Sharp $L^1$ estimates for singular transport equations

Sergiu Klainerman; Igor Rodnianski

doi:10.4171/jems/119

JournalsjemsVol. 10, No. 2pp. 477–505

Sharp $L^{1}$ estimates for singular transport equations

Sergiu Klainerman
Princeton University, USA
Igor Rodnianski
Massachusetts Institute of Technology, Cambridge, USA

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Abstract

We provide _L_1 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 verifying 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.

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Sergiu Klainerman, Igor Rodnianski, Sharp $L^{1}$ estimates for singular transport equations. J. Eur. Math. Soc. 10 (2008), no. 2, pp. 477–505

DOI 10.4171/JEMS/119