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

Abstract

Let $G$ be a countable discrete group with an orthogonal representation $\alpha$ on a real Hilbert space $H$. We prove $L_p$ Poincaré inequalities for the group measure space $L_{\infty }({\Omega }_H,\gamma )⋊G$, where both the group action and the Gaussian measure space $({\Omega }_H,\gamma )$ are associated with the representation $\alpha$. 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.

28 pages; revised

References in corpus (2)

• A Matrix Convexity Approach to Some Celebrated Quantum Inequalities
• Poincaré type inequalities on the discrete cube and in the CAR algebra

Cited by in corpus (4)

• Harmonic analysis approach to Gromov--Hausdorff convergence for noncommutative tori
• Ultraproduct methods for mixed $q$-Gaussian algebras
• Noncommutative maximal ergodic theorems for spherical means on the Heisenberg group
• On some topics of analysis on noncommutative spaces