Sharp bounds for general commutators on weighted Lebesgue spaces

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.

arXiv commentTo appear in Trans. A.M.S

References in corpus (4)

• Sharp weighted estimates for classical operators
• Linear bound for the dyadic paraproduct on weighted Lebesgue space $L_2(w)$
• The sharp weighted bound for general Calderon-Zygmund operators
• Sharp norm inequalities for commutators of classical operators

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• Bloom weighted bounds for sparse forms associated to commutators
• $A_1$ theory of weights for rough homogeneous singular integrals and commutators
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• Variational inequalities for the commutators of rough operators with BMO functions
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• Sharp norm inequalities for commutators of classical operators
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• Weak Type Endpoint Estimates for the Commutators of Rough Singular Integral Operators
• Weighted vector-valued bounds for a class of multilinear singular integral operators
• Reverse Hölder Property for strong weights and general measures
• Boundedness of singular integral operators with variable kernels on weighted weak Hardy spaces
• Sparse dominations and weighted variation inequalities for singular integrals and commutators