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

Harrington, Phillip S.; Raich, Andrew S.

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

Nous définissons $Z\left(q\right)$ faible, une généralisation de $Z\left(q\right)$ sur les domaines bornés $\Omega$ dans une variété de Stein $M^n$ qui suffit à prouver que l’image de $\overline{\partial }$ est fermée. Sous l’hypothèse d’une $Z\left(q\right)$ faible, nous montrons également que (i) les $(0,q)$-formes harmoniques sont triviales et (ii) si $\partial \Omega$ satisfait une $Z\left(q\right)$ faible et une $Z(n-1-q)$ faible, alors ${\overline{\partial }}_b$ a une image fermée sur les $(0,q)$-formes sur $\partial \Omega$. 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 $Z\left(q\right)$ faible.

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

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

[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), Volume 26 (1972), pp. 325-363 | Numdam | MR | Zbl

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

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

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

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

[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, pp. viii+146 (Annals of Mathematics Studies, No. 75) | MR | Zbl

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

[9] Harrington, Phillip S.; Raich, Andrew Regularity results for ${\overline{\partial }}_b$ on CR-manifolds of hypersurface type, Comm. Partial Differential Equations, Volume 36 (2011) no. 1, pp. 134-161 | DOI | MR | Zbl

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

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

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

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

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

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

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

[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.), Volume 1295, Springer, Berlin, 1987, pp. 96-131 | MR | Zbl

[18] Nicoara, Andreea C. Global regularity for ${\overline{\partial }}_b$ on weakly pseudoconvex CR manifolds, Adv. Math., Volume 199 (2006) no. 2, pp. 356-447 | DOI | MR | Zbl

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

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

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

[22] Shaw, Mei-Chi $L^2$ estimates and existence theorems for ${\overline{\partial }}_b$ on Lipschitz boundaries, Math. Z., Volume 244 (2003) no. 1, pp. 91-123 | DOI | MR | Zbl

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

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

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

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

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