Closed Range for $\bar{\partial }$ and $\bar{\partial }_b$ on Bounded Hypersurfaces in Stein Manifolds

Harrington, Phillip S.; Raich, Andrew S.

doi:10.5802/aif.2972

Closed Range for

\bar{\partial}

and

{\bar{\partial}}_{b}

on Bounded Hypersurfaces in Stein Manifolds
[Image fermée pour

\bar{\partial}

{\bar{\partial}}_{b}

sur les hypersurfaces bornées dans les variétés de Stein]

Harrington, Phillip S. ; Raich, Andrew S.

Annales de l'Institut Fourier, Tome 65 (2015) no. 4, pp. 1711-1754.

Résumé
Abstract

Nous définissons $Z (q)$ faible, une généralisation de $Z (q)$ sur les domaines bornés $Ω$ dans une variété de Stein $M^{n}$ qui suffit à prouver que l’image de $\bar{\partial}$ est fermée. Sous l’hypothèse d’une $Z (q)$ faible, nous montrons également que (i) les $(0, q)$ -formes harmoniques sont triviales et (ii) si $\partial Ω$ satisfait une $Z (q)$ faible et une $Z (n - 1 - q)$ faible, alors ${\bar{\partial}}_{b}$ a une image fermée sur les $(0, q)$ -formes sur $\partial Ω$ . Nous fournissons des exemples pour montrer que notre condition contient des exemples qui sont exclus de la $(q - 1)$ -pseudoconvexité et la notion précédente des auteurs de $Z (q)$ faible.

We define weak $Z (q)$ , a generalization of $Z (q)$ on bounded domains $Ω$ in a Stein manifold $M^{n}$ that suffices to prove closed range of $\bar{\partial}$ . Under the hypothesis of weak $Z (q)$ , we also show (i) that harmonic $(0, q)$ -forms are trivial and (ii) if $\partial Ω$ satisfies weak $Z (q)$ and weak $Z (n - 1 - q)$ , then ${\bar{\partial}}_{b}$ has closed range on $(0, q)$ -forms on $\partial Ω$ . We provide examples to show that our condition contains examples that are excluded from $(q - 1)$ -pseudoconvexity and the authors’ previous notion of weak $Z (q)$ .

Reçu le : 2012-01-10
Accepté le : 2015-01-13
Publié le : 2015-11-09

DOI : https://doi.org/10.5802/aif.2972
Classification : 32W05, 32W10, 32Q28, 35N15
Mots clés : variété de Stein,

{\bar{\partial}}_{b}

, tangentielle opérateur de Cauchy-Riemann, image fermée,

\bar{\partial}

-Neumann, faible

Z (q)

q

-pseudoconvexité

@article{AIF_2015__65_4_1711_0,
     author = {Harrington, Phillip S. and Raich, Andrew S.},
     title = {Closed {Range} for $\bar{\partial }$ and $\bar{\partial }_b$ on {Bounded} {Hypersurfaces} in {Stein} {Manifolds}},
     journal = {Annales de l'Institut Fourier},
     pages = {1711--1754},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {65},
     number = {4},
     year = {2015},
     doi = {10.5802/aif.2972},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2972/}
}

Harrington, Phillip S.; Raich, Andrew S. Closed Range for $\bar{\partial }$ and $\bar{\partial }_b$ on Bounded Hypersurfaces in Stein Manifolds. Annales de l'Institut Fourier, Tome 65 (2015) no. 4, pp. 1711-1754. doi : 10.5802/aif.2972. https://aif.centre-mersenne.org/articles/10.5802/aif.2972/

Bibliographie

[1] Andreotti, Aldo; Grauert, Hans Théorème de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France, Tome 90 (1962), pp. 193-259 | Numdam | MR 150342 | Zbl 0106.05501

[2] Andreotti, Aldo; Hill, C. Denson E. E. Levi convexity and the Hans Lewy problem. I. Reduction to vanishing theorems, Ann. Scuola Norm. Sup. Pisa (3), Tome 26 (1972), pp. 325-363 | Numdam | MR 460725 | Zbl 0256.32007

[3] Andreotti, Aldo; Norguet, François Problème de Levi pour les classes de cohomologie, C. R. Acad. Sci. Paris, Tome 258 (1964), pp. 778-781 | MR 159960 | Zbl 0124.38803

[4] Brinkschulte, Judith Local solvability of the $\bar{\partial}$ -equation with boundary regularity on weakly $q$ -convex domains, Math. Ann., Tome 334 (2006) no. 1, pp. 143-152 | Article | MR 2208952 | Zbl 1156.32303

[5] Chen, So-Chin; Shaw, Mei-Chi Partial differential equations in several complex variables, AMS/IP Studies in Advanced Mathematics, Tome 19, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2001, xii+380 pages | MR 1800297 | Zbl 0963.32001

[6] Eastwood, Michael G.; Suria, Giuseppe Vigna Cohomologically complete and pseudoconvex domains, Comment. Math. Helv., Tome 55 (1980) no. 3, pp. 413-426 | Article | MR 593056 | Zbl 0464.32010

[7] Folland, G. B.; Kohn, J. J. The Neumann problem for the Cauchy-Riemann complex, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972, viii+146 pages (Annals of Mathematics Studies, No. 75) | MR 461588 | Zbl 0247.35093

[8] Harrington, Phillip S. Sobolev estimates for the Cauchy-Riemann complex on $C^{1}$ pseudoconvex domains, Math. Z., Tome 262 (2009) no. 1, pp. 199-217 | Article | MR 2491606 | Zbl 1165.32020

[9] Harrington, Phillip S.; Raich, Andrew Regularity results for ${\bar{\partial}}_{b}$ on CR-manifolds of hypersurface type, Comm. Partial Differential Equations, Tome 36 (2011) no. 1, pp. 134-161 | Article | MR 2763350 | Zbl 1216.32025

[10] Harvey, F. Reese; Lawson, H. Blaine Jr. On boundaries of complex analytic varieties. I, Ann. of Math. (2), Tome 102 (1975) no. 2, pp. 223-290 | Article | MR 425173 | Zbl 0317.32017

[11] Henkin, Gennadi M.; Leiterer, Jürgen Global integral formulas for solving the $\bar{\partial}$ -equation on Stein manifolds, Ann. Polon. Math., Tome 39 (1981), pp. 93-116 | MR 617453 | Zbl 0477.32020

[12] Ho, Lop-Hing $\bar{\partial}$ -problem on weakly $q$ -convex domains, Math. Ann., Tome 290 (1991) no. 1, pp. 3-18 | Article | MR 1107660 | Zbl 0714.32006

[13] Hörmander, Lars $L^{2}$ estimates and existence theorems for the $\bar{\partial}$ operator, Acta Math., Tome 113 (1965), pp. 89-152 | Article | MR 179443 | Zbl 0158.11002

[14] Hörmander, Lars An introduction to complex analysis in several variables, North-Holland Mathematical Library, Tome 7, North-Holland Publishing Co., Amsterdam, 1990, xii+254 pages | MR 1045639 | Zbl 0271.32001

[15] Hörmander, Lars The null space of the $\bar{\partial}$ -Neumann operator, Ann. Inst. Fourier (Grenoble), Tome 54 (2004) no. 5, p. 1305-1369, xiv, xx | Article | Numdam | MR 2127850 | Zbl 1083.32033

[16] Krantz, Steven G.; Parks, Harold R. Distance to $C^{k}$ hypersurfaces, J. Differential Equations, Tome 40 (1981) no. 1, pp. 116-120 | Article | MR 614221 | Zbl 0431.57009

[17] Laurent-Thiébaut, Christine Transformation de Bochner-Martinelli dans une variété de Stein, Séminaire d’Analyse P. Lelong–P. Dolbeault–H. Skoda, Années 1985/1986 (Lecture Notes in Math.) Tome 1295, Springer, Berlin, 1987, pp. 96-131 | MR 1047723 | Zbl 0691.32003

[18] Nicoara, Andreea C. Global regularity for ${\bar{\partial}}_{b}$ on weakly pseudoconvex CR manifolds, Adv. Math., Tome 199 (2006) no. 2, pp. 356-447 | Article | MR 2189215 | Zbl 1091.32017

[19] Raich, Andrew Compactness of the complex Green operator on CR-manifolds of hypersurface type, Math. Ann., Tome 348 (2010) no. 1, pp. 81-117 | Article | MR 2657435 | Zbl 1238.32032

[20] Shaw, Mei-Chi Global solvability and regularity for $\bar{\partial}$ on an annulus between two weakly pseudoconvex domains, Trans. Amer. Math. Soc., Tome 291 (1985) no. 1, pp. 255-267 | MR 797058 | Zbl 0594.35010

[21] Shaw, Mei-Chi $L^{2}$ -estimates and existence theorems for the tangential Cauchy-Riemann complex, Invent. Math., Tome 82 (1985) no. 1, pp. 133-150 | Article | MR 808113 | Zbl 0581.35057

[22] Shaw, Mei-Chi $L^{2}$ estimates and existence theorems for ${\bar{\partial}}_{b}$ on Lipschitz boundaries, Math. Z., Tome 244 (2003) no. 1, pp. 91-123 | Article | MR 1981878 | Zbl 1039.32049

[23] Shaw, Mei-Chi The closed range property for $\bar{\partial}$ on domains with pseudoconcave boundary, Complex analysis (Trends Math.), Birkhäuser/Springer Basel AG, Basel, 2010, pp. 307-320 | MR 2885124 | Zbl 1204.32028

[24] Siu, Yum Tong Complex-analyticity of harmonic maps, vanishing and Lefschetz theorems, J. Differential Geom., Tome 17 (1982) no. 1, pp. 55-138 | MR 658472 | Zbl 0497.32025

[25] Straube, Emil J. Lectures on the $L^{2}$ -Sobolev theory of the $\bar{\partial}$ -Neumann problem, ESI Lectures in Mathematics and Physics, European Mathematical Society (EMS), Zürich, 2010, viii+206 pages | MR 2603659 | Zbl 1247.32003

[26] Straube, Emil J. The complex Green operator on CR-submanifolds of $ℂ^{n}$ of hypersurface type: compactness, Trans. Amer. Math. Soc., Tome 364 (2012) no. 8, pp. 4107-4125 | Article | MR 2912447 | Zbl 1278.32027

[27] Zampieri, Giuseppe Complex analysis and CR geometry, University Lecture Series, Tome 43, American Mathematical Society, Providence, RI, 2008, viii+200 pages | MR 2400390 | Zbl 1160.32001

ANNALES DE L'INSTITUT FOURIER

ARTICLES À PARAÎTRE

FEUILLETER

PRESENTATION

COMITÉ DE RÉDACTION

SOUMETTRE UN ARTICLE

ABONNEMENT