A nicely behaved singular integral on a purely unrectifiable set

Author:
Petri Huovinen

Journal:
Proc. Amer. Math. Soc. **129** (2001), 3345-3351

MSC (2000):
Primary 28A75, 42B20; Secondary 30E20

Published electronically:
April 2, 2001

MathSciNet review:
1845012

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

We construct an example of a purely 1-unrectifiable AD-regular set in the plane such that the limit

exists and is finite for almost every for some class of antisymmetric Calderón-Zygmund kernels. Moreover, the singular integral operators associated with these kernels are bounded in , where has a positive measure.

**1.**A.-P. Calderón,*Cauchy integrals on Lipschitz curves and related operators*, Proc. Nat. Acad. Sci. U.S.A.**74**(1977), no. 4, 1324–1327. MR**0466568****2.**Michael Christ,*Lectures on singular integral operators*, CBMS Regional Conference Series in Mathematics, vol. 77, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. MR**1104656****3.**Michael Christ,*A 𝑇(𝑏) theorem with remarks on analytic capacity and the Cauchy integral*, Colloq. Math.**60/61**(1990), no. 2, 601–628. MR**1096400****4.**Guy David,*Unrectifiable 1-sets have vanishing analytic capacity*, Rev. Mat. Iberoamericana**14**(1998), no. 2, 369–479 (English, with English and French summaries). MR**1654535**, 10.4171/RMI/242**5.**G. David, P. Mattila,*Removable sets for Lipschitz harmonic functions in the plane*, Rev. Mat. Iberoamericana**16**2000, pp. 137-215. CMP**2000:15****6.**Guy David and Stephen Semmes,*Analysis of and on uniformly rectifiable sets*, Mathematical Surveys and Monographs, vol. 38, American Mathematical Society, Providence, RI, 1993. MR**1251061****7.**Pertti Mattila,*Geometry of sets and measures in Euclidean spaces*, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR**1333890****8.**Pertti Mattila, Mark S. Melnikov, and Joan Verdera,*The Cauchy integral, analytic capacity, and uniform rectifiability*, Ann. of Math. (2)**144**(1996), no. 1, 127–136. MR**1405945**, 10.2307/2118585**9.**F. Nazarov, S. Treil, A. Volberg,*Pulling ourselves up by the hair*, Preprint.**10.**X. Tolsa,*Principal values for the Cauchy integral and rectifiability*, Proc. Amer. Math. Soc.**128**2000, pp. 2111-2119. CMP**2000:14**

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Additional Information

**Petri Huovinen**

Affiliation:
Department of Mathematics, University of Jyväskylä, P.O. Box 35, FIN-40351 Jyväskylä, Finland

Email:
pjh@math.jyu.fi

DOI:
http://dx.doi.org/10.1090/S0002-9939-01-05955-X

Keywords:
Singular integrals,
rectifiability

Received by editor(s):
August 31, 1999

Received by editor(s) in revised form:
March 22, 2000

Published electronically:
April 2, 2001

Additional Notes:
The author was supported by EU TMR Grant #ERBFMBICT972410

Communicated by:
David Preiss

Article copyright:
© Copyright 2001
American Mathematical Society