Principal values for the Cauchy integral and rectifiability

Tolsa, Xavier

doi:10.1090/S0002-9939-00-05264-3

Principal values for the Cauchy integral and rectifiability
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Proc. Amer. Math. Soc. 128 (2000), 2111-2119 Request permission

Abstract:

We give a geometric characterization of those positive finite measures $\mu$ on ${\mathbb C}$ with the upper density $\limsup _{r \to 0} \frac {\mu (\{\xi :|\xi -z|\leq r\})}{r}$ finite at $\mu$-almost every $z\in {\mathbb C}$, such that the principal value of the Cauchy integral of $\mu$, \[ \lim _{\varepsilon \to 0} \int _{|\xi -z|>\varepsilon } \frac {1}{\xi -z} d\mu (\xi ),\] exists for $\mu$-almost all $z\in {\mathbb C}$. This characterization is given in terms of the curvature of the measure $\mu$. In particular, we get that for $E\subset {\mathbb C}$, ${\mathcal H}^1$-measurable (where ${\mathcal H}^1$ is the Hausdorff $1$-dimensional measure) with $0<{\mathcal H}^1(E)<\infty$, if the principal value of the Cauchy integral of ${\mathcal H}^1_{\mid E}$ exists ${\mathcal H}^1$-almost everywhere in $E$, then $E$ is rectifiable.

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