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 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 in the flag little bmo space by establishing the “exponential-logarithmic” bridge between this flag little bmo space and the Muckenhoupt 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.
"Commutators of multiparameter flag singular integrals and applications." Anal. PDE 12 (5) 1325 - 1355, 2019. https://doi.org/10.2140/apde.2019.12.1325