Sobolev inequalities, the Poisson semigroup, and analysis on the sphere Sn.
William Beckner · Proceedings of the National Academy of Sciences · 1992
Hypercontractive estimates are obtained for the Poisson semigroup on Lp(Sn). Such estimates are determined by a sharp logarithmic Sobolev inequality that relates the entropy of a function on Sn to its smoothness.