Sharp bounds for general commutators on weighted Lebesgue spaces

Abstract

Abstract. We show that if a linear 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 BM O will obey a quadratic bound with respect to the A 2 constant of the weight. We also prove that the kth-order] 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 < ∞, all k, and all dimensions, as examples involving the Riesz transforms, power functions and power weights show.

On a Two Weights Estimate for the Commutator

“…It has been shown that the Calderón-Zgymund singular integral operator with smooth kernel [b, T ]f := bT (f )−T (bf ) is a bounded operator on L p , 1 < p < ∞, when b is a BMO function. Weighted estimates for the commutator have been studied in [1], [4], [6], [7], and elsewhere. Also [6] and [7] provide the sharp version of one weighted estimates for the commutators.…”

Weighted Inequalities and Dyadic Harmonic Analysis

“…A refinement of the argument in [11] shows that Theorem 15. If a linear operator T obeys a power bound in L r (w) for all w ∈ A r ,…”

“…Extrapolation gives bounds on L p (w), they are sharp for all 1 < p < ∞, all k ≥ 1 and all dimensions, as examples involving the Riesz transforms show [11].…”

“…The following transference theorem holds, Theorem 14 (Chung, Pereyra, Pérez [11]). If a linear operator T obeys linear bounds in…”

“…We observe that the sharp bounds for the commutator of the Hilbert transform imply that Beznosova's bounds [4] are the sharp bounds for the dyadic paraproduct in L p (w), which was not known until now. The author in collaboration with Chung and Carlos Pérez established a transference theorem that states that if a linear operator T obeys a linear bound on L 2 (w) then its commutator with a BMO function obeys a quadratic bound [11]. In light of Hytönen's theorem this means that all commutators of Calderón-Zygmund singular operators with BMO functions obey a quadratic bound as in inequality (4).…”