Extension of CR functions to «wedge type» domains

D'Agnolo, Andrea ; D'Ancona, Piero ; Zampieri, Giuseppe

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 2 (1991) no. 1, pp. 35-42

Résumé

Let \( X \) be a complex manifold, \( S \) a generic submanifold of \( X^{\mathbb{R}} \), the real underlying manifold to \( X \). Let \( \Omega \) be an open subset of \( S \) with \( \partial \Omega \) analytic, \( Y \) a complexification of \( S \). We first recall the notion of \( \Omega \)-tuboid of \( X \) and of \( Y \) and then give a relation between; we then give the corresponding result in terms of microfunctions at the boundary. We relate the regularity at the boundary for \( \bar{\partial}_{b} \) to the extendability of \( CR \) functions on \( \Omega \) to \( \Omega \)-tuboids of \( X \). Next, if \( X \) has complex dimension 2, we give results on extension for some classes of hypersurfaces (which correspond to some \( \bar{\partial}_{b} \) whose Poisson bracket between real and imaginary part is \( \ge 0 \)). The main tools of the proof are the complex \( \mathcal{C}_{\Omega \mid Y} \) by Schapira and the theorem of \( \Omega \)-regularity of Schapira-Zampieri and Uchida-Zampieri.