Moser-Trudinger and Beckner-Onofri's inequalities on the CR sphere

@article{Branson2007MoserTrudingerAB,
  title={Moser-Trudinger and Beckner-Onofri's inequalities on the CR sphere},
  author={Thomas P. Branson and Luigi Fontana and Carlo Morpurgo},
  journal={Annals of Mathematics},
  year={2007},
  volume={177},
  pages={1-52}
}
We derive sharp Moser-Trudinger inequalities on the CR sphere. The rst type is in the Adams form, for powers of the sublaplacian and for general spectrally dened operators on the space of CR-pluriharmonic functions. We will then obtain the sharp Beckner-Onofri inequality for CRpluriharmonic functions on the sphere and, as a consequence, a sharp logarithmic Hardy-Littlewood-Sobolev inequality in the form given by Carlen and Loss. 
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