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

TL;DR

This paper establishes multi-parameter Leibniz rules for complex flag paraproducts in mixed-norm spaces, providing endpoint estimates and broad applicability to various multipliers using advanced harmonic analysis techniques.

Contribution

It introduces a comprehensive proof of Leibniz rules for arbitrary complexity flag paraproducts in mixed-norm spaces, extending previous results to endpoint cases and a wide class of multipliers.

Findings

Proved Leibniz rules for flag paraproducts of arbitrary complexity.

Established endpoint estimates in mixed-norm spaces.

Demonstrated robustness and applicability to various multipliers.

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.