# 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}
}```
• Published 23 December 2007
• Mathematics, Materials Science
• Annals of Mathematics
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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