PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
1995 | 31 | 1 | 311-320
Tytuł artykułu

On ∂̅-problems on (pseudo)-convex domains

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this survey we shall tour the area of multidimensional complex analysis which centers around ∂̅-problems (i.e., the Cauchy-Riemann equations) on pseudoconvex domains. Along the way we shall highlight some of the classical milestones as well as more recent landmarks, and we shall discuss some of the major open problems and conjectures. For the sake of simplicity we will only consider domains in $ℂ^n$; intriguing phenomena occur already in the simple setting of (Euclidean) convex domains. We will not discuss at all the closely related theory of the induced Cauchy-Riemann equations on boundaries of domains or on submanifolds of higher codimension. The reader interested in such $∂̅_b$-problems may consult the recent monograph of Boggess [Bo].
Słowa kluczowe
Rocznik
Tom
31
Numer
1
Strony
311-320
Opis fizyczny
Daty
wydano
1995
Twórcy
  • Department of Mathematics, State University of New York at Albany, Albany, New York 12222, U.S.A.
Bibliografia
  • [Ba1] D. Barrett, Irregularity of the Bergman projection on a smooth bounded domain in $ℂ^2$, Ann. of Math. 119 (1984), 431-436.
  • [Ba2] D. Barrett, Behavior of the Bergman projection on the Diederich-Fornaess Worm, preprint.
  • [BL] S. R. Bell and E. Ligocka, A simplification and extension of Fefferman's Theorem on biholomorphic mappings, Invent. Math. 57 (1980), 283-289.
  • [Be1] B. Berndtsson, Weighted estimates for the $∂̅$-equation in domains in $ℂ$, Duke Math. J. 66 (1992), 239-255.
  • [Be2] B. Berndtsson, A smoothly bounded pseudoconvex domain in $ℂ^2$ where $L^∞$ estimates don't hold, Ark. Mat., to appear.
  • [Be3] B. Berndtsson, Some recent results on estimates for the ∂̅-equation, preprint, 1992.
  • [Br] J. Bertrams, Randregularität von Lösungen der ∂̅-Gleichung auf den Polyzylinder und zweidimensionalen analytischen Polyedern, Bonner Math. Schr. 176 (1986).
  • [BD] P. Bonneau and K. Diederich, Integral solution operators for the Cauchy-Riemann equations on pseudoconvex domains, Math. Ann. 286 (1990), 77-100.
  • [BS] H. P. Boas and E. J. Straube, Sobolev estimates for the ∂̅-Neumann operator on domains admitting a defining function that is plurisubharmonic on the boundary, Math. Z. 206 (1991), 81-88.
  • [Bo] A.Boggess, CR Manifolds and the Tangential Cauchy Riemann Complex, CRC Press, Boca Raton, FL, 1991.
  • [Bu] K. Burke, Duality of the Bergman Spaces on Some Weakly Pseudoconvex Domains, Ph.D. Dissertation, SUNY at Albany, New York, 1991.
  • [Ca1] D. Catlin, Necessary conditions for subellipticity of the ∂̅-Neumann problem, Ann. of Math. 117 (1983), 147-171.
  • [Ca2] D. Catlin, Subelliptic estimates for the ∂̅-Neumann problem on pseudoconvex domains, ibid. 126 (1987), 131-191.
  • [Ca3] D. Catlin, Estimates of invariant metrics on pseudoconvex domains of dimension two, Math. Z. 200 (1989), 429-466.
  • [CNS] D. C. Chang, A. Nagel and E. M. Stein, Estimates for the ∂̅-Neumann problem in pseudoconvex domains of finite type in $ℂ^2$, Acta Math. 169 (1992), 153-228.
  • [CC] J. Chaumat et A.-M. Chollet, Estimations hölderiennes pour les équations de Cauchy-Riemann dans les convexes compacts de $ℂ^n$, Math. Z. 207 (1991), 501-534.
  • [Ch] S. C. Chen, Global regularity of the ∂̅-Neumann problem in dimension two, in: Proc. Sympos. Pure Math. 52 (1991), 55-61.
  • [Da1] J. D'Angelo, Real hypersurfaces, orders of contact, and applications, Ann. of Math. 115 (1982), 615-637.
  • [Da2] J. D'Angelo, Several Complex Variables and the Geometry of Real Hypersurfaces, CRC Press, Boca Raton, FL, 1993.
  • [DF] K. Diederich and J. E. Fornaess, Pseudoconvex domains with real analytic boundary, Ann. of Math. 107 (1978), 371-384.
  • [DFW] K. Diederich, J. E. Fornaess and J. Wiegerinck, Sharp Hölder estimates for ∂̅ on ellipsoids, Manuscripta Math. 56 (1986), 399-413.
  • [FeK] C. L. Fefferman and J. J. Kohn, Hölder estimates on domains in two complex dimensions and on three dimensional CR manifolds, Adv. in Math. 69 (1988), 233-303.
  • [FKM] C. L. Fefferman, J. J. Kohn, and M. Machedon, Hölder estimates on CR manifolds with a diagonalizable Levi form, ibid. 84 (1990), 1-90.
  • [FoK] G. Folland and J. Kohn, The Neumann Problem for the Cauchy-Riemann complex, Ann. of Math. Stud., Princeton, 1972.
  • [FS1] J. E. Fornaess and N. Sibony, $L^p$ estimates for ∂̅, in: Proc. Sympos. Pure Math. 52 (1991), 129-163.
  • [FS2] J. E. Fornaess and N. Sibony, Smooth pseudoconvex domains in $ℂ^2$ for which the Corona theorem and $L^p$ estimates for ∂̅ fail, preprint, 1992.
  • [GL] H. Grauert und I. Lieb, Das Ramirezsche Integral und die Lösung der Gleichung ∂̅f=α im Bereich der beschränkten Formen, Rice Univ. Stud. 56 (1970), 29-50.
  • [Gr] P. Greiner, Subelliptic estimates of ∂̅-Neumann problem $ℂ^2$, J. Differential Geom. 9 (1974), 239-260.
  • [He] G. M. Henkin, Integral representations in strictly pseudoconvex domains and applications to the ∂̅-problem, Mat. Sb. 82 (1970), 300-308; Math. USSR-Sb. 11 (1970), 273-281.
  • [HL] G. M. Henkin and J. Leiterer, Theory of Functions on Complex Manifolds, Akademie-Verlag, Berlin, 1984.
  • [Hö] L. Hörmander, $L^2$-estimates and existence theorems for the ∂̅-operators, Acta Math. 113 (1965), 89-152.
  • [Ko1] J. J. Kohn, Harmonic integrals on strongly pseudoconvex manifolds I, II, Ann. of Math. 2 (1963), 112-148; ibid. 79 (1964), 450-472.
  • [Ko2] J. J. Kohn, Global regularity for ∂̅ on weakly pseudoconvex manifolds, Trans. Amer. Math. Soc. 181 (1973), 273-292.
  • [Ko3] J. J. Kohn, Boundary behavior of ∂̅ on weakly pseudoconvex manifolds of dimension two, J. Differential Geom. 6 (1972), 523-542.
  • [Ko4] J. J. Kohn, Subellipticity of the ∂̅-Neumann problem on pseudoconvex domains: Sufficient conditions, Acta Math. 142 (1979), 79-122.
  • [LR1] I. Lieb and R. M. Range, Integral representations and estimates in the theory of the ∂̅-Neumann problem, Ann. of Math. 123 (1986), 265-301.
  • [LR2] I. Lieb and R. M. Range, The kernel of the ∂̅-Neumann operator on strictly pseudoconvex domains, Math. Ann. 278 (1987), 151-173.
  • [Mc1] J. McNeal, Boundary behaviour of the Bergman kernel function in $ℂ^2$, Duke Math. J. 58 (1989), 499-512.
  • [Mc2] J. McNeal, Estimates for the Bergman kernel on convex domains, preprint.
  • [PS] D. H. Phong and E. M. Stein, Hilbert integrals, singular integrals, and Radon transforms, I, Acta Math. 157 (1986), 99-157; II, Invent. Math. 86 (1986), 75-113.
  • [Po] J. Polking, The Cauchy-Riemann equation on convex sets, in: Proc. Sympos. Pure Math. 52 (1993), 309-322.
  • [Ra1] R. M. Range, On Hölder estimates for ∂̅u=f on weakly pseudoconvex domains, in: Proc. Int. Conf. Cortona, 1976-77, Scuola Norm. Sup. Pisa, 1978, 247-267.
  • [Ra2] R. M. Range, Holomorphic Functions and Integral Representations in Several Complex Variables, Springer, New York, 1986.
  • [Ra3] R. M. Range, Integral kernels and Hölder estimates for ∂̅ on pseudoconvex domains of finite type in $ℂ^2$, Math. Ann. 288 (1990), 63-74.
  • [Ra4] R. M. Range, On Hölder and BMO estimates for ∂̅ on convex domains in $ℂ^2$, J. Geom. Anal. 2 (1992), 575-584.
  • [Si] N. Sibony, Un exemple de domaine pseudoconvexe régulier où l'équation ∂̅u=f n'admet pas de solution bornée pour f bornée, Invent. Math. 62 (1980), 235-242.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-bcpv31z1p311bwm
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.