Sobolev inequalities, the Poisson semigroup, and analysis on the sphere Sn.
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.
Beckner
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Department of Mathematics, University of Texas, Austin, TX 78712, USA.