Quadratic $A^1$ bounds for commutators of singular integrals with BMO functions

Abstract

For any Calderón-Zygmund operator T and any BM O function b we prove the following quadratic estimatewith constant c = c(n, T ) being the estimate optimal on p and the exponent of the weight constant. As an endpoint estimate we provewhich was used to derive vector-valued extensions of the classical estimates for M.1991 Mathematics Subject Classification. Primary 42B20, 42B25. Secondary 46B70, 47B38.

Borderline weighted estimates for commutators of singular integrals

“…We will see that this choice of symbols will be reflected in the maximal operator on the right hand side of the inequality. As a consequence of these type of estimates we can recover, among other results, the following endpoint A 1 result from C. Ortiz [18] (see Corollary 1):…”

“…Using the preceding lemma and following the proof of Lemma 3.1 in [18] we can derive the following improvement.…”

Sharp $$A_{1}$$ Weighted Estimates for Vector-Valued Operators

“…. In the case of commutators, for b ∈ BM O and T a Calderón-Zygmund operator, C. Ortiz-Caraballo [29] proved that…”

Improving Bounds for Singular Operators via Sharp Reverse Höolder Inequality for $$ A_\infty $$

“…As another main purpose in the present work, we will show how to extend this results on mixed A 1 -A ∞ bounds to the case of the commutator and its higher order analogue. The analogues of Theorem 1.2 and Theorem 1.3 for the commutators we consider were already proved by the first author in [18]: Theorem 1.6 (Quadratic A 1 bound for commutators). Let T be a Calderón-Zygmund operator and let b be in BM O.…”