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.

arXiv comment27 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

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Cited by in corpus (6)

• Fourier multipliers and their applications to PDE on the quantum Euclidean space
• Sobolev inequality and its applications to nonlinear PDE on noncommutative Euclidean spaces
• Optimal Semiclassical Regularity of Projection Operators and Strong Weyl Law
• Energy preserving evolutions over Bosonic systems
• Heisenberg-smooth operators from the phase space perspective
• Enhanced Stability in Quantum Optimal Transport Pseudometrics: From Hartree to Vlasov-Poisson