On Quantum Sobolev Inequalities

TL;DR

This paper develops quantum analogues of Sobolev inequalities using Schatten norms, establishing uncertainty principles and bounds for quantum phase space operators, and introduces quantum Besov spaces with explicit constants.

Contribution

It introduces quantum Sobolev inequalities based on Schatten norms, linking them to uncertainty principles and defining quantum Besov spaces, with explicit constant estimates.

Findings

Quantum Sobolev inequalities relate Schatten norms of commutators to phase space properties.

New bounds on Schatten norms of Weyl quantization are established.

Explicit estimates for optimal constants in quantum inequalities are provided.

Abstract

We investigate the quantum analogue of the classical Sobolev inequalities in the phase space, with the quantum Sobolev norms defined in terms of Schatten norms of commutators. These inequalities provide an uncertainty principle for the Wigner-Yanase skew information, and also lead to new bounds on the Schatten norms of the Weyl quantization in terms of its symbol. As an intermediate tool, we obtain the analogue of Hardy-Littlewood-Sobolev's inequalities for a semiclassical analogue of the convolution, and introduce quantum Besov spaces. Explicit estimates are obtained on the optimal constants.