Boundary regularity for ¯∂ on q-pseudoconvex wedges of CN

For a wedge Ω of CN, we refine the notion of weak q-pseudoconvexity of [G. Zampieri, Solvability of the ¯∂ problem with C∞ regularity up to the boundary on wedges of CN, Israel J. Math. 115 (2000) 321–331]. This is an intrinsic property which can be expressed in terms of q-subharmonicity both of a defining function or an exhaustion function of Ω. Under this condition we prove solvability of the ∂¯ system for forms with C∞(Ω¯ )-coefficients of degree q + 1. Our method relies on the L2-estimates by Hörmander [L. Hörmander, L2 estimates and existence theorems for the ¯∂ operator, Acta Math. 113 (1965) 89–152] and by Kohn [J.J. Kohn, Global regularity for ¯∂ on weakly pseudoconvex manifolds, Trans. Amer. Math. Soc. 181 (1973) 273–292]. For solvability with regularity up to the boundary in a domain without corners, we refer to [J.J. Kohn, Methods of partial differential equations in complex analysis, Proc. Sympos. Pure Math. 30 (1977) 215–237] in case of classical pseudoconvexity, that is for q = 0 in our notation (or else for strong pseudoconvexity which means that the number of the positive Levi eigenvalues of the boundary is N − 1 − q), and we refer to [L. Baracco, G. Zampieri, Global regularity for ¯∂ on q-pseudoconvex domains, 2003] for general q-pseudoconvexity. For local existence on wedge-type domains we refer to [G. Zampieri, Solvability of the ¯∂ problem with C∞ regularity up to the boundary on wedges of CN, Israel J. Math. 115 (2000) 321–331].  2005 Elsevier Inc. All rights reserved.