Transactions of the American Mathematical Society

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

 

 

Holomorphic martingales and interpolation between Hardy spaces: the complex method


Author: P. F. X. Müller
Journal: Trans. Amer. Math. Soc. 347 (1995), 1787-1792
MSC: Primary 60G46; Secondary 42B30, 46E15, 46M35
MathSciNet review: 1264825
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Abstract: A probabilistic proof is given to identify the complex interpolation space of $ {H^1}(\mathbb{T})$ and $ {H^\infty }(\mathbb{T})$ as $ {H^p}(\mathbb{T})$.


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

  • [B-L] J. Berg and J. Löfström, Interpolation spaces, Springer Verlag, New York, 1976.
  • [B1] J. Bourgain, New Banach space properties of the disc algebra and 𝐻^{∞}, Acta Math. 152 (1984), no. 1-2, 1–48. MR 736210, 10.1007/BF02392189
  • [B2] -, Some consequences of Pisier's approach to interpolations, preprint 1991.
  • [D] Richard Durrett, Brownian motion and martingales in analysis, Wadsworth Mathematics Series, Wadsworth International Group, Belmont, CA, 1984. MR 750829
  • [J] Peter W. Jones, 𝐿^{∞} estimates for the ∂ problem in a half-plane, Acta Math. 150 (1983), no. 1-2, 137–152. MR 697611, 10.1007/BF02392970
  • [Ja-Jo] Svante Janson and Peter W. Jones, Interpolation between 𝐻^{𝑝} spaces: the complex method, J. Funct. Anal. 48 (1982), no. 1, 58–80. MR 671315, 10.1016/0022-1236(82)90061-1
  • [K1] S. V. Kislyakov, Truncating functions in weighted $ {H^p}$ and two theorems of J. Bourgain, Uppsala University, Dept. of Math., Report 1989:10.
  • [K2] S. V. Kislyakov, Absolutely summing operators on the disc algebra, Algebra i Analiz 3 (1991), no. 4, 1–77 (Russian); English transl., St. Petersburg Math. J. 3 (1992), no. 4, 705–774. MR 1152601
  • [K-X] S. V. Kislyakov and Q, Xu, Interpolation of weighted and vector valued Hardy spaces, Publ. Inst. Recherche. Math. Lille 25 (1991).
  • [M] Paul F. X. Müller, Holomorphic martingales and interpolation between Hardy spaces, J. Anal. Math. 61 (1993), 327–337. MR 1253447, 10.1007/BF02788847
  • [P] Gilles Pisier, Interpolation between 𝐻^{𝑝} spaces and noncommutative generalizations. I, Pacific J. Math. 155 (1992), no. 2, 341–368. MR 1178030
  • [V] Nicholas Th. Varopoulos, The Helson-Szegő theorem and 𝐴_{𝑝}-functions for Brownian motion and several variables, J. Funct. Anal. 39 (1980), no. 1, 85–121. MR 593791, 10.1016/0022-1236(80)90022-1
  • [X1] Q. Xu, Elementary proofs of two theorems of P.W. Jones on interpolation between Hardy spaces, Ann. Inst. Fourier (Grenoble) 42 (1992), 875-889.
  • [X2] -, Notes on interpolation of Hardy spaces, preprint 1990.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1995-1264825-8
Keywords: Holomorphic martingales, complex interpolation method, Hardy spaces
Article copyright: © Copyright 1995 American Mathematical Society