zbMATH — the first resource for mathematics

The sharp weighted bound for the Riesz transforms. (English) Zbl 1142.42005
The present work is concerned with the sharp bound for the operator norm of the Riesz transforms in \(L^2(\omega)\). The sharp bound for other \(p\) are also derived by means of extrapolation. S. Buckley [Mich. Math. J. 40, No. 1, 153–170 (1993; Zbl 0794.42011)] proved an \(L^2\) bound in terms of the square of the classical \(A_2\) characteristic of the weight, while A. K. Lerner [J. Funct. Anal. 232, No. 2, 477–494 (2006; Zbl 1140.42005)] established an improvement to \(3/2\). In this paper, the author establishes the linear and best possible bound.

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
44A15 Special integral transforms (Legendre, Hilbert, etc.)
46E15 Banach spaces of continuous, differentiable or analytic functions
Full Text: DOI
[1] Kari Astala, Tadeusz Iwaniec, and Eero Saksman, Beltrami operators in the plane, Duke Math. J. 107 (2001), no. 1, 27 – 56. · Zbl 1009.30015
[2] Stephen M. Buckley, Summation conditions on weights, Michigan Math. J. 40 (1993), no. 1, 153 – 170. · Zbl 0794.42011
[3] R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241 – 250. · Zbl 0291.44007
[4] Oliver Dragičević, Loukas Grafakos, María Cristina Pereyra, and Stefanie Petermichl, Extrapolation and sharp norm estimates for classical operators on weighted Lebesgue spaces, Publ. Mat. 49 (2005), no. 1, 73 – 91. · Zbl 1081.42007
[5] R. A. Fefferman, C. E. Kenig, and J. Pipher, The theory of weights and the Dirichlet problem for elliptic equations, Ann. of Math. (2) 134 (1991), no. 1, 65 – 124. · Zbl 0770.35014
[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. · Zbl 0972.42011
[7] Richard Hunt, Benjamin Muckenhoupt, and Richard Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227 – 251. · Zbl 0262.44004
[8] A. LERNER, On some weighted norm inequalities for Littlewood-Paley operators, preprint 2006.
[9] Benjamin Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207 – 226. · Zbl 0236.26016
[10] F. Nazarov, S. Treil, and A. Volberg, The Bellman functions and two-weight inequalities for Haar multipliers, J. Amer. Math. Soc. 12 (1999), no. 4, 909 – 928. · Zbl 0951.42007
[11] Stefanie Petermichl, Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 6, 455 – 460 (English, with English and French summaries). · Zbl 0991.42003
[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. · Zbl 1139.44002
[13] S. Petermichl, S. Treil, and A. Volberg, Why the Riesz transforms are averages of the dyadic shifts?, Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial, 2000), 2002, pp. 209 – 228. · Zbl 1031.47021
[14] Stefanie Petermichl and Alexander Volberg, Heating of the Ahlfors-Beurling operator: weakly quasiregular maps on the plane are quasiregular, Duke Math. J. 112 (2002), no. 2, 281 – 305. · Zbl 1025.30018
[15] S. Petermichl and J. Wittwer, A sharp estimate for the weighted Hilbert transform via Bellman functions, Michigan Math. J. 50 (2002), no. 1, 71 – 87. · Zbl 1040.42008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.