Abstract
AbstractIt is well known that elliptic estimates fail for the $${\bar{\partial }}$$
∂
¯
-Neumann problem. Instead, the best that one can hope for is that derivatives in every direction but one can be estimated by the associated Dirichlet form, and when this happens, we say that the $${\bar{\partial }}$$
∂
¯
-Neumann problem satisfies maximal estimates. In the pseudoconvex case, a necessary and sufficient geometric condition for maximal estimates has been derived by Derridj (for (0, 1)-forms) and Ben Moussa (for (0, q)-forms when $$q\ge 1$$
q
≥
1
). In this paper, we explore necessary conditions and sufficient conditions for maximal estimates in the non-pseudoconvex case. We also discuss when the necessary conditions and sufficient conditions agree and provide examples. Our results subsume the earlier known results from the pseudoconvex case.
Publisher
Springer Science and Business Media LLC
Reference24 articles.
1. Adams, R., Fournier, J.: Sobolev Spaces, 2nd edn. Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam (2003)
2. Ahn, H., Baracco, L., Zampieri, G.: Non-subelliptic estimates for the tangential Cauchy–Riemann system. Manuscr. Math. 121(4), 461–479 (2006)
3. Andreotti, A., Grauert, H.: Théorème de finitude pour la cohomologie des espaces complexes. Bull. Soc. Math. Fr. 90, 193–259 (1962)
4. Ben Moussa, B.: Analyticité semi-globale pour le $${\overline{\partial }}$$-Neumann dans des domaines pseudoconvexes. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29, 51–100 (2000)
5. Brinkschulte, J.: Local solvability of the $${\overline{\partial }}$$-equation with boundary regularity on weakly $$q$$-convex domains. Math. Ann. 334(1), 143–152 (2006)