On the unbounded of a class of Fourier integral operator on $L^2(R^n)$

TL;DR

This paper presents an example of a Fourier integral operator with a specific symbol class that cannot be extended as a bounded operator on L^2(R^n), highlighting limitations in boundedness properties.

Contribution

The paper provides a counterexample of a Fourier integral operator with a symbol in the intersection of certain symbol classes that is not bounded on L^2(R^n).

Findings

Counterexample of unbounded Fourier integral operator on L^2(R^n)

Symbol belongs to intersection of S_{ρ,1}^0 classes for 0<ρ<1

Highlights limitations of boundedness in Fourier integral operators.

Abstract

In this paper, we give an example of Fourier integral operator with a symbol belongs to $\bigcap\limits_{0<\rho <1}S_{\rho,1}^{0}$ that cannot be extended as a bounded operator on $L^{2}(\mathbb{R}^{n}).$