, then we can find a u that is smooth up to the boundary also and u satisfies (0.1) (Theorems 1, 2 and 3).Our method is to use the 3-Neumann problem with weights which was used by Hormander [3] and Kohn [4] to solve the equation (0.1) on weakly pseudo-convex domains. The 3-Neumann problem was a method to solve the equation 3 with solutions smooth up to the boundary. If one can show that the subelliptic estimate holds for the 3-Neumann problem, then one can conclude that the harmonic forms are finite dimensional and one can solve (0.1) provided a has L2 coefficients and satisfies (0.2) and a is orthogonal to the harmonic space. One can find a solution u

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We establish the L 2 theory for the Cauchy-Riemann equations on product domains provided that the Cauchy-Riemann operator has closed range on each factor. We deduce regularity of the canonical solution on ( p, 1)-forms in special Sobolev spaces represented as tensor products of Sobolev spaces on the factors of the product. This leads to regularity results for smooth data.

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