A natural localization of Hardy spaces in several complex variables

Mihai Putinar; Roland Wolff

Annales Polonici Mathematici (1997)

  • Volume: 66, Issue: 1, page 183-201
  • ISSN: 0066-2216

Abstract

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Let H²(bΩ) be the Hardy space of a bounded weakly pseudoconvex domain in ℂ n . The natural resolution of this space, provided by the tangential Cauchy-Riemann complex, is used to show that H²(bΩ) has the important localization property known as Bishop’s property (β). The paper is accompanied by some applications, previously known only for Bergman spaces.

How to cite

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Mihai Putinar, and Roland Wolff. "A natural localization of Hardy spaces in several complex variables." Annales Polonici Mathematici 66.1 (1997): 183-201. <http://eudml.org/doc/269954>.

@article{MihaiPutinar1997,
abstract = {Let H²(bΩ) be the Hardy space of a bounded weakly pseudoconvex domain in $ℂ^n$. The natural resolution of this space, provided by the tangential Cauchy-Riemann complex, is used to show that H²(bΩ) has the important localization property known as Bishop’s property (β). The paper is accompanied by some applications, previously known only for Bergman spaces.},
author = {Mihai Putinar, Roland Wolff},
journal = {Annales Polonici Mathematici},
keywords = {weakly pseudoconvex domain; Hardy space; quasi-coherent module; Bishop's property (β); Toeplitz operators; bounded weakly pseudoconvex domain; tangential Cauchy-Riemann complex; localization property; Bishop’s property ; Bergman spaces},
language = {eng},
number = {1},
pages = {183-201},
title = {A natural localization of Hardy spaces in several complex variables},
url = {http://eudml.org/doc/269954},
volume = {66},
year = {1997},
}

TY - JOUR
AU - Mihai Putinar
AU - Roland Wolff
TI - A natural localization of Hardy spaces in several complex variables
JO - Annales Polonici Mathematici
PY - 1997
VL - 66
IS - 1
SP - 183
EP - 201
AB - Let H²(bΩ) be the Hardy space of a bounded weakly pseudoconvex domain in $ℂ^n$. The natural resolution of this space, provided by the tangential Cauchy-Riemann complex, is used to show that H²(bΩ) has the important localization property known as Bishop’s property (β). The paper is accompanied by some applications, previously known only for Bergman spaces.
LA - eng
KW - weakly pseudoconvex domain; Hardy space; quasi-coherent module; Bishop's property (β); Toeplitz operators; bounded weakly pseudoconvex domain; tangential Cauchy-Riemann complex; localization property; Bishop’s property ; Bergman spaces
UR - http://eudml.org/doc/269954
ER -

References

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  1. [1] E. Bishop, A duality theorem for an arbitrary operator, Pacific J. Math. 9 (1959), 379-394. Zbl0086.31702
  2. [2] A. Boggess, CR Manifolds and the Tangential Cauchy-Riemann Complex, CRC Press, Boca Raton, Fla., 1991. 
  3. [3] A. Douady, Le problème des modules pour les sous-espaces analytiques compacts d'un espace analytique donné, Ann. Inst. Fourier (Grenoble) 16 (1) (1966), 1-95. Zbl0146.31103
  4. [4] R. G. Douglas and V. Paulsen, Hilbert Modules over Function Algebras, Pitman Res. Notes Math. Ser. 219, Harlow, 1989. Zbl0686.46035
  5. [5] R. G. Douglas, V. Paulsen, C. H. Sah, and K. Yan, Algebraic reduction and rigidity for Hilbert modules, Amer. J. Math. 117 (1995), 75-92. Zbl0833.46040
  6. [6] N. Dunford and J. T. Schwartz, Linear Operators, Part III, Wiley-Interscience, New York, 1971. 
  7. [7] J. Eschmeier and M. Putinar, Spectral Decompositions and Analytic Sheaves, London Math. Soc. Monographs, Oxford Univ. Press, Oxford, 1996. Zbl0855.47013
  8. [8] C. Foiaş, Spectral maximal spaces and decomposable operators in Banach space, Arch. Math. (Basel) 14 (1963), 341-349. Zbl0176.43802
  9. [9] G. M. Henkin, H. Lewy's equation and analysis on pseudoconvex manifolds, Russian Math. Surveys 32 (1977), 59-130 (transl. from Uspekhi Mat. Nauk 32 (3) (1977), 57-118). Zbl0382.35038
  10. [10] J. J. Kohn, The range of the tangential Cauchy-Riemann operator, Duke Math. J. 53 (1986), 525-545. Zbl0609.32015
  11. [11] S. G. Krantz, Function Theory of Several Complex Variables, Wadsworth, Belmont, Calif., 1992. Zbl0776.32001
  12. [12] S. G. Krantz, Partial Differential Equations and Complex Analysis, CRC Press, Boca Raton, Fla., 1992. Zbl0852.35001
  13. [13] M. Putinar, Quasi-similarity of tuples with Bishop's property (β), Integral Equations Operator Theory 15 (1992), 1047-1052. Zbl0773.47011
  14. [14] R. M. Range, Holomorphic Functions and Integral Representations in Several Complex Variables, Grad. Texts in Math. 108, Springer, New York, 1986. Zbl0591.32002
  15. [15] M.-C. Shaw, Local solvability and estimates for ∂ ̅ b on CR manifolds, in: Proc. Sympos. Pure Math. 52, Amer. Math. Soc., Providence, R.I., 1991, 335-345. 
  16. [16] F. Treves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York, 1967. Zbl0171.10402
  17. [17] F. H. Vasilescu, Analytic Functional Calculus and Spectral Decompositions, Reidel, Dordrecht, 1982. Zbl0495.47013
  18. [18] R. Wolff, Spectra of analytic Toeplitz tuples on Hardy spaces, Bull. London Math. Soc., to appear Zbl0865.47003

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