# On Quantum Sobolev Inequalities

@inproceedings{Lafleche2022OnQS, title={On Quantum Sobolev Inequalities}, author={Laurent Lafleche}, year={2022} }

We investigate the quantum analogue of the classical Sobolev inequalities in the phase space. These inequalities can be seen as a many body uncertainty principle, and also lead to new bounds on the Schatten norms of the Weyl quantization in terms of its symbol. As an intermediate tool, we define a semiclassical analogue of the convolution together with the corresponding Young's and Hardy-Littlewood-Sobolev's inequalities, and introduce quantum Besov spaces. Explicit estimates are obtained on…

## References

SHOWING 1-10 OF 47 REFERENCES

### Optimal Semiclassical Regularity of Projection Operators and Strong Weyl Law

- Mathematics
- 2023

Projection operators arise naturally as one particle density operators associated to Slater determinants in fields such as quantum mechanics and the study of determinantal processes. In the context…

### Quantum harmonic analysis on phase space

- Physics
- 1984

Relative to an irreducible representation of the canonical commutation relations, convolutions between quantum mechanical operators and between functions and operators are defined, for which the…

### TRACE INEQUALITIES AND QUANTUM ENTROPY: An introductory course

- Physics
- 2009

We give an elementary introduction to the subject of trace inequalities and related topics in analysis, with a special focus on results that are relevant to quantum statistical mechanics. This is a…

### On the Bourgain, Brezis, and Mironescu Theorem Concerning Limiting Embeddings of Fractional Sobolev Spaces

- Mathematics
- 2003

The article is concerned with the Bourgain, Brezis and Mironescu theorem on the asymptotic behaviour of the norm of the Sobolev-type embedding operator: Ws,p ? Lpn/(n-sp) as s ? 1 and s ? n/p. Thei…

### On the Convergence of Time Splitting Methods for Quantum Dynamics in the Semiclassical Regime

- PhysicsFound. Comput. Math.
- 2021

The pseudo-metric introduced in Golse and Paul is used and it is proved that the convergence of time splitting algorithms for the von Neumann equation of quantum dynamics is uniform in the Planck constant.

### Semiclassical evolution with low regularity

- MathematicsJournal de Mathématiques Pures et Appliquées
- 2021

### Best constant in Sobolev inequality

- Mathematics, Materials Science
- 1976

SummaryThe best constant for the simplest Sobolev inequality is exhibited. The proof is accomplished by symmetrizations (rearrangements in the sense of Hardy-Littlewood) and one-dimensional calculus…

### Noncommutative Riesz transforms -- Dimension free bounds and Fourier multipliers

- Mathematics
- 2014

We obtain dimension free estimates for noncommutative Riesz transforms associated to conditionally negative length functions on group von Neumann algebras. This includes Poisson semigroups, beyond…

### Semiclassical Limit to the Vlasov Equation with Inverse Power Law Potentials

- MathematicsCommunications in Mathematical Physics
- 2019

We consider mixed quasi-free states describing N fermions in the mean-field limit. In this regime, the time evolution is governed by the nonlinear Hartree equation. In the large N limit, we study the…