Weighted little bmo and two-weight inequalities for Journé commutators

Holmes, Irina; Petermichl, Stefanie; Wick, Brett D.

doi:10.2140/apde.2018.11.1693

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Weighted little bmo and two-weight inequalities for Journé commutators

Irina Holmes, Stefanie Petermichl and Brett D. Wick

Vol. 11 (2018), No. 7, 1693–1740

DOI: 10.2140/apde.2018.11.1693

Abstract

We characterize the boundedness of the commutators $[b, T]$ with biparameter Journé operators $T$ in the two-weight, Bloom-type setting, and express the norms of these commutators in terms of a weighted little bmo norm of the symbol $b$ . Specifically, if $μ$ and $λ$ are biparameter $A_{p}$ weights, $ν : = μ^{1 ∕ p} λ^{- 1 ∕ p}$ is the Bloom weight, and $b$ is in $bmo (ν)$ , then we prove a lower bound and testing condition $∥ b ∥_{bmo (ν)} ≲ sup ∥ [b, R_{k}^{1} R_{l}^{2}] : L^{p} (μ) \to L^{p} (λ) ∥$ , where $R_{k}^{1}$ and $R_{l}^{2}$ are Riesz transforms acting in each variable. Further, we prove that for such symbols $b$ and any biparameter Journé operators $T$ , the commutator $[b, T] : L^{p} (μ) \to L^{p} (λ)$ is bounded. Previous results in the Bloom setting do not include the biparameter case and are restricted to Calderón–Zygmund operators. Even in the unweighted, $p = 2$ case, the upper bound fills a gap that remained open in the multiparameter literature for iterated commutators with Journé operators. As a by-product we also obtain a much simplified proof for a one-weight bound for Journé operators originally due to R. Fefferman.

Keywords

commutators, Calderón–Zygmund operators, bounded mean oscillation, weights, Journé operators, little BMO, multiparameter harmonic analysis, singular integrals, weighted inequalities

Mathematical Subject Classification 2010

Primary: 42B20, 42B25, 42A50

Milestones

Received: 14 June 2017

Revised: 10 September 2017

Accepted: 24 October 2017

Published: 20 May 2018

Authors

Irina Holmes
Department of Mathematics Michigan State University East Lansing, MI United States

Stefanie Petermichl
Institut de Mathématiques de Toulouse Université Paul Sabatier Toulouse France

Brett D. Wick
Department of Mathematics Washington University - St. Louis St. Louis, MO United States

Vol. 11, No. 7, 2018