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1995 | 113 | 3 | 211-221
Tytuł artykułu

The Bourgain algebra of the disk algebra A(𝔻) and the algebra QA

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Abstrakty
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It is shown that the Bourgain algebra $A(𝔻)_b$ of the disk algebra A(𝔻) with respect to $H^{∞}(𝔻)$ is the algebra generated by the Blaschke products having only a finite number of singularities. It is also proved that, with respect to $H^{∞}(𝔻)$, the algebra QA of bounded analytic functions of vanishing mean oscillation is invariant under the Bourgain map as is $A(𝔻)_b$.
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autor
  • Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27514, U.S.A.
  • Mathematisches Institut I, Universität Karlsruhe, 76128 Karlsruhe, Germany
Bibliografia
  • [1] S.-Y. Chang and D. E. Marshall, Some algebras of bounded analytic functions containing the disc algebra, in: Lecture Notes in Math. 604, Springer, 1977, 12-20.
  • [2] J. Cima, S. Janson and K. Yale, Completely continuous Hankel operators on $H^∞$ and Bourgain algebras, Proc. Amer. Math. Soc. 105 (1989), 121-125.
  • [3] J. Cima, K. Stroethoff and K. Yale, Bourgain algebras on the unit disk, Pacific J. Math. 160 (1993), 27-41.
  • [4] J. Cima, S. Janson and K. Yale, The Bourgain algebra of the disk algebra, Proc. Roy. Irish Acad. 94A (1994), 19-23.
  • [5] J. Cima and R. Timoney, The Dunford-Pettis property for certain planar uniform algebras, Michigan Math. J. 34 (1987), 99-104.
  • [6] K. F. Clancey and J. A. Gosselin, The local theory of Toeplitz operators, Illinois J. Math. 22 (1978), 449-458.
  • [7] A. Davie, T. W. Gamelin and J. B. Garnett, Distance estimates and pointwise bounded density, Trans. Amer. Math. Soc. 175 (1973), 37-68.
  • [8] J. B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981.
  • [9] I. M. Gelfand, D. A. Raikov and G. E. Shilov, Kommutative normierte Algebren, Deutscher Verlag Wiss., Berlin, 1964.
  • [10] P. Gorkin and K. Izuchi, Bourgain algebras on the maximal ideal space of $H^∞$, Rocky Mountain J. Math., to appear.
  • [11] P. Gorkin, K. Izuchi and R. Mortini, Bourgain algebras of Douglas algebras, Canad. J. Math. 44 (1992), 797-804.
  • [12] P. Helmer and J. Pym, Approximation by functions with finitely many discontinuities, Quart. J. Math. Oxford 43 (1992), 223-226.
  • [13] K. Hoffman, Banach Spaces of Analytic Functions, Prentice Hall, Englewood Cliffs, N.J., 1962.
  • [14] K. Izuchi, Bourgain algebras of the disk, polydisk, and ball algebras, Duke Math. J. 66 (1992), 503-520.
  • [15] K. Izuchi, K. Stroethoff and K. Yale, Bourgain algebras of spaces of harmonic functions, Michigan Math. J. 41 (1994), 309-321.
  • [16] D. E. Marshall and K. Stephenson, Inner divisors and composition operators, J. Funct. Anal. 46 (1982), 131-148.
  • [17] R. Mortini and M. von Renteln, Strong extreme points and ideals in uniform algebras, Arch. Math. (Basel) 52 (1989), 465-470.
  • [18] D. Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391-405.
  • [19] C. Sundberg and T. H. Wolff, Interpolating sequences for $QA_B$, ibid. 276 (1983), 551-581.
  • [20] T. H. Wolff, Some theorems of vanishing mean oscillation, thesis, Univ. of California, Berkeley, 1979.
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Bibliografia
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