Commutators of multi-parameter flag singular integrals and applications

TL;DR

This paper develops new bounds for iterated and big commutators of Riesz transforms in multi-parameter flag settings, using advanced harmonic analysis techniques, and applies these results to div-curl lemmas in Hardy spaces.

Contribution

It introduces the iterated and big commutators in the multi-parameter flag setting and establishes their boundedness in flag BMO and little BMO spaces, connecting these to Muckenhoupt weights.

Findings

Boundedness of iterated commutators with flag BMO symbols.

Boundedness of big commutator with flag little-BMO symbols.

Application to div-curl lemmas in multi-parameter Hardy spaces.

Abstract

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.