Existence and regularity of solutions of the \( \bar{\delta} \)-system on wedges of \( \mathbb{C}^{N} \)

Zampieri, Giuseppe

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 10 (1999) no. 4, pp. 271-278

Résumé

For a wedge \( W \) of \( \mathbb{C}^{N} \), we introduce two conditions of weak \( q \)-pseudoconvexity, and prove that they entail solvability of the \( \bar{\delta} \)-system for forms of degree \( \ge q + 1 \) with coefficients in \( C^{\infty} (W) \) and \( C^{\infty} (\bar{W}) \) respectively. Existence and regularity for \( \bar{\delta} \) in \( W \) is treated by Hörmander [5, 6] (and also by Zampieri [9, 11] in case of piecewise smooth boundaries). Regularity in \( W \) is treated by Henkin [4] (strong \( q \)-pseudoconvexity by the method of the integral representation), Dufresnoy [3] (full pseudoconvexity), Michel [8] (constant number of negative eigenvalues), and Zampieri [10] (more general \( q \)-pseudoconvexity and wedge type domains). This is an announcement of our papers [10, 11]; it contains refinements both in statements and proofs and, mainly, a parallel treatement of regularity in \( W \) and \( \bar{W} \). All our techniques strongly rely on the method of \( L^{2} \) estimates by Hörmander [5, 6].