New results on Fourier multipliers on $L^p$: a perspective through unimodular symbols

TL;DR

This paper investigates unimodular Fourier multipliers with exponential growth on weighted L^p spaces, establishing new boundedness results and extending classical theorems to analyze operator behavior.

Contribution

It demonstrates that the general theory of multipliers can be understood through unimodular multipliers and introduces bounds for multiplier norms with exponential growth.

Findings

Boundedness of multipliers characterized by exponential bounds on unimodular multipliers.

New results on the boundedness of rough operators, singular operators along curves, and oscillatory integrals.

Extension of Stein's theorem on analytic families of operators to study derivative behavior as a parameter approaches zero.

Abstract

The paper focuses on the behaviour of unimodular Fourier multipliers with exponential growth in the context of weighted $L^p$-spaces. Our main result shows that much of the general theory of multipliers is approachable through the theory of unimodular multipliers. Indeed, we show that a bounded measurable function $m$ is a multiplier on $L^p$ for $1\leq p<\infty$ provided that $e^{itm}$ is a multiplier on $L^p$ and its multiplier norm admits an exponential bound of the form $e^{c|t|^s}$ for suitable $c>0$ and $0<s<1$. We then apply this principle to obtain new results related to the boundedness of homogeneous rough operators, singular operators along curves and oscillatory integrals. A key ingredient in our study is an extension of the classical Stein's theorem on analytic families of operators that studies the behaviour of the derivative operator when $\theta \to 0$.