Noncommutative $L_p$-Poincaré inequalities

We derive $L_p$-Poincaré inequalities for quantum Markov semigroups on finite von Neumann algebras, from a spectral gap (PI) or from a modified logarithmic Sobolev inequality (MLSI). For $p\ge 2$, the corresponding coefficients have optimal order $O(p)$ and $O(\sqrt{p})$, respectively. Both estimates follow from Beckner inequalities by integration along semigroups. We also establish a non-tracial extension to quantum Markov semigroups satisfying GNS detailed balance. As a direct byproduct, we apply $L_p$-Poincaré inequalities to Schatten-valued functions on graphs in order to obtain the Schatten distortion of expanders.