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
DOI:
https://doi.org/10.1090/S0002-9939-00-05264-3
Published electronically:
February 25, 2000
MathSciNet review:
1654076
Full-text PDF
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












- [D] G. David, Unrectifiable 1-sets in the plane have vanishing analytic capacity. Rev. Mat. Iberoamericana 14 (1998), 369-479. CMP 99:04
- [DM] G. David, P. Mattila, Removable sets for Lipschitz harmonic functions in the plane. Preprint (1997).
- [L] J.-C. Léger, Courbure de Menger et rectifiabilité. PhD Thesis, Université de Paris-Sud, 1997.
- [Ma] P. Mattila, Cauchy Singular Integrals and Rectifiability of Measures in the Plane. Adv. Math. 115 (1995), 1-34. MR 96h:30078
- [MM] P. Mattila, M. S. Melnikov, Existence and weak type inequalities for Cauchy integrals of general measures on rectifiable curves and sets. Proc. Amer. Math. 120(1994), 143-149. MR 94b:30047
- [MMV] P. Mattila, M. S. Melnikov, J. Verdera, The Cauchy integral, analytic capacity, and uniform rectifiability. Ann. of Math. 144 (1996), 127-136. MR 97k:31004
- [Me] M. S. Melnikov, Analytic capacity: discrete approach and curvature of a measure. Sbornik: Mathematics 186:6 (1995), 827-846. MR 96f:30020
- [MV]
M. S. Melnikov, J. Verdera, A geometric proof of the
boundedness of the Cauchy integral on Lipschitz graphs. Int. Math. Res. Not. 7 (1995), 325-331. MR 96f:45011
- [NTV] F. Nazarov, S. Treil, A. Volberg, Pulling ourselves up by the hair. Preprint, 1997.
- [T] X. Tolsa, Cotlar's inequality and existence of principal values for the Cauchy integral without the doubling condition. J. Reine Angew. Math. 502 (1998), 199-235. CMP 99:02
Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 30E20, 42B20, 30E25, 30C85
Retrieve articles in all journals with MSC (1991): 30E20, 42B20, 30E25, 30C85
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