On two weight estimates for iterated commutators

Abstract

In this paper we extend the bump conjecture and a particular case of the separated bump conjecture with logarithmic bumps to iterated commutators T m b . Our results are new even for the first order commutator T 1 b . A new bump type necessary condition for the two-weighted boundedness of T m b is obtained as well. We also provide some results related to a converse to Bloom's theorem.

Bump conditions and two-weight inequalities for commutators of fractional integrals

“…This result was recently improved by Lerner et al [19], who provided a wider class of weights (µ, ν):…”

“…In [13], Cruz-Uribe et al only proved this conjecture is true for Φ(t) = t p ′ [log(e + t)] p ′ −1+δ and Ψ(t) = t p [log(e + t)] p−1+δ for some δ > 0. This conjecture is still open, and we refer readers to see [16,19,20] for more recent works about it. Analogue to the case of singular integral operators, Pérez [25] gave the following sufficient condition:…”

“…Very recently, Cruz-Uribe et al [9] generalized the work in [19] by assuming the Young functions Ā, C ∈ B p ′ , B, D ∈ B p and (µ, ν)…”

“…Inspired by the works in [3,9,19], in this paper, we mainly consider two weight inequalities for I b,m α . Our first main result can be formulated as follows.…”

New oscillation classes and two weight bump conditions for commutators

“…then inequality (1.5) holds. Recently, Lerner, Ombrosi, and Riviera-Ríos [28] have developed the first techniques for the iterated commutators T m b . They showed that if b ∈ BMO, Ā ∈ B p ′ and B ∈ B p , and…”