Mixed A_p-A_∞estimates with one supremum

Abstract

We establish several mixed A p -A ∞ bounds for Calderón-Zygmund operators that only involve one supremum. We address both cases when the A ∞ part of the constant is measured using the exponential-logarithmic definition and using the Fujii-Wilson definition. In particular, we answer a question of the first author and provide an answer, up to a logarithmic factor, of a conjecture of Hytönen and Lacey. Moreover, we give an example to show that our bounds with the logarithmic factors can be arbitrarily smaller than the previously known bounds (both one supremum and two suprema).2010 Mathematics Subject Classification. 42B20, 42B25.

Sharp weighted estimates involving one supremum

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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.

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Two weight inequalities for bilinear forms

“…Therefore, our result improves the bump theorem. In [19], Lerner and Moen also proved the following estimate, for any Calderón-Zygmund operator T , there holds…”

A new Ap–A∞ estimate for Calderón–Zygmund operators in spaces of homogeneous type

“…For any P ∈ P a , set S a (P ) := {Q ∈ S a : π(Q) = P }, where π(Q) is the minimal cube in P a which contains Q. We have the following Lemma, which was proved in [9] (see also in [13]) in the Euclidean setting and it is still valid for the spaces of homogeneous type with no change of the proof. Bp .…”