Variation for singular integrals on Lipschitz graphs: $L^p$ and endpoint estimates
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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 $L^{1,\infty }(\mu )$.References
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Additional Information
- Albert Mas
- Affiliation: Departamento de Matemáticas, Universidad del País Vasco, 48080 Bilbao, Spain
- MR Author ID: 852137
- Email: amasblesa@gmail.com
- 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).
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3091264