Geometric conditions which imply compactness of the $\overline{\partial }$-Neumann operator
Annales de l'Institut Fourier, Volume 54 (2004) no. 3, p. 699-710

For smooth bounded pseudoconvex domains in ${ℂ}^{2}$, we provide geometric conditions on the boundary which imply compactness of the $\overline{\partial }$-Neumann operator. It is noteworthy that the proof of compactness does not proceed via verifying the known potential theoretic sufficient conditions.

On donne, pour les domaines lisses bornés pseudoconvexes de ${ℂ}^{2}$, des conditions géométriques concernant le bord qui entraînent la compacité de l’opérateur $\overline{\partial }$-Neumann. Il est remarquable que la preuve de la compacité ne procède pas par verification des conditions suffisantes bien connues de type théorie du potentiel.

DOI : https://doi.org/10.5802/aif.2030
Classification:  32W05
Keywords: $\overline{\partial }$-Neumann operator, compactness, geometric conditions
@article{AIF_2004__54_3_699_0,
author = {Straube, Emil},
title = {Geometric conditions which imply compactness of the ${\overline{\partial }}$-Neumann operator},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {54},
number = {3},
year = {2004},
pages = {699-710},
doi = {10.5802/aif.2030},
zbl = {1061.32028},
mrnumber = {2097419},
language = {en},
url = {http://www.numdam.org/item/AIF_2004__54_3_699_0}
}

Straube, Emil. Geometric conditions which imply compactness of the ${\overline{\partial }}$-Neumann operator. Annales de l'Institut Fourier, Volume 54 (2004) no. 3, pp. 699-710. doi : 10.5802/aif.2030. http://www.numdam.org/item/AIF_2004__54_3_699_0/

[1] Harold P. Boas Small sets of infinite type are benign for the $\overline{\partial }$-Neumann problem, Proceedings of the American Math. Soc, Tome 103 (1988), pp. 569-578 | MR 943086 | Zbl 0652.32016

[2] Harold P. Boas; Emil J. Straube; M. Schneider And Y.-T. Siu, Eds Global regularity of the $\overline{\partial }$-Neumann problem: a survey of the ${L}^{2}$-Sobolev theory, Several Complex Variables, Cambridge University Press (MSRI Publications) Tome 37 (1999) | Zbl 0967.32033

[3] David W. Catlin Boundary behavior of holomorphic functions on weakly pseudoconvex domains (1978) (Princeton University Ph.D. Thesis) | Zbl 0484.32005

[4] David W. Catlin; Y.-T. Siu Ed. Global regularity of the $\overline{\partial }$-Neumann problem, Complex Analysis of Several Variables (Proc. Symp. Pure Math.) Tome 41 (1984), pp. 39-49 | Zbl 0578.32031

[5] David W. Catlin Subelliptic estimates for the $\overline{\partial }$-Neumann problem on pseudoconvex domains, Annals of Mathematics (2), Tome 126 (1987), pp. 131-191 | MR 898054 | Zbl 0627.32013

[6] So-Chin Chen; Mei-Chi Shaw Partial Differential Equations in Several Complex Variables, American Mathematical Society/International Press, Studies in Advanced Mathematics (2001) | MR 1800297 | Zbl 0963.32001

[7] Michael Christ; Siqi Fu Compactness in the $\overline{\partial }$-Neumann problem, magnetic Schrödinger operators, and the Aharonov--Bohm effect (2003) (preprint) | MR 2166176 | Zbl 1098.32020

[8] John P. D'Angelo Several Complex Variables and the Geometry of Real Hypersurfaces, CRC Press, Boca Raton, FL, Studies in Advanced Mathematics (1993) | MR 1224231 | Zbl 0854.32001

[9] M. Derridj Regularité pour $\overline{\partial }$ dans quelques domaines faiblement pseudo-convexes, Journal of Differential Geometry, Tome 13 (1978), pp. 559-576 | MR 570218 | Zbl 0435.35057

[10] G.B. Folland; J.J. Kohn The Neumann Problem for the Cauchy--Riemann Complex, Princeton University Press, Annals of Mathematics Studies, Tome 75 (1972) | MR 461588 | Zbl 0247.35093

[11] Siqi Fu; Emil J. Straube Compactness of the $\overline{\partial }$-Neumann problem on convex domains, Journal of Functional Analysis, Tome 159 (1998), pp. 629-641 | MR 1659575 | Zbl 0959.32042

[12] Siqi Fu; Emil J. Straube; J. Mcneal Ed. Compactness in the $\overline{\partial }$-Neumann problem, Complex Analysis and Geometry, Math. Research Inst. Publ., Ohio State Univ., Tome 9 (2001), pp. 141-160 | Zbl 1011.32025

[13] Siqi Fu; Emil J. Straube Semi-classical analysis of Schrödinger operators and compactness in the $\overline{\partial }$-Neumann problem, Journal of Math. Analysis and Applications, Tome 271 (2002) no. 1, pp. 267-282 | MR 1923760 | Zbl 01836752

[13] Siqi Fu; Emil J. Straube Semi-classical analysis of Schrödinger operators and compactness in the $\overline{\partial }$-Neumann problem. (correction), J. Math. Anal. Appl., Tome 280 (2003) no. 1, p. 195-196 | MR 1972203 | Zbl 01836752

[14] Torsten Hefer; Ingo Lieb On the compactness of the $\overline{\partial }$-Neumann operator, Ann. Fac. Sci. Toulouse Math (6), Tome 9 (2000), pp. 415-432 | Numdam | MR 1842025 | Zbl 1017.32025

[15] Peter Matheos A Hartogs domain with no analytic discs in the boundary for which the $\overline{\partial }$-Neumann problem is not compact (1997) (University of California Los Angeles Ph.D. Thesis)

[16] Jeffery D. Mcneal A sufficient condition for compactness of the $\overline{\partial }$-Neumann problem, Journal of Functional Analysis, Tome 195 (2002), pp. 190-205 | MR 1934357 | Zbl 1023.32029

[17] Nessim Sibony Une classe de domaines pseudoconvexes, Duke Math. Journal, Tome 55 (1987), pp. 299-319 | MR 894582 | Zbl 0622.32016

[18] Emil J. Straube Plurisubharmonic functions and subellipticity of the $\overline{\partial }$-Neumann problem on non-smooth domains, Mathematical Research Letters, Tome 4 (1997), pp. 459-467 | MR 1470417 | Zbl 0887.32005

[19] William P. Ziemer Weakly Differentiable Functions, Springer-Verlag, Graduate Texts in Mathematics, Tome 120 (1989) | MR 1014685 | Zbl 0692.46022