### 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· ∇_{x}f = g belong to L_{x} ^{r} L_{v} ^{r'}, where 2n/(n+1) < r ≤ 2 and n ≥ 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/(4(n-1),4/3} in this case, in any dimension n ≥ 2, which strongly indicates that velocity averages should almost belong to W_{x} ^{1/2,2n/(n+1)} whenever the densities belong to L_{x} ^{2n/(n+1)} 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.

Original language | English (US) |
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Pages (from-to) | 333-388 |

Number of pages | 56 |

Journal | Analysis and PDE |

Volume | 12 |

Issue number | 2 |

DOIs | |

State | Published - 2019 |

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### Keywords

- Kinetic theory
- Kinetic transport equation
- Velocity averaging lemmas

### ASJC Scopus subject areas

- Analysis
- Numerical Analysis
- Applied Mathematics