Kato-Ponce inequalities on weighted and variable Lebesgue spaces

TL;DR

This paper establishes fractional Leibniz rules and commutator estimates in weighted and variable Lebesgue spaces, utilizing uniform weighted estimates and bilinear extrapolation, with applications to bilinear multipliers and singular integrals.

Contribution

It introduces new fractional Leibniz rules and commutator estimates in weighted and variable Lebesgue spaces, expanding the analytical toolkit for these function spaces.

Findings

Proved fractional Leibniz rules in weighted and variable Lebesgue spaces.

Developed a bilinear extrapolation theorem for these spaces.

Applied results to boundedness of bilinear multipliers and singular integrals.

Abstract

We prove fractional Leibniz rules and related commutator estimates in the settings of weighted and variable Lebesgue spaces. Our main tools are uniform weighted estimates for sequences of square-function-type operators and a bilinear extrapolation theorem. We also give applications of the extrapolation theorem to the boundedness on variable Lebesgue spaces of certain bilinear multiplier operators and singular integrals.