Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-2010-10645-7
Published electronically: November 15, 2010
MathSciNet review: 2775389
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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)$.


References [Enhancements On Off] (What's this?)

  • 1. L. V. Ahlfors: Lectures on Quasiconformal Mappings, second edition, University Lecture Series 38, American Mathematical Society, 2006. MR 2241787 (2009d:30001)
  • 2. K. Astala: Area distortion of quasiconformal mappings, Acta Math. 173 (1994), 37-60. MR 1294669 (95m:30028b)
  • 3. K. Astala, T. Iwaniec, G. Martin: Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton Mathematical Series 48, Princeton University Press, 2009. MR 2472875
  • 4. K. Astala, T. Iwaniec, E. Saksman: Beltrami operators in the plane, Duke Math. J. 107 (2001), no. 1, 27-56. MR 1815249 (2001m:30021)
  • 5. R. Bañuelos, P. Janakiraman: $ L^p$-bounds for the Beurling-Ahlfors transform, Trans. Amer. Math. Soc. 360 (2008), 3603-3612. MR 2386238 (2009d:42032)
  • 6. O. Dragičević: Some remarks on the $ L^p$ estimates for powers of the Ahlfors-Beurling operator, submitted.
  • 7. O. Dragičević, L. Grafakos, M. C. Pereyra, S. Petermichl: Extrapolation and sharp norm estimates for classical operators on weighted Lebesgue spaces, Publ. Mat. 49 (2005), no. 1, 73-91. MR 2140200 (2006d:42019)
  • 8. O. Dragičević, S. Petermichl, A. Volberg: A rotation method which gives linear $ L^p$ estimates for powers of the Ahlfors-Beurling operator, J. Math. Pures Appl. 86 (2006), 492-509. MR 2281449 (2007k:30074)
  • 9. O. Dragičević, A. Volberg: Sharp estimate of the Ahlfors-Beurling operator via averaging martingale transforms, Michigan Math. J. 51 (2003), no. 2, 415-435. MR 1992955 (2004c:42030)
  • 10. T. Iwaniec: Extremal inequalities in Sobolev spaces and quasiconformal mappings, Z. Anal. Anwend. 1 (1982), 1-16. MR 719167 (85g:30027)
  • 11. T. Iwaniec, G. Martin: Riesz transforms and related singular integrals, J. Reine Angew. Math. 473 (1996), 25-57. MR 1390681 (97k:42033)
  • 12. S. Petermichl, A. Volberg: Heating of the Ahlfors-Beurling operator: weakly quasiregular maps on the plane are quasiregular, Duke Math. J. 112 (2002), no. 2, 281-305. MR 1894362 (2003d:42025)
  • 13. J. Wittwer: A sharp estimate on the norm of the martingale transform, Math. Res. Lett. 7 (2000), no. 1, 1-12. MR 1748283 (2001e:42022)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 42B20, 47A10

Retrieve articles in all journals with MSC (2010): 42B20, 47A10


Additional Information

Oliver Dragičević
Affiliation: Faculty of Mathematics and Physics and Institute of Mathematics, Physics and Mechanics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia
Email: oliver.dragicevic@fmf.uni-lj.si

DOI: https://doi.org/10.1090/S0002-9939-2010-10645-7
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.

American Mathematical Society