On Fourier multipliers with rapidly oscillating symbols

TL;DR

This paper establishes precise bounds for the $L_p$ norms of Fourier multipliers with rapidly oscillating symbols, depending on a large parameter, advancing understanding of their behavior in harmonic analysis.

Contribution

It provides asymptotically sharp bounds for Fourier multipliers with oscillatory symbols, a novel result in the analysis of such operators.

Findings

Derived sharp bounds for $L_p$ norms of oscillatory Fourier multipliers

Analyzed dependence of bounds on the large parameter $\lambda$

Enhanced understanding of oscillatory behavior in Fourier analysis

Abstract

We provide asymptotically sharp bounds for the $L_p$ norms of the Fourier multipliers with the symbols $e^{i\lambda \varphi(\xi/|\xi|)}$, where $\lambda\in \mathbb{R}$ is a large parameter.