Variation for singular integrals on Lipschitz graphs: and endpoint estimates

Author:
Albert Mas

Journal:
Trans. Amer. Math. Soc. **365** (2013), 5759-5781

MSC (2010):
Primary 42B20, 42B25

DOI:
https://doi.org/10.1090/S0002-9947-2013-05815-1

Published electronically:
June 6, 2013

MathSciNet review:
3091264

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Abstract: Let be integers and let denote the -dimensional Hausdorff measure restricted to an -dimensional Lipschitz graph in with slope strictly less than . For , we prove that the -variation and oscillation for Calderón-Zygmund singular integrals with odd kernel are bounded operators in for , from to , and from to . Concerning the first endpoint estimate, we actually show that such operators are bounded from the space of finite complex Radon measures in to .

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Additional Information

**Albert Mas**

Affiliation:
Departamento de Matemáticas, Universidad del País Vasco, 48080 Bilbao, Spain

Email:
amasblesa@gmail.com

DOI:
https://doi.org/10.1090/S0002-9947-2013-05815-1

Keywords:
$\rho$-variation and oscillation,
Calder\'on-Zygmund singular integrals.

Received by editor(s):
September 22, 2011

Published electronically:
June 6, 2013

Additional Notes:
The author was partially supported by grants AP2006-02416 (FPU program, Spain), MTM2010-16232 (Spain), and 2009SGR-000420 (Generalitat de Catalunya, Spain).

Article copyright:
© Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.