Velocity Averaging Lemmas: Classical, Quantum and Semi-Classical

Averaging lemmas were introduced as a tool of the mathematical analysis of kinetic equations, i.e. PDEs for functions in phase space $(x,v)$ containing a transport ("advection") term. By integrating over $v$ in velocity space $\mathbb{R}_v^d$ (velocity averaging), one gains regularity for the density in position space $\mathbb{R}_x^d$. The concept was invented independently by V.I. Agoshkov and by F. Golse, B. Perthame, R. Sentis and P.-L. Lions, and successfully applied to the analysis of Vlasov or Boltzmann equations in "classical kinetic theory". In "quantum kinetic theory", the Schr\"odinger equation for the complex-valued "wave function" in the physical space is converted into the Wigner equation for the real-valued Wigner function in phase space (which can take negative values). The Wigner ("Quantum Vlasov") equation contains the transport term of classical kinetic equations plus a pseudo-differential operator containing the potential. We give answers to the long standing question of whether and to which extent averaging lemmas apply to the "quantum" case of the Wigner equation. The hard part are the "semi-classical" averaging lemmas, where one considers the asymptotics of vanishing Planck constant towards the non-negative Wigner measure. In that context "pure vs. mixed states" play a crucial role, as well as the connection between the Schr\"odinger equation and Quantum Hydrodynamics (QHD). We present the results for the classical and quantum cases and sketch the "semi-classical" case which is worked out in full detail in a follow-up article.