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arXiv · 1002.2396

Sharp bounds for general commutators on weighted Lebesgue spaces

Daewon Chung·Cristina Pereyra·Carlos Perez
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Abstract

We show that if an operator T is bounded on 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 kth-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. 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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BibTeXRIS

Daewon Chung, Cristina Pereyra, Carlos Perez. 2010-02-11. Sharp bounds for general commutators on weighted Lebesgue spaces. https://doi.org/10.1090/s0002-9947-2011-05534-0

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