Poincaré type inequalities for group measure spaces and related transportation cost inequalities

摘要

Let G be a countable discrete group with an orthogonal representation α on a real Hilbert space H. We prove L_p Poincaré inequalities for the group measure space L_∞(Ω_H,γ) G, where both the group action and the Gaussian measure space (Ω_H, γ) are associated with the representation α. The idea of proof comes from Pisier's method on the boundedness of Riesz transform and Lust-Piquard's work on spin systems. Then we deduce a transportation type inequality from the L_p Poincaré inequalities in the general noncommutative setting. This inequality is sharp up to a constant (in the Gaussian setting). Several applications are given, including Wiener/Rademacher chaos estimation and new examples of Rieffel's compact quantum metric spaces.