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Sharp bounds for general commutators on weighted Lebesgue spaces


Authors: Daewon Chung, M. Cristina Pereyra and Carlos Perez
Journal: Trans. Amer. Math. Soc. 364 (2012), 1163-1177
MSC (2010): Primary 42B20, 42B25; Secondary 46B70, 47B38
Published electronically: November 2, 2011
MathSciNet review: 2869172
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Abstract: We show that if a linear operator $ T$ is bounded on a weighted Lebesgue space $ L^2(w)$ and obeys a linear bound with respect to the $ A_2$ constant of the weight, then its commutator $ [b,T]$ with a function $ b$ in $ BMO$ will obey a quadratic bound with respect to the $ A_2$ constant of the weight. We also prove that the $ k$th-order commutator $ T^k_b=[b,T^{k-1}_b]$ will obey a bound that is a power $ (k+1)$ of the $ A_2$ constant of the weight. Sharp extrapolation provides corresponding $ L^p(w)$ estimates. In particular these estimates hold for $ T$ any Calderón-Zygmund singular integral operator. The results are sharp in terms of the growth of the operator norm with respect to the $ A_p$ constant of the weight for all $ 1<p<\infty $, all $ k$, and all dimensions, as examples involving the Riesz transforms, power functions and power weights show.


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Additional Information

Daewon Chung
Affiliation: Department of Mathematics and Statistics MSC01 1115, University of New Mexico, Albuquerque, New Mexico 87131-0001
Email: midiking@math.unm.edu

M. Cristina Pereyra
Affiliation: Department of Mathematics and Statistics, MSC01 1115, University of New Mexico, Albuquerque, New Mexico 87131-0001
Email: crisp@math.unm.edu

Carlos Perez
Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad De Sevilla, 41080 Sevilla, Spain
Email: carlosperez@us.es

DOI: https://doi.org/10.1090/S0002-9947-2011-05534-0
Keywords: Commutators, singular integrals, BMO, $A_{2}$, $A_{p}$
Received by editor(s): February 11, 2010
Published electronically: November 2, 2011
Additional Notes: The third author would like to acknowledge the support of the Spanish Ministry of Science and Innovation via grant MTM2009-08934.
Article copyright: © Copyright 2011 American Mathematical Society