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Proceedings of the London Mathematical Society Advance Access originally published online on August 22, 2008
Proceedings of the London Mathematical Society 2009 98(2):393-426; doi:10.1112/plms/pdn035
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© 2008 London Mathematical Society

Uniform rectifiability, Calderón–Zygmund operators with odd kernel, and quasiorthogonality

Xavier Tolsa

Institució Catalana de Recerca i Estudis Avançats (ICREA)
Passeig Lluís Companys
23 08010 Barcelona
Catalonia
and
Departament de Matemàtiques
Universitat Autònoma de Barcelona
08193 Bellaterra, Barcelona
Catalonia, Spain

Received 25 January 2007. Revision received 7 May 2008.

In this paper we study some questions in connection with uniform rectifiability and the L2 boundedness of Calderón–Zygmund operators (CZOs). We show that uniform rectifiability can be characterized in terms of some new adimensional coefficients that are related to the Jones’ β numbers. We also use these new coefficients to prove that n-dimensional CZOs with odd kernel of type C2 are bounded in L2(µ), if µ is an n-dimensional uniformly rectifiable measure.

2000 Mathematics Subject Classification 28A75 (primary), 42B20 (secondary).

Partially supported by grants MTM2007-62817 and 2005-SGR-00744 (Generalitat de Catalunya).


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