# Coifman-Meyer multipliers: Leibniz-type rules and applications to scattering of solutions to PDEs

## TL;DR

This paper develops Leibniz-type rules for Coifman-Meyer multipliers in various function spaces, improving existing estimates and applying these results to analyze scattering behavior in PDE solutions.

## Contribution

It introduces new Leibniz-type rules for Coifman-Meyer multipliers across diverse weighted and unweighted function spaces, enhancing previous theoretical bounds.

## Key findings

- Improved estimates for Leibniz-type rules in unweighted spaces.
- Extension of rules to weighted, Lorentz, Morrey, and variable-exponent spaces.
- Application of results to scattering analysis of PDE solutions involving fractional Laplacians.

## Abstract

Leibniz-type rules for Coifman-Meyer multiplier operators are studied in the settings of Triebel-Lizorkin and Besov spaces associated to weights in the Muckenhoupt classes. Even in the unweighted case, improvements on the currently known estimates are obtained. The flexibility of the methods of proofs allows to prove Leibniz-type rules in a variety of function spaces that include Triebel-Lizorkin and Besov spaces based on weighted Lebesgue, Lorentz and Morrey spaces as well as variable-exponent Lebesgue spaces. Applications to scattering properties of solutions to certain systems of partial differential equations involving fractional powers of the Laplacian are presented.