FATOU THEOREMS OLD AND NEW: AN OVERVIEW OF THE BOUNDARY BEHAVIOR OF HOLOMORPHIC FUNCTIONS

  • Published : 2000.03.01

Abstract

We consider the boundary behavior of a Hardy class holomorphic function, either on the disk D in the complex plane or on a domain in multi-dimensional complex space. Although the two theories are formally different, we postulate some unifying fearures, and we suggest some future directions for research.

Keywords

References

  1. Illinois J. Math. v.33 The comparability of the Kobayashi approach region and the admissible approach region G. Aladro
  2. J.Funct. Analysis v.132 homog type paper, A variant of the notion of a space of homogeneous type, A. Carberry, et al
  3. Math. Z. v.200 Estimates of invariant metrics on pseudoconvex domains of dimension two D. Catlin.
  4. Am. Math. Society Singular Integrals F. M. Christ
  5. Duke Math. Jour. v.50 The Lindel of principle and normal functions of several complex variables J. A. Cima;S. G. Krantz
  6. The Theory of Cluster Sets E. F. Collingwood;A. J. Lohwater
  7. Springer Lecture Notes Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes R. R. Coifman;G. Weiss
  8. personal communication F. di Biase
  9. Acta Math. v.30 Series trigonometriques et seeries de Taylor P. Fatou
  10. Trans. Am. Math. Soc. v.207 Boundary behavior of the Caratheodory and Kobayashi metrics on strongly pseudoconvex domains in Cn with smooth boundary I. Graham
  11. Math. Scand. v.20 Lp estimates for (pluri-)subharmonic functions L. Hormander
  12. Trans. A. M. S. v.135 Harmonic functions on Hermitian hyperbolic space A. Koranyi
  13. Trans. A.M.S. v.140 Boundary behavior of Poisson integrals on symmetric spaces A. Koranyi
  14. Function Theory of Several Complex Variables(2nd. ed.) S. G. Krantz
  15. A Carus Monograph of the Mathematical Association of America A Panorama of Harmonic Analysis S. G. Krantz
  16. Proceedings of the 1994 Conference in Cetraro(D. Struppa, ed.) Geometric Foundations for Analysis on Complex Domains S. G. Krantz
  17. Proceedings of Symposia in Pure Mathematics Fundamentals of Harmonic Analysis on Domains in Complex Space S. G. Krantz
  18. Jour. Geometric. Anal. v.1 Invariant metrics and the boundary behavior of holomorphic functions on domains in Cn S. G. Krantz
  19. Partial Dierential Equations and Complex Analysis S. G. Krantz
  20. Rocky Mountain J. Math. v.22 On the boundary behavior of the Kobayashi metric S. G. Krantz
  21. Proceedings of Symposia in Pure Mathematics v.52 Convexity in complex analysis E. Bedford(ed.);J. D'Angelo(ed.);R. Greene(ed.);S. Krantz(ed.)
  22. Michigan Jour. Math. v.41 A Note on Hardy Spaces and Functions of Bounded Mean Oscillation on Domains in Cn S. G. Krantz;S. Y. Li
  23. Ann. Inst. Fourier Grenoble v.45 Duality theorems for Hardy and Bergman spaces on convex domains of fnite type in Cn S. G. Krantz;S. Y. Li
  24. Hardy Classes, Integral Operators, and Duality on Spaces of Homogeneous Type S. G. Krantz;S. Y. Li
  25. Complex Variables v.32 Area integral characterizations of functions in Hardy spaces on domains in Cn S. G. Krantz
  26. Integral Eq. and Op. Thy. v.28 The effect of boundary geometry on Hankel operators belonging to the trace ideals of Bergman spaces S. G. Krantz;S.Y. Li;R. Rochberg
  27. J. Math. v.41 Analysis of Some Function Spaces Associated to Hankel Operators Ill. S. G. Krantz;S.Y. Li;R. Rochberg
  28. Michigan Math. Jour. v.43 The effect of boundary regularity on the singular numbers of Friedrichs operators on Bergman spaces S. G. Krantz;S.Y. Li;R. Rochberg
  29. The Geometry of Domains in Space S. G. Krantz;H. R. Parks
  30. Acta Math. L. Lempert
  31. Complex Analytic Geometry S. Lojaciewicz
  32. D. Ma, thesis
  33. J.Funct. Analysis v.108 Convex domains of Fnite type J. McNeal
  34. Can. J. Math. v.30 Local boundary behavior of bounded holomorphic functions A. Nagel;W. Rudin
  35. thesis A. Neff
  36. Proc. Nat. Acad. Sci. USA v.78 Boundary behavior of functions holomorphic in domains of fnite type A. Nagel;E. M. Stein;S. Wainger
  37. Acta Math. v.155 Balls and metrics defined by vector fields I. Basic properties A. Nagel;E. M. Stein;S. Wainger
  38. Boundary Properties of Analytic Functions I. I. Privalov
  39. Functional Analysis W. Rudin
  40. Can. Jour. Math. v.8 A generalization of an inequality of Hardy and Littlewood K. T. Smith
  41. The Boundary Behavior of Holomorphic Functions of Several Complex Variables E. M. Stein
  42. Introduction to Fourier Analysis on Euclidean Space E. M. Stein;G. Weiss