Abstract
We prove non-subelliptic estimates for the tangential Cauchy-Riemann system over a weakly “q-pseudoconvex” higher codimensional submanifold M of \(\mathbb{C}^{n}\). Let us point out that our hypotheses do not suffice to guarantee subelliptic estimates, in general. Even more: hypoellipticity of the tangential C-R system is not in question (as shows the example by Kohn of (Trans AMS 181:273–292,1973) in case of a Levi-flat hypersurface). However our estimates suffice for existence of smooth solutions to the inhomogeneous C-R equations in certain degree. The main ingredients in our proofs are the weighted L 2 estimates by Hörmander (Acta Math 113:89–152,1965) and Kohn (Trans AMS 181:273–292,1973) of Sect. 2 and the tangential \(\bar\partial\)-Neumann operator by Kohn of Sect 4; for this latter we also refer to the book (Adv Math AMS Int Press 19,2001). As for the notion of q pseudoconvexity we follow closely Zampieri (Compositio Math 121:155–162,2000). The main technical result, Theorem 2.1, is a version for “perturbed” q-pseudoconvex domains of a similar result by Ahn (Global boundary regularity of the \(\bar\partial\)-equation on q-pseudoconvex domains, Preprint, 2003) who generalizes in turn Chen-Shaw (Adv Math AMS Int Press 19, 2001).
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To Prof. Giovanni Zacher in his 80th birthday.
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Ahn, H., Baracco, L. & Zampieri, G. Non-subelliptic estimates for the tangential Cauchy–Riemann system. manuscripta math. 121, 461–479 (2006). https://doi.org/10.1007/s00229-006-0049-z
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DOI: https://doi.org/10.1007/s00229-006-0049-z