Fourier restriction for Schatten class operators and functions on phase space

TL;DR

This paper introduces a novel Fourier restriction framework for Schatten class operators using the Fourier-Wigner transform, linking it to symplectic Fourier transforms on phase space, and confirms a conjecture related to the Weyl transform of measures.

Contribution

It develops a new Fourier restriction theory for Schatten class operators via the Fourier-Wigner transform and establishes its equivalence to symplectic Fourier restriction, advancing quantum harmonic analysis.

Findings

Fourier-Wigner restriction is equivalent to symplectic Fourier restriction.

Established Schatten class results for quantum Fourier extension operator.

Confirmed conjecture on Weyl transform of measures.

Abstract

We formulate a variant of Fourier restriction for operators in Schatten classes, where the Fourier-Wigner transform of a bounded operator replaces the Fourier transform of a function. The Fourier-Wigner transform is closely related to the group Fourier transform of the Heisenberg group. The first result shows that Fourier-Wigner restriction for Schatten class operators is equivalent to the restriction of the symplectic Fourier transform of functions on phase space. We deduce various Schatten class results for the quantum Fourier extension operator and answer a conjecture by Mishra and Vemuri concerning the Weyl transform of measures in the affirmative.