Vol. 11, No. 7, 2018

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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
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 Ap weights, ν := μ1pλ1p is the Bloom weight, and b is in bmo(ν), then we prove a lower bound and testing condition bbmo(ν) sup[b,Rk1Rl2] : Lp(μ) Lp(λ), where Rk1 and Rl2 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] : Lp(μ) Lp(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