Fourier-Wigner multipliers and the Bochner-Riesz conjecture for Schatten class operators
Helge Jørgen Samuelsen
TL;DR
The paper develops Fourier-Wigner multipliers as a quantum analogue of classical Fourier multipliers acting on Schatten classes ${\mathcal{S}}^p$, defining $\mathfrak{T}_m(T)=\mathcal{F}_W^{-1}(m\mathcal{F}_W(T))$ and establishing a tight link between classical multiplier spaces $\mathcal{M}_{p,q}$ and their Schatten counterparts $\mathfrak{M}_{p,q}$ for compactly supported $m$. It proves a fundamental equivalence (Theorem) between $\mathcal{M}_{p,q}$ and $\mathfrak{M}_{p,q}$ in the compactly supported case, enabling a reformulation of the Bochner-Riesz conjecture in phase space and extending Fefferman-type ball results to the quantum setting. The work builds on a comprehensive framework of time-frequency analysis, Weyl quantisation, and operator convolutions, and it also analyzes trace-class multipliers, showing inclusions $\mathcal{M}_1\subseteq\mathfrak{M}_1$ with open questions about complete characterisations. Overall, the paper extends classical harmonic analysis results to quantum harmonic analysis, linking phase-space multiplier theory with Schatten-class operator theory and informing quantum versions of restriction and Bochner-Riesz phenomena.
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.