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Principal values for the Cauchy integral and rectifiability. (English) Zbl 0944.30022

The author gives a geometric characterization of those positive finite measures \(\mu\) on \(\mathbb C\) with finite upper density at \(\mu\)-almost every \(z\in \mathbb C\), such that the principal value of the Cauchy integral of \(\mu\) exists for \(\mu\)-almost all \(z\in \mathbb C\). This characterization is given in terms of the Menger curvature of \(\mu\). In particular, one gets the following theorem: For \(E\subset \mathbb C\), \(H^1\)-measurable (where \(H^1\) is the Hausdorff 1-dimensional measure) with \(0<H^1(E)<\infty\), if the principal value of the Cauchy integral of the measure \(H^1_{|E}\) exists \(H^1\)-almost everywhere in \(E\), then \(E\) is rectifiable. This theorem answers a question raised by P. Mattila [Adv. Math. 115, No. 1, 1-34 (1995; Zbl 0842.30029)].

MSC:

30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
28A78 Hausdorff and packing measures
30E25 Boundary value problems in the complex plane
30C85 Capacity and harmonic measure in the complex plane

Citations:

Zbl 0842.30029
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