## The planar Cantor sets of zero analytic capacity and the local $T(b)$-Theorem

HTML articles powered by AMS MathViewer

- by Joan Mateu, Xavier Tolsa and Joan Verdera
- J. Amer. Math. Soc.
**16**(2003), 19-28 - DOI: https://doi.org/10.1090/S0894-0347-02-00401-0
- Published electronically: July 10, 2002
- PDF | Request permission

## Abstract:

In this paper we obtain rather precise estimates for the analytic capacity of a big class of planar Cantors sets. In fact, we show that analytic capacity and positive analytic capacity are comparable for these sets. The main tool for the proof is an appropriate version of the $T(b)$-Theorem.## References

- 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** - 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**, DOI 10.4064/cm-60-61-2-601-628 - Guy David,
*Analytic capacity, Calderón-Zygmund operators, and rectifiability*, Publ. Mat.**43**(1999), no. 1, 3–25. MR**1697514**, DOI 10.5565/PUBLMAT_{4}3199_{0}1 - Alexander M. Davie and Bernt Øksendal,
*Analytic capacity and differentiability properties of finely harmonic functions*, Acta Math.**149**(1982), no. 1-2, 127–152. MR**674169**, DOI 10.1007/BF02392352 - V. Ya. Èĭderman,
*Hausdorff measure and capacity associated with Cauchy potentials*, Mat. Zametki**63**(1998), no. 6, 923–934 (Russian, with Russian summary); English transl., Math. Notes**63**(1998), no. 5-6, 813–822. MR**1679225**, DOI 10.1007/BF02312776 - John Garnett,
*Positive length but zero analytic capacity*, Proc. Amer. Math. Soc.**24**(1970), 696–699; errata, ibid. 26 (1970), 701. MR**0276456**, DOI 10.1090/S0002-9939-1970-0276456-5 - John Garnett,
*Analytic capacity and measure*, Lecture Notes in Mathematics, Vol. 297, Springer-Verlag, Berlin-New York, 1972. MR**0454006**, DOI 10.1007/BFb0060912
[GY]gyJ. Garnett, S. Yoshinobu, Large sets of zero analytic capacity, Proc. Amer. Math. Soc. - L. D. Ivanov,
*Variatsii mnozhestv i funktsiĭ*, Izdat. “Nauka”, Moscow, 1975 (Russian). Edited by A. G. Vituškin. MR**0476953**
[I2]iL. D. Ivanov, On sets of analytic capacity zero, in - Peter W. Jones,
*Square functions, Cauchy integrals, analytic capacity, and harmonic measure*, Harmonic analysis and partial differential equations (El Escorial, 1987) Lecture Notes in Math., vol. 1384, Springer, Berlin, 1989, pp. 24–68. MR**1013815**, DOI 10.1007/BFb0086793 - Pertti Mattila,
*On the analytic capacity and curvature of some Cantor sets with non-$\sigma$-finite length*, Publ. Mat.**40**(1996), no. 1, 195–204. MR**1397014**, DOI 10.5565/PUBLMAT_{4}0196_{1}2 - Takafumi Murai,
*Construction of $H^1$ functions concerning the estimate of analytic capacity*, Bull. London Math. Soc.**19**(1987), no. 2, 154–160. MR**872130**, DOI 10.1112/blms/19.2.154 - Xavier Tolsa,
*$L^2$-boundedness of the Cauchy integral operator for continuous measures*, Duke Math. J.**98**(1999), no. 2, 269–304. MR**1695200**, DOI 10.1215/S0012-7094-99-09808-3
[T2]t2X. Tolsa, On the analytic capacity $\gamma ^+$, - Joan Verdera,
*A weak type inequality for Cauchy transforms of finite measures*, Publ. Mat.**36**(1992), no. 2B, 1029–1034 (1993). MR**1210034**, DOI 10.5565/PUBLMAT_{3}62B92_{1}9 - Joan Verdera,
*Removability, capacity and approximation*, Complex potential theory (Montreal, PQ, 1993) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 439, Kluwer Acad. Publ., Dordrecht, 1994, pp. 419–473. MR**1332967** - Joan Verdera,
*$L^2$ boundedness of the Cauchy integral and Menger curvature*, Harmonic analysis and boundary value problems (Fayetteville, AR, 2000) Contemp. Math., vol. 277, Amer. Math. Soc., Providence, RI, 2001, pp. 139–158. MR**1840432**, DOI 10.1090/conm/277/04543
[Vi1]vi1A. G. Vitushkin, Estimate of the Cauchy integral,

**129**(2001), 3543-3548.

*“Linear and Complex Analysis Problem Book 3"*(part II), Lectures Notes in Mathematics

**1574**, Springer-Verlag, Berlin, 1994, pp. 150–153.

*Indiana Univ. Math. J.*

**(51)**(2) (2002), 317–344. [T3]T3X. Tolsa, Painlevé’s problem and the semiadditivity of analytic capacity, Acta Math. (to appear). [Uy]uyN. X. Uy, Removable sets of analytic functions satisfying a Lipschitz condition,

*Ark. Mat.*

**17**(1979), 19–27.

*Mat. Sb.*

**71(4)**(1966), 515–534. [Vi2]viA. G. Vitushkin, Analytic capacity of sets in problems of approximation theory,

*Russian Math. Surveys*

**22**(1967), 139–200.

## Bibliographic Information

**Joan Mateu**- Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, (Barcelona), Spain
**Xavier Tolsa**- Affiliation: Département de Mathématiques, Université de Paris Sud 91405 Orsay, cedex, France
- MR Author ID: 639506
- ORCID: 0000-0001-7976-5433
**Joan Verdera**- Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, (Barcelona), Spain
- Received by editor(s): August 7, 2001
- Published electronically: July 10, 2002
- Additional Notes: The authors were partially supported by the grants BFM 2000-0361, HPRN-2000-0116 and 2001- SGR-00431. The second author was supported by a Marie Curie Fellowship of the European Union under contract HPMFCT-2000-00519.
- © Copyright 2002 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**16**(2003), 19-28 - MSC (2000): Primary 30C85; Secondary 42B20, 30E20
- DOI: https://doi.org/10.1090/S0894-0347-02-00401-0
- MathSciNet review: 1937197