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Sharp Logarithmic Sobolev Inequalities for q-Araki-Woods Ornstein-Uhlenbeck Semigroups

  • Jinsong Wu
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Abstract

In this paper, we establish the sharp $2$-logarithmic Sobolev inequality for q-Araki-Woods Ornstein-Uhlenbeck semigroups with −1<q<1 and obtain hypercontractivity along the Gross curve with the optimal coefficient, and determine the exact hypercontractive time for 1<p≤2≤ r<∞. Reverse hypercontractivity holds in the two separate ranges 0<p<r<1 and −∞<p<r<0. An explicit counterexample rules out the proposed universal reverse curve across zero. We also obtain the complete modified logarithmic Sobolev inequality via the intertwining property.

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Cite

BibTeXRIS
BibTeX
@misc{wu2026sharp,
  title = {{Sharp Logarithmic Sobolev Inequalities for $q$-Araki-Woods Ornstein-Uhlenbeck Semigroups}},
  author = {Jinsong Wu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.33686},
  eprint = {2610.33686},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.33686},
}
RIS
TY  - UNPB
TI  - Sharp Logarithmic Sobolev Inequalities for q-Araki-Woods Ornstein-Uhlenbeck Semigroups
AU  - Jinsong Wu
PY  - 2026
DA  - 2026/10/24
PB  - arXiv
DO  - 10.48550/arxiv.2610.33686
UR  - https://arxiv.org/abs/2610.33686
AB  - In this paper, we establish the sharp $2$-logarithmic Sobolev inequality for q-Araki-Woods Ornstein-Uhlenbeck semigroups with −1<q<1 and obtain hypercontractivity along the Gross curve with the optimal coefficient, and determine the exact hypercontractive time for 1<p≤2≤ r<∞. Reverse hypercontractivity holds in the two separate ranges 0<p<r<1 and −∞<p<r<0. An explicit counterexample rules out the proposed universal reverse curve across zero. We also obtain the complete modified logarithmic Sobolev inequality via the intertwining property.
ER  -