Wigner analysis of operators. Part III: Controlling ghost frequencies

TL;DR

This paper develops a new framework using Gaussian and Sobolev regularization to control ghost frequencies in Wigner analysis, improving the analysis of Fourier integral operators for PDEs.

Contribution

It introduces a novel method combining regularization techniques to mitigate ghost frequencies in Wigner kernels, especially for non-quadratic phase FIOs.

Findings

Established rigorous estimates for Wigner kernels with non-quadratic phases.

Demonstrated improved bounds for Schrödinger equations with complex symbol classes.

Mitigated interference from ghost frequencies in time-frequency analysis.

Abstract

The integration of operator kernels with the Wigner distribution, first conceptualized by E. Wigner in 1932 and later extended by L. Cohen and others, has opened new avenues in time-frequency analysis and operator calculus. Despite substantial advancements, the presence of ``ghost frequencies" in Wigner kernels continues to pose significant challenges, particularly in the analysis of Fourier integral operators (FIOs) and their applications to partial differential equations (PDEs). In this work, we build on the foundational concepts of Wigner analysis to introduce a novel framework for controlling ghost frequencies through the combined use of Gaussian and Sobolev regularization techniques. By focusing on FIOs with non-quadratic phase functions, we develop rigorous estimates for the Wigner kernels that are crucial for their applicability to Schr\"odinger equations with non-trivial symbol classes. Unlike previous approaches, our methodology not only mitigates the interference caused by ghost frequencies but also establishes robust bounds in the context of generalized symplectic mappings.