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

Giuseppe Zampieri

Existence and regularity of solutions of the $\bar{δ}$ -system on wedges of $C^{N}$

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1999)

Volume: 10, Issue: 4, page 271-278
ISSN: 1120-6330

Access Full Article

top

Access to full text

Full (PDF)

Full (DjVu)

Abstract

top

For a wedge

W

of

C^{N}

, we introduce two conditions of weak

q

-pseudoconvexity, and prove that they entail solvability of the

\bar{δ}

-system for forms of degree

\geq q + 1

with coefficients in

C^{\infty} (W)

and

C^{\infty} (\bar{W})

respectively. Existence and regularity for

\bar{δ}

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].

How to cite

top

MLA
BibTeX
RIS

Zampieri, Giuseppe. "Existence and regularity of solutions of the $ \bar{\delta} $-system on wedges of $ \mathbb{C}^{N} $." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 10.4 (1999): 271-278. <http://eudml.org/doc/252311>.

@article{Zampieri1999,
abstract = {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].},
author = {Zampieri, Giuseppe},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {L2 estimates; Cauchy-Riemann system; C; R; structures; wedge; -pseudoconvex; -system; -estimates},
language = {eng},
month = {12},
number = {4},
pages = {271-278},
publisher = {Accademia Nazionale dei Lincei},
title = {Existence and regularity of solutions of the $ \bar\{\delta\} $-system on wedges of $ \mathbb\{C\}^\{N\} $},
url = {http://eudml.org/doc/252311},
volume = {10},
year = {1999},
}

TY - JOUR
AU - Zampieri, Giuseppe
TI - Existence and regularity of solutions of the $ \bar{\delta} $-system on wedges of $ \mathbb{C}^{N} $
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1999/12//
PB - Accademia Nazionale dei Lincei
VL - 10
IS - 4
SP - 271
EP - 278
AB - 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].
LA - eng
KW - L2 estimates; Cauchy-Riemann system; C; R; structures; wedge; -pseudoconvex; -system; -estimates
UR - http://eudml.org/doc/252311
ER -

References

top

Airapetyan, R.A. - Henkin, G.M., Integral representation of differential forms on Cauchy-Riemann manifolds and the theory of $C R$ -functions. Uspekhi Mat. Nauk., 39 (3), 1984, 39-106. Zbl0589.32035 MR747791
Andreotti, A. - Grauert, H., Théorèmes de finitude pour la cohomologie des espaces complexes. Bull. Soc. Math. France, 90, 1962, 193-259. Zbl0106.05501 MR150342
Dufresnoy, A., Sur l’operateur $\bar{δ}$ et les fonctions différentiables au sens de Whitney. Ann. Inst. Fourier, 29 (1), 1979, 229-238. Zbl0387.32011 MR526786
Henkin, G.M., H. Lewy’s equation and analysis on pseudoconvex manifolds (Russian). I. Uspehi Mat. Nauk., 32 (3), 1977, 57-118. Zbl0358.35057 MR454067
Hörmander, L., $L^{2}$ estimates and existence theorems for the $\bar{δ}$ operator. Acta Math., 113, 1965, 89-152. Zbl0158.11002 MR179443
Hörmander, L., An introduction to complex analysis in several complex variables. Van Nostrand, Princeton, N.J.1966. Zbl0138.06203
Kohn, J.J., Regularity at the boundary of the $\bar{δ}$ -Neumann problem. Proceedings of the National Academy of Sciences of the United States of America, 49, 1963, 206-213. Zbl0118.31101 MR149510
Michel, V., Sur la regularité $C^{\infty}$ du $\bar{δ}$ au bord d’un domaine de $C^{n}$ dont la forme de Levi a exactement $s$ valeurs propres strictement negatives. Math. Ann., 195, 1993, 131-165. MR1198845 DOI10.1007/BF01444880
Zampieri, G., $L^{2}$ -estimates with Levi-singular weight, and existence for $\bar{δ}$ . J. d’Analyse Math., 74, 1998, 99-112. Zbl0954.32025 MR1631646 DOI10.1007/BF02819447
Zampieri, G., Solvability of $\bar{δ}$ with $C^{\infty}$ regularity up to the boundary on wedges of $C^{n}$ . Israel J. of Math., 1999, in press. MR1767934
Zampieri, G., $C^{\infty}$ solvability of the $\bar{δ}$ system on wedges of $C^{n}$ . Preprint 1998. MR1737843 DOI10.1006/jmaa.1999.6629

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Language to use for this widget.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Number of notes per page

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.