Geodesic
Derivatives at the boundary for analytic Lipschitz functions
Sbornik. Mathematics, Tome 207 (2016) no. 10, pp. 1471-1490

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We consider the behaviour of holomorphic functions on a bounded open subset of the plane, satisfying a Lipschitz condition with exponent $\alpha$, with $0\alpha1$, 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 show that whenever such a bounded point derivation exists at a boundary point $b$, it may be evaluated by taking a limit of classical difference quotients, approaching from a set having full area density at $b$. Bibliography: 13 titles.
Keywords: analytic function, boundary, Lipschitz condition, point derivation, difference quotient, capacity
Mots-clés : Hausdorff content. 
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     author = {A. G. O'Farrell},
     title = {Derivatives at the boundary for analytic {Lipschitz} functions},
     journal = {Sbornik. Mathematics},
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     url = {http://geodesic.mathdoc.fr/item/SM_2016_207_10_a7/}
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A. G. O'Farrell. Derivatives at the boundary for analytic Lipschitz functions. Sbornik. Mathematics, Tome 207 (2016) no. 10, pp. 1471-1490. http://geodesic.mathdoc.fr/item/SM_2016_207_10_a7/