Maximal gain of regularity in velocity averaging Lemmas

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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· ∇xf = g belong to Lx r Lv r', where 2n/(n+1) < r ≤ 2 and n ≥ 1 is the dimension, then the velocity averages belong to Hx 1/2. We further explore the setting where the densities belong to Lx 4/3 Lv 2 and show, by completing the work initiated by Pierre-Emmanuel Jabin and Luis Vega on the subject, that velocity averages almost belong to Wx n/(4(n-1),4/3 in this case, in any dimension n ≥ 2, which strongly indicates that velocity averages should almost belong to Wx 1/2,2n/(n+1) whenever the densities belong to Lx 2n/(n+1) Lv 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 languageEnglish (US)
Pages (from-to)333-388
Number of pages56
JournalAnalysis and PDE
Issue number2
StatePublished - 2019


  • Kinetic theory
  • Kinetic transport equation
  • Velocity averaging lemmas

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Applied Mathematics


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