We derive sharp Moser-Trudinger inequalities on the CR sphere. The first type
is in the Adams form, for powers of the sublaplacian and for general spectrally
defined operators on the space of CR-pluriharmonic functions. We will then
obtain the sharp Beckner-Onofri inequality for CR-pluriharmonic functions on
the sphere, and, as a consequence, a sharp logarithmic ... [Show full abstract] Hardy-Littlewood-Sobolev
inequality in the form given by Carlen and Loss.