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Riesz transforms and harmonic Lip1-capacity in Cantor sets

Published online by Cambridge University Press:  05 November 2004

Joan Mateu  and
Xavier Tolsa
Joan Mateu
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain. E-mail: mateu@mat.uab.es
Xavier Tolsa
Affiliation:
Institució Catalana de Recerca i Estudis Avançats (ICREA) and Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain. E-mail: xtolsa@mat.uab.es
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Abstract

We estimate the $L^2$-norm of the $s$-dimensional Riesz transforms on some Cantor sets in ${\mathbb R}^d$. Towards this end, we show that the Riesz transforms truncated at different scales behave in a quasiorthogonal way. As an application, we obtain some precise numerical estimates for the Lipschitz harmonic capacity of these sets.

Keywords

Riesz transformsanalytic capacity42B2042B25
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 89 , Issue 3 , November 2004 , pp. 676 - 696
Copyright
2004 London Mathematical Society

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Footnotes

The authors were partially supported by grants BFM2000-0361, MTM2004-00519, HPRN-2000-0116, and 2001-SGR-00431. X. Tolsa was also supported by the program Ramón y Cajal, MCYT (Spain).