Global regularity of ∂¯ on an annulus between a q-pseudoconvex and a p-pseudoconcave boundary

Let Ω1 and Ω2 be two domains of Cn with Ω2⊂⊂Ω1⊂⊂Cn and let Ω be the annulus Ω=Ω1∖Ω¯¯¯2. We assume that Ω is q-pseudoconvex at ∂Ω1 and p-pseudoconcave at ∂Ω2, where the indices p and q satisfy 0≤q+2≤p≤n−1. For an antiholomorphic form f in C∞(Ω¯¯¯) of degree k with q+1≤k≤p−1 and which satisfies the compatibility condition ∂¯f=0, the authors look for solutions u of degree k−1 of the inhomogeneous Cauchy-Riemann equation ∂¯u=f. Using the ∂¯-Neumann method of Kohn, in this paper the authors prove solvability of this equation in C∞(Ω¯¯¯), modulo harmonic forms H, that is, solutions of (∂¯,∂¯∗). If we make a stronger hypothesis on Ω1 and Ω2, say strong q-pseudoconvexity at ∂Ω1 and strong p-pseudoconcavity at ∂Ω2, then the local hypoellipticity at the boundary for (∂¯,∂¯∗) follows: a solution u which is orthogonal to ker(∂¯) is smooth precisely in the part of ∂Ω where f is. In particular, when f is in C∞(Ω¯¯¯), then the so-called "canonical'' solution is also in C∞(Ω¯¯¯).