Derivatives at the boundary for analytic Lipschitz functions

作者：O'Farrell A G^*

来源：Sbornik: Mathematics , 2016, 207(10): 1471-1490.

DOI：10.1070/SM8629

• analytic function
• boundary
• Lipschitz condition
• point derivation
• difference quotient
• capacity
• Hausdorff content

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 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.

• 出版日期2016