Vol. 12, No. 5, 2019

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Commutators of multiparameter flag singular integrals and applications

Xuan Thinh Duong, Ji Li, Yumeng Ou, Jill Pipher and Brett D. Wick

Vol. 12 (2019), No. 5, 1325–1355
Abstract

We introduce the iterated commutator for the Riesz transforms in the multiparameter flag setting, and prove the upper bound of this commutator with respect to the symbol $b$ in the flag BMO space. Our methods require the techniques of semigroups, harmonic functions and multiparameter flag Littlewood–Paley analysis. We also introduce the big commutator in this multiparameter flag setting and prove the upper bound with symbol $b$ in the flag little bmo space by establishing the “exponential-logarithmic” bridge between this flag little bmo space and the Muckenhoupt ${A}_{p}$ weights with flag structure. As an application, we establish the div-curl lemmas with respect to the appropriate Hardy spaces in the multiparameter flag setting.

Keywords
multiparameter flag setting, flag commutator, Hardy space, BMO space, div-curl lemma
Mathematical Subject Classification 2010
Primary: 42B30, 42B20, 42B35