Cauchy's integral formula in domains of arbitrary connectivity

M. V. Samokhin

Sbornik. Mathematics, Tome 191 (2000) no. 8, pp. 1215-1231

Résumé

It is shown that a straightforward generalization of Cauchy's integral formula is possible only in domains with boundary of finite length (in some sense or other). An example of a simply connected domain with boundary of infinite length is constructed such that for fairly general functionals on $H^\infty$ no extremal function (including the Ahlfors function) can be represented as a Cauchy potential.

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