Sharp weighted estimates involving one supremum

Abstract

Abstract. In this note, we study the sharp weighted estimate involving one supremum. In particular, we give a positive answer to an open question raised by Lerner and Moen [13]. We also extend the result to rough homogeneous singular integral operators.

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

“…We observe that we recover again the linear dependence already available in the scalar case. We wonder whether it is possible to provide some estimate analogous to the one supremmum estimates obtained in [23] and [34].…”

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

“…where in the last step we use the Carleson embedding theorem [8,Theorem 4.5] and the sparsity of S. Now, we turn our attention to T * S,1,1 (b, f, gw). We observe that for any r > 1, and from this point it suffices to follow the proof of [19,Theorem 3.1] to obtain the following estimate:…”

“…Calculating the norm by duality we have that First, we work on T Sj ,1,1 (b, f, gw). Following ideas in [19] we have that…”

“…Very recently, a conjecture left open by Moen and Lerner in [14] was solved by Li in [19]. Actually, he obtained a more general result.…”

“…

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 A q − A exp ∞ constant is proved, providing a counterpart for commutators of the result obained in [19]. Both of the preceding results rely upon a sparse domination result in terms of bilinear forms which is established using techniques from [13].2010 Mathematics Subject Classification.

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