Estimates for the kinetic transport equation in hyperbolic Sobolev spaces

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Colleges, School and Institutes

External organisations

  • Saitama University
  • Seoul National University


We establish smoothing estimates in the framework of hyperbolic Sobolev spaces for the velocity averaging operator ρ of the solution of the kinetic transport equation. If the velocity domain is either the unit sphere or the unit ball, then, for any exponents q and r, we find a characterisation of the exponents β+ and β-, except possibly for an endpoint case, for which Dβ++ Dβ-ρ is bounded from space-velocity L2x,v to space-time LqtLrx. Here, D+ and D- are the classical and hyperbolic derivative operators, respectively. In fact, we shall provide an argument which unifies these velocity domains and the velocity averaging estimates in either case are shown to be equivalent to mixed-norm bounds on the cone multiplier operator acting on L2. We develop our ideas further in several ways, including estimates for initial data lying in certain Besov spaces, for which a key tool in the proof is the sharp ℓP decoupling theorem recently established by Bourgain and Demeter. We also show that the level of permissible smoothness increases significantly if we restrict attention to initial data which are radially symmetric in the spatial variable.


Original languageEnglish
Pages (from-to)1-28
Number of pages28
JournalJournal de Mathematiques Pures et Appliquees
Early online date30 Mar 2018
Publication statusPublished - Jun 2018


  • kinetic transport equation , averaging lemmas , hyperbolic Sobolev spaces , cone multiplier operator