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The sharp weighted bound for the Riesz transforms


Author: Stefanie Petermichl
Journal: Proc. Amer. Math. Soc. 136 (2008), 1237-1249
MSC (2000): Primary 42-XX
DOI: https://doi.org/10.1090/S0002-9939-07-08934-4
Published electronically: December 7, 2007
MathSciNet review: 2367098
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Abstract: We establish the best possible bound on the norm of the Riesz transforms as operators in the weighted space $ L^p_{\mathbb{R}^n}(\omega)$ for $ 1 < p<\infty$ in terms of the classical $ A_p$ characteristic of the weight.


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  • 1. K. ASTALA, T. IWANIEC, E. SAKSMAN, Beltrami operators, Duke Math. J. 107 (2001), no. 1, pp. 27-56. MR 1815249 (2001m:30021)
  • 2. S. BUCKLEY, Summation condition on weights, Mich. Math. J. 40(1) (1993), pp. 153-170. MR 1214060 (94d:42021)
  • 3. R. R. COIFMAN, C. FEFFERMAN, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), pp. 241-250. MR 0358205 (50:10670)
  • 4. O. DRAGICEVIĆ, 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, pp. 73-91. MR 2140200 (2006d:42019)
  • 5. R. FEFFERMAN, C. KENIG, J. PIPHER, The theory of weights and the Dirichlet problem for elliptic equations, Annals of Math. 134 (1991), pp. 65-124. MR 1114608 (93h:31010)
  • 6. S. Hucovic, S. Treil, A. Volberg, The Bellman functions and the sharp square estimates for square functions, Operator Theory: Advances and Applications, the volume in memory of S. A. Vinogradov, v. 113, Birkhauser Verlag, 2000.
  • 7. R. A. HUNT, B. MUCKENHOUPT, R. L. WHEEDEN, Weighted norm inequalities for the conjugate function and the Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), pp. 227-251. MR 0312139 (47:701)
  • 8. A. LERNER, On some weighted norm inequalities for Littlewood-Paley operators, preprint 2006.
  • 9. B. MUCKENHOUPT, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), pp. 207-226. MR 0293384 (45:2461)
  • 10. F. NAZAROV, S. TREIL, A. VOLBERG, The Bellman functions and two weight inequalities for Haar multipliers, J. of Amer. Math. Soc. 12 (1999), no. 4, pp. 909-928. MR 1685781 (2000k:42009)
  • 11. S. PETERMICHL, Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol, Comptes Rendus Acad. Sci. Paris, t. 330 (2000), no. 1, pp. 455-460. MR 1756958 (2000m:42016)
  • 12. S. PETERMICHL, The sharp bound for the Hilbert transform in weighted Lebesgue spaces in terms of the classical $ A_p$ characteristic, to appear in Amer. J. Math.
  • 13. S. PETERMICHL, S. TREIL, A. VOLBERG, Why are the Riesz transforms averages of the dyadic shift?, Proceedings of the 6th international conference on harmonic analysis (El Escorial), Publ. Mat. (2002), Extra Vol., pp. 209-228. MR 1964822 (2003m:42028)
  • 14. S. PETERMICHL, A. VOLBERG, Heating of the Beurling operator: weakly quasiregular maps on the plane are quasiregular, Duke Math. J. 112, No. 2 (2002), pp. 281-305. MR 1894362 (2003d:42025)
  • 15. S. PETERMICHL, J. WITTWER, A sharp weighted estimate on the norm of Hilbert transform via invariant $ A_2$ characteristic of the weight, Mich. Math. J. 50 (2002), pp. 71-87. MR 1897034 (2003e:42016)

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Additional Information

Stefanie Petermichl
Affiliation: Institut de Mathématiques de Bordeaux, 351, cours de la Libération, F-33405 Talence Cedex, France
Email: Stefanie.Petermichl@math.u-bordeaux1.fr

DOI: https://doi.org/10.1090/S0002-9939-07-08934-4
Received by editor(s): September 19, 2006
Published electronically: December 7, 2007
Additional Notes: The author was supported by NSF grant #DMS 9729992
Communicated by: Michael Lacey
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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