A_1 theory of weights for rough homogeneous singular integrals and commutators

Abstract

Quantitative A 1 − A ∞ estimates for rough homogeneous singular integrals T Ω and commutators of BMO symbols and T Ω are obtained. In particular the following estimates are proved:

ImprovedA1−A∞and Related Estimates for Commutators of Rough Singular Integrals

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An A 1 − A ∞ estimate improving a previous result in [22] for [b, T Ω ] with Ω ∈ L ∞ (S n−1 ) and b ∈ BMO is obtained. Also a new result in terms of the A ∞ constant and the one supremum∞ constant is proved, providing a counterpart for commutators of the result obained in [19].

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“…An A 1 − A ∞ estimate improving a previous result in [22] for [b, T Ω ] with Ω ∈ L ∞ (S n−1 ) and b ∈ BMO is obtained. Also a new result in terms of the A ∞ constant and the one supremum…”

Weighted Norm Inequalities for Rough Singular Integral Operators

“…At this point we conjecture that (1.17) should hold with pp ′ instead of (pp ′ ) 2 . We can also derive an improvement of some results obtained in [47] concerning the A 1 constant.…”

“…In the last decade, plenty of works about weighted estimates have been devoted to the study of the quantitative dependence on the A p constant, on the A 1 constant, and also on mixed constants involving the A ∞ constant, of the weighted L p boundedness constant of several operators. Quite recently, some results in that direction for rough singular integrals have appeared in works such as [9,30,47]. Motivated by the latter (in particular by the most recent [9]), in this section we present several results showing improvements on the dependence on the A p and A 1 constant of T , where T is either a rough homogeneous singular integral or the Bochner-Riesz multiplier at the critical index.…”

“…It is well known that T Ω , with Ω ∈ L ∞ (S n−1 ), is bounded on L p (w), 1 < p < ∞, when w ∈ A p (first proved in the celebrated paper by J. Duoandikoetxea and J. L. Rubio de Francia [21], later improved in [19] and [53], and with quantitative version in [30]). In view of these works, the second author conjectured after [45] that (1.3) would hold for rough singular integral operators T Ω in the case Ω ∈ L ∞ (S n−1 ) (the conjecture is made explicit in [47]). Indeed, the method mentioned above could not be used since (1.2) was not available.…”