Fourier-Wigner multipliers and the Bochner-Riesz conjecture for Schatten class operators

TL;DR

This paper introduces Fourier-Wigner multipliers for Schatten class operators, extending time-frequency analysis tools, and reformulates the Bochner-Riesz conjecture within this operator framework.

Contribution

It defines Fourier-Wigner multipliers for Schatten classes and connects them to the Bochner-Riesz conjecture, providing new analytical tools and reformulations.

Findings

Established properties of Fourier-Wigner multipliers via convolution and inequalities

Proved an equivalence for compactly supported Fourier multipliers

Reformulated the Bochner-Riesz conjecture in the context of Schatten class operators

Abstract

In this paper we introduce the notion of Fourier-Wigner multipliers for the Schatten class operators $\mathcal{S}^p$, which acts as an extension of classical localisation operators in time-frequency analysis. We establish results about Fourier-Wigner multipliers through convolution with rank-one operators and Werner-Young's inequality. We are also able to prove an equivalence relation for compactly supported Fourier multipliers. This allows us to reformulate the Bochner-Riesz conjecture in terms of Schatten class operators.