Noncommutative $L_p$-Poincaré inequalities

Li Gao; Marius Junge; Zhendong Xu; Hao Zhang

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arXiv · 2610.25277

Noncommutative $L_p$-Poincaré inequalities

Li Gao·Marius Junge·Zhendong Xu·Hao Zhang

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Abstract

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 $p\ge2$, 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.

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BibTeX RIS

Li Gao, Marius Junge, Zhendong Xu, Hao Zhang. 2026-10-18. Noncommutative $L_p$-Poincaré inequalities. https://arxiv.org/abs/2610.25277

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