Duke Mathematical Journal

The $\overline\partial$-Neumann operator on Lipschitz pseudoconvex domains with plurisubharmonic defining functions

Joachim Michel and Mei-Chi Shaw

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

On a bounded pseudoconvex domain Ω in ℂn with a plurisubharmonic Lipschitz defining function, we prove that the $\overline\partial$-Neumann operator is bounded on Sobolev (1/2)-spaces.

Article information

Source
Duke Math. J., Volume 108, Number 3 (2001), 421-447.

Dates
First available in Project Euclid: 5 August 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1091737180

Digital Object Identifier
doi:10.1215/S0012-7094-01-10832-6

Mathematical Reviews number (MathSciNet)
MR1838658

Zentralblatt MATH identifier
1020.32030

Subjects
Primary: 32W05: $\overline\partial$ and $\overline\partial$-Neumann operators
Secondary: 32U10: Plurisubharmonic exhaustion functions 35N15: $\overline\partial$-Neumann problem and generalizations; formal complexes [See also 32W05, 32W10, 58J10]

Citation

Michel, Joachim; Shaw, Mei-Chi. The $\overline\partial$-Neumann operator on Lipschitz pseudoconvex domains with plurisubharmonic defining functions. Duke Math. J. 108 (2001), no. 3, 421--447. doi:10.1215/S0012-7094-01-10832-6. https://projecteuclid.org/euclid.dmj/1091737180


Export citation

References

  • \lccD. Barrett, Behavior of the Bergman projection on the Diederich-Fornaess worm, Acta Math. 168 (1992), 1–10. MR 93c:32033
  • \lccB. Berndtsson, “$\db_b$ and Carleson type inequalities” in Complex Analysis (College Park, Md., 1985/86), II, Lecture Notes in Math. 1276, Springer, New York, 1987, 42–54. MR 89b:32004
  • \lccB. Berndtsson and P. Charpentier, A Sobolev mapping property of the Bergman kernel, Math. Z. 235 (2000), 1–10. MR CMP 1785069
  • \lccH. P. Boas and E. J. Straube, Equivalence of regularity for the Bergman projection and the $\db$-Neumann operator, Manuscripta Math. 67 (1990), 25–33. MR 90k:32057
  • ––––, Sobolev estimates for the $\overline\partial$-Neumann operator on domains in $\bold C^n$ admitting a defining function that is plurisubharmonic on the boundary, Math. Z. 206 (1991), 81–88. MR 92b:32027
  • ––––, “Global regularity of the $\overline\partial$-Neumann problem: A survey of the $L^2$-Sobolev theory” in Several Complex Variables (Berkeley, 1995–1996), ed. M. Schneider and Y.-T. Siu, Math. Sci. Res. Inst. Publ. 37, Cambridge Univ. Press, Cambridge, 1999, 79–111. MR CMP 1748601
  • \lccA. Bonami and P. Charpentier, Une estimation Sobolev $1/2$ pour le projecteur de Bergman, C. R. Acad. Sci. Paris Ser. I Math. 307 (1988), 173–176. MR 90b:32048
  • ––––, Boundary values for the canonical solution to $\bar\partial$-equation and $W^{1/2}$ estimates, preprint, Bordeaux, 1990.
  • \lccP. Charpentier, “Sur les valeurs au bord de solutions de l'équation $\bar\partial u=f$” in Analyse complexe multivariable: Récents développements (Guadeloupe,1988), Sem. Conf. 5, EditEl, Rende, Italy, 1991, 51–81. MR 94f:32029
  • \lccM. Christ, Global $C^\infty$ irregularity of the $\bar\partial$-Neumann problem for worm domains, J. Amer. Math. Soc. 9 (1996), 1171–1185. MR 96m:32014
  • \lccB. E. J. Dahlberg, Weighted norm inequalities for the Lusin area integral and the nontangential maximal functions for functions harmonic in a Lipschitz domain, Studia Math. 67 (1980), 297–314. MR 82f:31003
  • \lccK. Diederich and J. E. Fornaess, Pseudoconvex domains: Bounded strictly plurisubharmonic exhaustion functions, Invent. Math. 39 (1977), 129–141. MR 55:10728
  • ––––, Pseudoconvex domains: An example with nontrivial Nebenhülle, Math. Ann. 225 (1977), 275–292. MR 55:3320
  • \lccL. E. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Stud. Adv. Math., CRC, Boca Raton, 1992. MR 93f:28001
  • \lccP. Grisvard, Elliptic Problems in Nonsmooth Domains, Monogr. Stud. Math. 24, Pitman, Boston, 1985. MR 86m:35044
  • \lccG. M. Henkin and A. Iordan, Compactness of the Neumann operator for hyperconvex domains with non-smooth B-regular boundary, Math. Ann. 307 (1997), 151–168. MR 98a:32018
  • \lccG. M. Henkin, A. Iordan and J. J. Kohn, Estimations sous-elliptiques pour le problème $\db$-Neumann dans un domaine strictement pseudoconvexe à frontière lisse par morceaux, C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), 17–22. MR 97g:32014
  • \lccL. Hörmander, $L^2$ estimates and existence theorems for the $\bar\partial$ operator, Acta Math. 113 (1965), 89–152. MR 31:3691
  • ––––, An Introduction to Complex Analysis in Several Variables, 3d ed., North-Holland Math. Library 7, North-Holland, Amsterdam, 1990. MR 91a:32001
  • \lccD. Jerison and C. E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal. 130 (1995), 161–219. MR 96b:35042
  • \lccJ. J. Kohn, Harmonic integrals on strongly pseudoconvex manifolds, I, Ann. of Math. (2) 78 (1963), 112–148. MR 27:2999
  • ––––, Subellipticity of the $\bar\partial$-Neumann problem on pseudo-convex domains: Sufficient conditions, Acta Math. 142 (1979), 79–122. MR 80d:32020
  • ––––, “Quantitative estimates for global regularity” in Analysis and Geometry in Several Complex Variables (Katata, Japan, 1997), Birkhäuser, Boston, 1999, 97–128. MR 2000f:32046
  • \lccJ.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, I, Grundlehren Math. Wiss. 181, Springer, New York, 1972. MR 50:2670
  • \lccJ. Michel and M.-C. Shaw, Subelliptic estimates for the $\db$-Neumann operator on piecewise smooth strictly pseudoconvex domains, Duke Math. J. 93 (1998), 115–128. MR 99b:32019
  • \lccR. M. Range, A remark on bounded strictly plurisubharmonic exhaustion functions, Proc. Amer. Math. Soc. 81 (1981), 220–222. MR 82e:32030
  • \lccR. M. Range and Y. T. Siu, Uniform estimates for the $\db$-equation on domains with piecewise smooth strictly pseudoconvex boundaries, Math. Ann. 206 (1973), 325–354. MR 49:3214
  • \lccM.-C. Shaw, Local existence theorems with estimates for $\overline\partial_b$ on weakly pseudo-convex $CR$ manifolds, Math. Ann. 294 (1992), 677–700. MR 94b:32026
  • \lccE. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. 30, Princeton Univ. Press, Princeton, 1970. MR 44:7280
  • \lccE. J. Straube, Plurisubharmonic functions and subellipticity of the $\db$-Neumann problem on nonsmooth domains, Math. Res. Lett. 4 (1997), 459–467. MR 98m:32024
  • \lccS. Vassiliadou, $L^2$ existence and subelliptic estimates for the $\db$-Neumann operator on certain piecewise smooth domains in $\mathbb C^n$, to appear in Complex Variables Theory Appl.