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

Let G be a countable discrete group with an orthogonal representation α on a real Hilbert space H  . We prove LpLp Poincaré inequalities for the group measure space L∞(ΩH,γ)⋊GL∞(ΩH,γ)⋊G, where both the group action and the Gaussian measure space (ΩH,γ)(ΩH,γ) are associated with the representation α  . The idea of proof comes from Pisierʼs method on the boundedness of Riesz transform and Lust-Piquardʼs work on spin systems. Then we deduce a transportation type inequality from the LpLp 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 Rieffelʼs compact quantum metric spaces.