Maximal gain of regularity in velocity averaging lemmas

Abstract

We investigate new settings of velocity averaging lemmas in kinetic theory where a maximal gain of half a derivative is obtained. Specifically, we show that if the densities $f$ and $g$ in the transport equation $v\cdot {\nabla }_{x}f=g$ belong to $L_x^r{L}_v^{r^{\prime }}$, where $2n∕\left(n+1\right)<r\le 2$ and $n\ge 1$ is the dimension, then the velocity averages belong to $H_x^{1∕2}$.

We further explore the setting where the densities belong to $L_x^{4∕3}{L}_v^2$ and show, by completing the work initiated by Pierre-Emmanuel Jabin and Luis Vega on the subject, that velocity averages almost belong to $W_x^{n∕\left(4\left(n-1\right)\right),4∕3}$ in this case, in any dimension $n\ge 2$, which strongly indicates that velocity averages should almost belong to $W_x^{1∕2,2n∕\left(n+1\right)}$ whenever the densities belong to $L_x^{2n∕\left(n+1\right)}{L}_v^2$.

These results and their proofs bear a strong resemblance to the famous and notoriously difficult problems of boundedness of Bochner–Riesz multipliers and Fourier restriction operators, and to smoothing conjectures for Schrödinger and wave equations, which suggests interesting links between kinetic theory, dispersive equations and harmonic analysis.