On Quantum Sobolev Inequalities
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.
Keywords
Cite
@article{arxiv.2210.03013,
title = {On Quantum Sobolev Inequalities},
author = {Laurent Lafleche},
journal= {arXiv preprint arXiv:2210.03013},
year = {2024}
}
Comments
27 pages. v2: added references, Morrey inequalities and comments on Riesz transforms. v3: typos corrected, added parallel and references to the skew information (Cor 2.2), comparison with commutators with complex exponentials (Prop 2.3) and an appendix on the quantum fractional Laplacian