New oscillation classes and two weight bump conditions for commutators

2021/03/11 by David Cruz-Uribe, Kabe Moen, Cruz-Uribe, David +3
Mathematics · #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2103.06821

openalex publication_date 2021/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we consider two weight bump conditions for higher order commutators. Given b and a Calderón-Zygmund operator T, define the commutator T¹_bf=[T,b]f= bTf-T(bf), and for m≥ 2 define the iterated commutator T^m_b f = [b,T_b^m-1]f. Traditionally, commutators are defined for functions b∈ BMO, but we show that if we replace BMO by an oscillation class first introduced by Pérez [31], we can give a range of sufficient conditions on a pair of weights (u,v) for T^m_b : L^p(v)→ L^p(u) to be bounded. Our results generalize work of the first two authors in [10], and more recent work by Lerner, et al. [28]. We also prove necessary conditions for the iterated commutators to be bounded, generalizing results of Isralowitz, et al.[20].

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