Beckner Inequalities For Moebius Measures On Spheres

Poincare,transportation and logarithmic Sobolev inequalities are essential tools in the study of concentration of measure and in the estimation of all relaxation time of various ergodic system.In this paper,we consider the Moebius measures ?xn-indexed by dimension n>3 and |x|<1 on the unit sphere Sn-1 on Rn(n>3).Taking the advantage of the fact that the density of ?xn depends only on one coordinate,we transfer the estimate on Cp(?xn)to that on the optimal Becknar constant of v|x|,n and then Therorem 11 in Barthe and Roberto helps us to finally offer a a two-sides estimate of precise order on the Beckner inequality constant with exponent p E[1,2).As special cases for p = 1 and p tending to 2,our estimates cover those in[5]for n>3.