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 -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.

**[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