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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Principal values for the Cauchy integral and rectifiability


Author: Xavier Tolsa
Journal: Proc. Amer. Math. Soc. 128 (2000), 2111-2119
MSC (1991): Primary 30E20; Secondary 42B20, 30E25, 30C85
Published electronically: February 25, 2000
MathSciNet review: 1654076
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Abstract:

We give a geometric characterization of those positive finite measures $\mu$on ${\mathbb C}$ with the upper density $\limsup_{r \to 0} \frac{\mu(\{\xi:\vert\xi-z\vert\leq r\})}{r}$ finite at $\mu$-almost every $z\in{\mathbb C}$, such that the principal value of the Cauchy integral of $\mu$,


\begin{displaymath}\lim_{\varepsilon \to 0} \int_{\lvert\xi-z\rvert>{\varepsilon}} \frac{1}{\xi-z}\, d\mu(\xi),\end{displaymath}

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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Additional Information

Xavier Tolsa
Affiliation: Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, Barcelona 08071, Spain
Email: tolsa@cerber.mat.ub.es

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05264-3
PII: S 0002-9939(00)05264-3
Keywords: Cauchy integral, principal values, curvature of measures, rectifiability
Received by editor(s): June 2, 1998
Received by editor(s) in revised form: September 3, 1998
Published electronically: February 25, 2000
Additional Notes: This research was partially supported by DGICYT PB94-0879.
Communicated by: Albert Baernstein II
Article copyright: © Copyright 2000 American Mathematical Society