We give a review of results on the initial value problem for the Vlasov--Poisson system, concentrating on the main ingredients in the proof of global existence of classical solutions.
We provide estimates for a transport equation which contains singular integral operators. The form of the equation was motivated by the study of Kirchhoff–Sobolev parametrices in a Lorentzian space-time satisfying the Einstein equations. While our main application is for a specific problem in General Relativity we believe that the phenomenon which our result illustrates is of a more general interest.