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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

We give a geometric characterization of those positive finite measures on with the upper density finite at -almost every , such that the principal value of the Cauchy integral of ,

exists for -almost all . This characterization is given in terms of the curvature of the measure . In particular, we get that for , -measurable (where is the Hausdorff -dimensional measure) with , if the principal value of the Cauchy integral of exists -almost everywhere in , then 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:
https://doi.org/10.1090/S0002-9939-00-05264-3

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