Polynomially oscillatory multipliers on Gelfand-Shilov spaces

TL;DR

This paper investigates the conditions under which polynomially oscillatory multipliers act continuously on Gelfand-Shilov spaces, revealing a dependence on polynomial degree and space parameters, with implications for linear evolution equations.

Contribution

It identifies the parameter regions where the multiplier operator is continuous on Gelfand-Shilov spaces and demonstrates non-continuity in certain cases, extending understanding of well-posedness for related PDEs.

Findings

Operator is continuous in a specific wedge depending on polynomial degree.

Operator is not continuous in a large part of the complement region in dimension one.

Results inform well-posedness of generalized Schrödinger equations.

Abstract

We study continuity of the multiplier operator $e^{i q}$ acting on Gelfand--Shilov spaces, where $q$ is a polynomial on $\mathbf R^d$ of degree at least two with real coefficients. In the parameter quadrant for the spaces we identify a wedge that depends on the polynomial degree for which the operator is continuous. We also show that in a large part of the complement region the operator is not continuous in dimension one. The results give information on well-posedness for linear evolution equations that generalize the Schr\"odinger equation for the free particle.