Commutators of multi-parameter flag singular integrals and applications

We introduce the iterated commutator for the Riesz transforms in the multi-parameter 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 multi-parameter flag Littlewood-Paley analysis. We also introduce the big commutator in this multi-parameter 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 multi-parameter flag setting.