Multi-parameter flag Leibniz rules of arbitrary complexity in mixed-norm spaces

Abstract

We prove multi-parameter Leibniz rules corresponding to flag paraproducts of arbitrary complexity in mixed-norm spaces, including endpoint estimates. The proof relies on multi-linear harmonic analysis techniques and a quantitative treatment of the commutators introduced by Bourgain and Li. The argument is robust and applicable to a generic class of multipliers, including (symmetric) Mikhlin multipliers of positive order and asymmetric variants of partial differential operators and Mikhlin multipliers of positive order.