Multilinear Fourier multipliers on variable Lebesgue spaces

TL;DR

This paper investigates multilinear Fourier multipliers on variable Lebesgue spaces, establishing necessary conditions, localization properties, and a Mihlin-H"ormander type theorem with applications.

Contribution

It introduces new necessary conditions and a localization theorem for multipliers, and extends the Mihlin-H"ormander theorem to weighted variable Lebesgue spaces.

Findings

Necessary condition for bilinear multipliers identified

Localization theorem proved for variable Lebesgue spaces

Mihlin-H"ormander type theorem established with applications

Abstract

In this paper, we study properties of the bilinear multiplier space. We give a necessary condition for a continuous integrable function to be a bilinear multiplier on variable exponent Lebesgue spaces. And we prove the localization theorem of multipliers on variable exponent Lebesgue spaces. Moreover, we present a Mihlin-H\"ormander type theorem for multilinear Fourier multipliers on weighted variable Lebesgue spaces and give some applications.