Sobolev inequalities, the Poisson semigroup, and analysis on the sphere Sn.

Beckner, W · Proc Natl Acad Sci U S A · 1992

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Abstract

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.