Geodesic
The planar Cantor sets of zero analytic capacity and the local 𝑇(𝑏)-Theorem
Mateu, Joan1 ; Tolsa, Xavier2 ; Verdera, Joan1
1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, (Barcelona), Spain
2 Département de Mathématiques, Université de Paris Sud 91405 Orsay, cedex, France
Journal of the American Mathematical Society, Tome 16 (2003) no. 1, pp. 19-28

Voir la notice de l'article provenant de la source American Mathematical Society

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.
DOI : 10.1090/S0894-0347-02-00401-0

Mateu, Joan 1 ; Tolsa, Xavier 2 ; Verdera, Joan 1

1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, (Barcelona), Spain
2 Département de Mathématiques, Université de Paris Sud 91405 Orsay, cedex, France 
@article{10_1090_S0894_0347_02_00401_0,
     author = {Mateu, Joan and Tolsa, Xavier and Verdera, Joan},
     title = {The planar {Cantor} sets of zero analytic capacity and the local {{\dh}‘‡({\dh}‘)-Theorem}},
     journal = {Journal of the American Mathematical Society},
     pages = {19--28},
     publisher = {mathdoc},
     volume = {16},
     number = {1},
     year = {2003},
     doi = {10.1090/S0894-0347-02-00401-0},
     url = {http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-02-00401-0/}
}
TY  - JOUR
AU  - Mateu, Joan
AU  - Tolsa, Xavier
AU  - Verdera, Joan
TI  - The planar Cantor sets of zero analytic capacity and the local 𝑇(𝑏)-Theorem
JO  - Journal of the American Mathematical Society
PY  - 2003
SP  - 19
EP  - 28
VL  - 16
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-02-00401-0/
DO  - 10.1090/S0894-0347-02-00401-0
ID  - 10_1090_S0894_0347_02_00401_0
ER  - 
%0 Journal Article
%A Mateu, Joan
%A Tolsa, Xavier
%A Verdera, Joan
%T The planar Cantor sets of zero analytic capacity and the local 𝑇(𝑏)-Theorem
%J Journal of the American Mathematical Society
%D 2003
%P 19-28
%V 16
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-02-00401-0/
%R 10.1090/S0894-0347-02-00401-0
%F 10_1090_S0894_0347_02_00401_0
Mateu, Joan; Tolsa, Xavier; Verdera, Joan. The planar Cantor sets of zero analytic capacity and the local 𝑇(𝑏)-Theorem. Journal of the American Mathematical Society, Tome 16 (2003) no. 1, pp. 19-28. doi: 10.1090/S0894-0347-02-00401-0

[1] Christ, Michael Lectures on singular integral operators 1990

[2] Christ, Michael A 𝑇(𝑏) theorem with remarks on analytic capacity and the Cauchy integral Colloq. Math. 1990 601 628

[3] David, Guy Analytic capacity, Calderón-Zygmund operators, and rectifiability Publ. Mat. 1999 3 25

[4] Davie, Alexander M., ØKsendal, Bernt Analytic capacity and differentiability properties of finely harmonic functions Acta Math. 1982 127 152

[5] ÈĭDerman, V. Ya. Hausdorff measure and capacity associated with Cauchy potentials Mat. Zametki 1998 923 934

[6] Garnett, John Positive length but zero analytic capacity Proc. Amer. Math. Soc. 1970

[7] Garnett, John Analytic capacity and measure 1972

[8] Ivanov, L. D. Variatsii mnozhestv i funktsiÄ­ 1975 352

[9] Jones, Peter W. Square functions, Cauchy integrals, analytic capacity, and harmonic measure 1989 24 68

[10] Mattila, Pertti On the analytic capacity and curvature of some Cantor sets with non-𝜎-finite length Publ. Mat. 1996 195 204

[11] Murai, Takafumi Construction of 𝐻¹ functions concerning the estimate of analytic capacity Bull. London Math. Soc. 1987 154 160

[12] Tolsa, Xavier 𝐿²-boundedness of the Cauchy integral operator for continuous measures Duke Math. J. 1999 269 304

[13] Verdera, Joan A weak type inequality for Cauchy transforms of finite measures Publ. Mat. 1992

[14] Verdera, Joan Removability, capacity and approximation 1994 419 473

[15] Verdera, Joan 𝐿² boundedness of the Cauchy integral and Menger curvature 2001 139 158

Cité par Sources :