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Boundary smoothness of analytic functions

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Abstract

We consider the behaviour of holomorphic functions on a bounded open subset of the plane, satisfying a Lipschitz condition with exponent \(\alpha \), with \(0<\alpha <1\), in the vicinity of an exceptional boundary point where all such functions exhibit some kind of smoothness. Specifically, we consider the relation between the abstract idea of a bounded point derivation on the algebra of such functions and the classical complex derivative evaluated as a limit of difference quotients. We obtain a result which applies, for example, when the open set admits an interior cone at the special boundary point.

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References

  1. Dolzhenko, E.P.: On removal of singularities of analytic functions. Uspehi Mat. Nauk 18, no. 4 (112), 135–42 (1963) [English transl., AMS Translations (2) 97 (1971) 33–41]

  2. Federer, H.: Geometric Measure Theory. Springer, Berlin (1969)

  3. Hormander, L.: The Analysis of Linear Partial Differential Operators I. Springer, New York (1983)

  4. Lord, D., O’Farrell, A.G.: J. d’Analyse Math. (Jerusalem) 63, 103–19 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  5. O’Farrell, A.G.: Equiconvergence of Derivations. Pac. J. Math. 53, 539–554 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  6. O’Farrell, A.G.: Annihilators of rational modules. J. Funct. Anal. 19, 373–389 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  7. Schwartz, L.: Théorie des distributions, 2nd edn. Hermann, Paris (1966)

    MATH  Google Scholar 

  8. Sherbert, D.R.: The structure of ideals and point derivations in Banach algebras of Lipschitz functions. Trans. AMS 111, 240–272 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  9. Zalcman, L.: Null sets for a class of analytic functions. Am. Math. Mon. 75, 462–470 (1968)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematics and Statistics, NUI, Maynooth, Maynooth, Co.Kildare, Ireland

    Anthony G. O’Farrell

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  1. Anthony G. O’Farrell
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Correspondence to Anthony G. O’Farrell.

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Dedicated to Lawrence Zalcman on the occasion of his $$70$$ 70 th birthday.

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O’Farrell, A.G. Boundary smoothness of analytic functions. Anal.Math.Phys. 4, 131–144 (2014). https://doi.org/10.1007/s13324-014-0074-0

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  • DOI: https://doi.org/10.1007/s13324-014-0074-0

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