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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Weighted estimates for powers of the Ahlfors-Beurling operator

Author: Oliver Dragičević
Journal: Proc. Amer. Math. Soc. 139 (2011), 2113-2120
MSC (2010): Primary 42B20; Secondary 47A10
Published electronically: November 15, 2010
MathSciNet review: 2775389
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Abstract: We prove that for any $ n\in\mathbb{Z}\backslash\{0\}$, $ p>1$ and any weight $ w$ from the Muckenhoupt $ A_p$ class, the norm of the $ n$-th power of the Ahlfors-Beurling operator $ T$ on the weighted Lebesgue space $ L^p(w)$ is majorized by $ C(p) \vert n\vert^3 [w]_p^{\operatorname{max}\{1,1/(p-1)\}}$, where $ [w]_p$ is the $ A_p$ characteristic of $ w$. We apply this estimate for a result concerning the spectrum of $ T$ on $ L^p(w)$.

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Oliver Dragičević
Affiliation: Faculty of Mathematics and Physics and Institute of Mathematics, Physics and Mechanics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia

Received by editor(s): February 10, 2010
Received by editor(s) in revised form: June 5, 2010
Published electronically: November 15, 2010
Additional Notes: This work was partially supported by the Ministry of Higher Education, Science and Technology of Slovenia (research program Analysis and Geometry, contract no. P1-0291).
Communicated by: Franc Forstneric
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.