Fourier Multipliers and Littlewood-Paley For Modulation Spaces

TL;DR

This paper investigates Fourier multipliers and Littlewood-Paley theory within modulation spaces, establishing boundedness and extension properties of linear operators, thereby extending classical harmonic analysis results to this modern function space setting.

Contribution

It introduces new boundedness results for Fourier multipliers and Littlewood-Paley functions in modulation spaces, and proves an extension property for operators analogous to classical $L^p$ results.

Findings

Boundedness of Fourier multipliers in modulation spaces

Extension of linear operators to $l_2$-valued operators in modulation spaces

Analogue of Marcinkiewicz-Zygmund theorem for modulation spaces

Abstract

In this paper we have studied Fourier multipliers and Littlewood-Paley square functions in the context of modulation spaces. We have also proved that any bounded linear operator from modulation space $\mathcal{M}_{p,q}(\R^n), 1\leq p,q\leq \infty,$ into itself possesses an $l_2-$valued extension. This is an analogue of a well known result due to Marcinkiewicz and Zygmund on classical $L^p-$spaces.