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Loomis-Whitney inequalities on Reiter-Heisenberg groups

  • Sheng-Chen Mao
  • Ye Zhang
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Abstract

We establish a Loomis-Whitney inequality for the Reiter-Heisenberg groups ๐”พqp, a family of step-two Carnot groups that includes the Heisenberg groups when q=1. The proof is based on the duality between Brascamp-Lieb inequalities and entropy subadditivity: we first derive the result for ๐”พq1 from the known inequality on the first Heisenberg group, using conditional entropy and the invariance of differential entropy under volume-preserving diffeomorphisms; then we pass from ๐”พq1 to ๐”พqp via a stability principle for Loomis-Whitney inequalities under finite central sums, which generalizes the argument in (Zhang, 2024 arXiv:2402.02749v2). As consequences, we obtain the associated geometric projection inequality, a Gagliardo-Nirenberg-Sobolev inequality, and an isoperimetric inequality.

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Cite

BibTeXRIS
BibTeX
@misc{mao2026loomiswhitney,
  title = {{Loomis-Whitney inequalities on Reiter-Heisenberg groups}},
  author = {Sheng-Chen Mao and Ye Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.10091},
  eprint = {2609.10091},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.10091},
}
RIS
TY  - UNPB
TI  - Loomis-Whitney inequalities on Reiter-Heisenberg groups
AU  - Sheng-Chen Mao
AU  - Ye Zhang
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.10091
UR  - https://arxiv.org/abs/2609.10091
AB  - We establish a Loomis-Whitney inequality for the Reiter-Heisenberg groups ๐”พ_qp, a family of step-two Carnot groups that includes the Heisenberg groups when q=1. The proof is based on the duality between Brascamp-Lieb inequalities and entropy subadditivity: we first derive the result for ๐”พ_q1 from the known inequality on the first Heisenberg group, using conditional entropy and the invariance of differential entropy under volume-preserving diffeomorphisms; then we pass from ๐”พ_q1 to ๐”พ_qp via a stability principle for Loomis-Whitney inequalities under finite central sums, which generalizes the argument in (Zhang, 2024 arXiv:2402.02749v2). As consequences, we obtain the associated geometric projection inequality, a Gagliardo-Nirenberg-Sobolev inequality, and an isoperimetric inequality.
ER  -