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q-Pseudoconvexity and Regularity at the Boundary for Solutions of the $\bar \partial$-problem

Published online by Cambridge University Press:  04 December 2007

Giuseppe Zampieri
Affiliation:
Dip. Matematica-Universitè v. Belzoni 7, Padova, Italy. E-mail: zampieri@math.unipd.it
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Abstract

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For a domain Ω of ${\mathbb C}^N$ we introduce a fairly general and intrinsic condition of weak q-pseudoconvexity, and prove, in Theorem 4, solvability of the $\bar \partial$-complex for forms with $C^\infty (\bar \Omega)$-coefficients in degree $\geq q+1$. All domains whose boundary have a constant number of negative Levi eigenvalues are easily recognized to fulfill our condition of q-pseudoconvexity; thus we regain the result of Michel (with a simplified proof). Our method deeply relies on the $L^2$-estimates by Hörmander (with some variants). The main point of our proof is that our estimates (both in weightened-$L^2$ and in Sobolev norms) are sufficiently accurate to permit us to exploit the technique by Dufresnoy for regularity up to the boundary.

Type
Research Article
Copyright
© 2000 Kluwer Academic Publishers