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A Quantitative Estimate for Bounded Point Evaluations in Pt(μ)-spaces

  • Alexandru Aleman
  • Stefan Richter
  • Carl Sundberg
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 202)

Abstract

In this note we explain how X. Tolsa’s work on analytic capacity and an adaptation of Thomson’s coloring scheme can be used to obtain a quantitative version of J. Thomson’s theorem on bounded point evaluations for Pt(μ)-spaces.

Keywords

Bounded point evaluation Cauchy transform 

Mathematics Subject Classification (2000)

Primary 46E15 Secondary 47B20 

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References

  1. [1]
    Alexandru Aleman, Stefan Richter, and Carl Sundberg. Nontangential limits in Pt(μ)-spaces and the index of invariant subspaces. Ann. of Math., 169(2), 2009.Google Scholar
  2. [2]
    James E. Brennan. Thomson’s theorem on mean-square polynomial approximation. Algebra i Analiz, 17(2):1–32, 2005.MathSciNetGoogle Scholar
  3. [3]
    James E. Brennan. Invariant subspaces and rational approximation. J. Functional Analysis, 7:285–310, 1971.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    James E. Brennan. Point evaluations, invariant subspaces and approximation in the mean by polynomials. J. Funct. Anal., 34(3):407–420, 1979.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    James E. Brennan. The structure of certain spaces of analytic functions. Comput. Methods Funct. Theory, 8(1–2):625–640, 2008.zbMATHMathSciNetGoogle Scholar
  6. [6]
    Claes Fernstrøm. Bounded point evaluations and approximation in Lp by analytic functions. In Spaces of analytic functions (Sem. Functional Anal. and Function Theory, Kristiansand, 1975), pages 65–68. Lecture Notes in Math., Vol. 512. Springer, Berlin, 1976.Google Scholar
  7. [7]
    John Garnett. Analytic capacity and measure. Lecture Notes in Mathematics, Vol. 297. Springer-Verlag, Berlin, 1972.zbMATHGoogle Scholar
  8. [8]
    James E. Thomson. Approximation in the mean by polynomials. Ann. of Math. (2), 133(3):477–507, 1991.CrossRefMathSciNetGoogle Scholar
  9. [9]
    Xavier Tolsa. Painlevé’s problem and the semiadditivity of analytic capacity. Acta Math., 190(1):105–149, 2003.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Dragan Vukotić. A sharp estimate for Ap a functions in ℂn. Proc. Amer. Math. Soc., 117(3):753–756, 1993.zbMATHMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2010

Authors and Affiliations

  • Alexandru Aleman
    • 1
  • Stefan Richter
    • 2
  • Carl Sundberg
    • 2
  1. 1.Department of MathematicsLund UniversityLundSweden
  2. 2.Department of MathematicsUniversity of TennesseeKnoxvilleUSA

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