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Noncommutative Lp-Poincaré inequalities

  • Li Gao
  • Marius Junge
  • Zhendong Xu
  • Hao Zhang
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Abstract

We derive Lp-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≥2, the corresponding coefficients have optimal order O(p) and O(√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 Lp-Poincaré inequalities to Schatten-valued functions on graphs in order to obtain the Schatten distortion of expanders.

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Cite

BibTeXRIS
BibTeX
@misc{gao2026noncommutative,
  title = {{Noncommutative $L_p$-Poincaré inequalities}},
  author = {Li Gao and Marius Junge and Zhendong Xu and Hao Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25277},
  eprint = {2610.25277},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25277},
}
RIS
TY  - UNPB
TI  - Noncommutative L_p-Poincaré inequalities
AU  - Li Gao
AU  - Marius Junge
AU  - Zhendong Xu
AU  - Hao Zhang
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2610.25277
UR  - https://arxiv.org/abs/2610.25277
AB  - 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≥2, the corresponding coefficients have optimal order O(p) and O(√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.
ER  -