Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On a theorem of Stein


Author: Steven G. Krantz
Journal: Trans. Amer. Math. Soc. 320 (1990), 625-642
MSC: Primary 32H15; Secondary 32A40
DOI: https://doi.org/10.1090/S0002-9947-1990-0964899-0
MathSciNet review: 964899
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper the Kobayashi metric on a domain in $ {{\mathbf{C}}^n}$ is used to define a new function space. Elements of this space belong to a nonisotropic Lipschitz class. It is proved that if $ f$ is holomorphic on the domain and in the classical Lipschitz space $ {\Lambda _\alpha }$ then in fact $ f$ is in the new function space. The result contains the original result of Stein on this subject and provides the optimal result adapted to any domain. In particular, it recovers the Hartogs extension phenomenon in the category of Lipschitz spaces.


References [Enhancements On Off] (What's this?)

  • [CA] D. Catlin, Estimates of invariant metrics on pseudoconvex domains of dimension two, preprint. MR 978601 (90e:32029)
  • [CK] Der Chen E. Chang and S. G. Krantz, Holomorphic Lipschitz functions and applications to the $ \overline \partial $-problem, preprint.
  • [GR] I. Graham, Boundary behavior of the Carathéodory and Kobayashi metrics on strongly pseudoconvex domains in $ {{\mathbf{C}}^n}$ with smooth boundary, Trans. Amer. Math. Soc. 207 (1975), 219-240. MR 0372252 (51:8468)
  • [GK] R. E. Greene and S. G. Krantz, Stability of the Carathéodory and Kobayashi metrics and applications to biholomorphic mappings, Proc. Sympos. Pure Math., vol. 41, Amer. Math. Soc., Providence, R. I., 1984, pp. 77-93. MR 740874 (85k:32043)
  • [GRR] H. Grauert and H. Reckziegel, Hermitesche Metriken und normale Familien holomorpher Abbildungen, Math. Z. 89 (1965), 108-125. MR 0194617 (33:2827)
  • [KR1] S. G. Krantz, Fatou theorems on domains in $ {{\mathbf{C}}^n}$, Bull. Amer. Math. Soc., 16 (1987), 93-96. MR 866022 (88d:32028)
  • [KR2] -, Invariant metrics and boundary behavior of holomorphic functions, in preparation.
  • [KR3] -, Function theory of several complex variables, Wiley, New York, 1982. MR 635928 (84c:32001)
  • [KR4] -, Lipschitz spaces, smoothness of functions, and approximation theory, Expositiones Math. 3 (1983), 193-260. MR 782608 (86g:41001)
  • [KR5] -, Smoothness of harmonic and holomorphic functions, Proc. Sympos. Pure Math., vol. 35, Amer. Math. Soc., Providence, R. I., 1979, pp. 63-67. MR 545292 (81c:32020)
  • [KR6] -, Boundary values and estimates for holomorphic functions of several complex variables, Duke Math. J. 47 (1980), 81-98. MR 563368 (81g:32012)
  • [KR7] -, The boundary behavior of the Kobayashi metric, Rocky Mountain J. Math. (to appear). MR 1159955 (93e:32032)
  • [KM] S. G. Krantz and Daowei Ma, Bloch functions on strongly pseudoconvex domains, Indiana Math. J. (in press).
  • [RE] H. Reiffen, Die differentialgeometrischen Eigenschaften der invarianten Distanzfunktion von Carathéodory, Schr. Math. Inst. Univ. Münster 26 (1963). MR 0158093 (28:1320)
  • [RO] H. Royden, Remarks on the Kobayashi metric, Lecture Notes in Math., vol. 185, Springer-Verlag, Berlin, 1971. MR 0304694 (46:3826)
  • [ST] E. M. Stein, Singular integrals and estimates for the Cauchy-Riemann equations, Bull. Amer. Math. Soc. 79 (1973), 440-445. MR 0315302 (47:3851)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 32H15, 32A40

Retrieve articles in all journals with MSC: 32H15, 32A40


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1990-0964899-0
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society