Logarithmic Sobolev inequalities for Moebius measures on spheres

Abstract

Abstract In this paper, using the method in [1], i.e., reduce Moebius measures μ x n {\mu_{x}^{n}} indexed by | x | < 1 {|x|<1} on spheres S n - 1 {S^{n-1}} ( n ≥ 3 {n\geq 3} ) to one-dimensional diffusions on [ 0 , π ] {[0,\pi]} , we obtain that the optimal Poincaré constant is not greater than 2 n - 2 {\frac{2}{n-2}} and the optimal logarithmic Sobolev constant denoted by C LS ⁢ ( μ x n ) {C_{\rm LS}(\mu_{x}^{n})} behaves like 1 n ⁢ log ⁡ ( 1 + 1 1 - | x | ) {\frac{1}{n}\log(1+\frac{1}{1{-}|x|})} . As a consequence, we claim that logarithmic Sobolev inequalities are strictly stronger than L 2 {L^{2}} -transportation-information inequalities.