Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On complementation of vector-valued Hardy spaces

Author: Wolfgang Hensgen
Journal: Proc. Amer. Math. Soc. 104 (1988), 1153-1162
MSC: Primary 46E40; Secondary 30D55, 42B30, 46J15
MathSciNet review: 933514
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ X$ be a complex Banach space and $ 1 < p < \infty $. $ {H^p}(X)$ resp. $ {h^p}(X)$ denote the Hardy spaces of $ X$-valued analytic resp. harmonic functions on the disc. $ {L^p}(X)$ is the Lebesgue-Bochner space of $ X$-valued integrable functions on the circle and $ {{\mathbf{H}}^p}(X)$ its Hardy-type subspace $ \{ f \in {L^p}(X):\hat f(n) = 0\forall n < 0\} $. It is proved that the following four conditions are equivalent: $ {H^p}(X)$ is complemented in $ {h^p}(X)$; the canonical analytic (or Riesz) projection is a bounded operator $ {h^p}(X) \to {H^p}(X);{{\mathbf{H}}^p}(X)$ is complemented in $ {L^p}(X)$; analytic projection is a bounded operator $ {L^p}(X) \to {{\mathbf{H}}^p}(X)$. It is well known that the last condition, in turn, is equivalent to the UMD property of $ X$.

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

  • [1] J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat. 21 (1983), 163-168. MR 727340 (85a:46011)
  • [2] A. V. Bukhvalov, On an analytic representation of operators with abstract norm, Soviet Math. Dokl. 14 (1973), 197-201.
  • [3] -, Hardy spaces of vector-valued functions, J. Soviet Math. 16 (1981), 1051-1059.
  • [4] A. V. Bukhvalov and A. A. Danilevich, Boundary properties of analytic and harmonic functions with values in Banach space, Math. Notes Acad. Sci. USSR 31 (1982), 104-110. MR 649004 (84f:46032)
  • [5] D. L. Burkholder, A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions, Proc. Conf. Harmonic Analysis in Honour of A. Zygmund, Univ. of Chicago, 1981, 1983, pp. 270-286. MR 730072 (85i:42020)
  • [6] -, Martingales and Fourier analysis in Banach spaces, Probability and Analysis (Varenna, (Como), 1985), (G. Letta and M. Pratelli, eds.), Lecture Notes in Math., vol. 1206, Springer-Verlag, Berlin, Heidelberg and New York, 1986, pp. 61-108. MR 864712 (88c:42017)
  • [7] J. Diestel and J. J. Uhl, Jr., Vector measures, Math. Surveys, No. 15, Amer. Math. Soc., Providence, R. I., 1977. MR 0453964 (56:12216)
  • [8] N. Dinculeanu, Vector measures, VEB Deutscher Verlag der Wissenschaften, Berlin, 1966 (Hochschulbücher für Math. 64). MR 0206189 (34:6011a)
  • [9] C. Grossetête, Sur certaines classes de fonctions harmoniques dans le disque à valeur dans un espace vectoriel topologique localement convexe, C. R. Acad. Sci. Paris 273 (1971), 1048-1051.
  • [10] -, Classes de Hardy et de Nevanlinna pour les fonctions holomorphes à valeurs vectorielles, C. R. Acad. Sci. Paris 274 (1972), 251-253.
  • [11] J. A. Gutierrez and H. E. Lacey, On the Hilbert transform for Banach space valued functions, Martingale Theory in Harmonic Analysis and Banach Spaces (Cleveland, 1981), Lecture Notes in Math., vol. 939, Springer-Verlag, Berlin, Heidelberg and New York, 1982, pp. 73-80. MR 668538 (84a:42018)
  • [12] M. Heins, Vector-valued harmonic functions, Functions, Series, Operators, Colloq. Math. Soc. J. Bolyai 35 (1980), 621-632. MR 751028 (85k:31003)
  • [13] W. Hensgen, Hardy-Räume vektorwertiger Funktionen, Thesis, Munich, 1986.
  • [14] E. Hille and R. S. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R.I., 1957. MR 0089373 (19:664d)
  • [15] K. Hoffmann, Banach Spaces of Analytic Functions, Prentice Hall, Englewood Cliffs, N.J., 1962. MR 0133008 (24:A2844)
  • [16] A. Ionescu-Tulcea and C. Ionescu-Tulcea, Topics in the theory of lifting, Springer-Verlag, Berlin, Heidelberg and New York, 1969 (Ergebnisse... 48). MR 0276438 (43:2185)
  • [17] W. Kaballo, On Fredholm operator valued $ {H^p}$-functions, Proc. Toeplitz Mem. Conf., Tel Aviv, Birkhäuser, 1982, pp. 313-319. MR 669915 (84i:47024)
  • [18] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. II, Springer-Verlag, Berlin, Heidelberg and New York, 1979 (Ergebnisse... 97). MR 540367 (81c:46001)
  • [19] W. J. Ricker, Characterization of Poisson integrals of vector-valued functions and measures on the unit circle, Hokkaido Math. J. 16 (1987), 29-42. MR 878839 (88e:31004)
  • [20] R. Ryan, Boundary values of analytic vector-valued functions, Indag. Math. 65 (1962), 558-572. MR 0145086 (26:2621)
  • [21] -, The F. and M. Riesz theorem for vector measures, Indag. Math. 66 (1963), 408-412. MR 0152876 (27:2848)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46E40, 30D55, 42B30, 46J15

Retrieve articles in all journals with MSC: 46E40, 30D55, 42B30, 46J15

Additional Information

Keywords: Vector-valued Hardy spaces, analytic (or Riesz) projection, UMD Banach spaces
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society