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Variation for singular integrals on Lipschitz graphs: $ L^p$ and endpoint estimates

Author: Albert Mas
Journal: Trans. Amer. Math. Soc. 365 (2013), 5759-5781
MSC (2010): Primary 42B20, 42B25
Published electronically: June 6, 2013
MathSciNet review: 3091264
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Abstract: Let $ 1\leq n<d$ be integers and let $ \mu $ denote the $ n$-dimensional Hausdorff measure restricted to an $ n$-dimensional Lipschitz graph in $ \mathbb{R}^d$ with slope strictly less than $ 1$. For $ \rho >2$, we prove that the $ \rho $-variation and oscillation for Calderón-Zygmund singular integrals with odd kernel are bounded operators in $ L^{p}(\mu )$ for $ 1<p<\infty $, from $ L^1(\mu )$ to $ L^{1,\infty }(\mu )$, and from $ L^\infty (\mu )$ to $ BMO(\mu )$. Concerning the first endpoint estimate, we actually show that such operators are bounded from the space of finite complex Radon measures in $ \mathbb{R}^d$ to $ L^{1,\infty }(\mu )$.

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Additional Information

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

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.

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