A Complete Solution of Optimal Hypercontractivity and Log--Sobolev inequalities on Cyclic Groups $\mathbb{Z}_{n}$ for $n\geq 4$
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866911738167820288 |
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| author | Yao, Gan |
| author_facet | Yao, Gan |
| contents | For $1<p\le q<\infty$ and $n\geq 4$, we prove that the Poisson-like semigroup $(P_t)_{t\in \mathbb{R}_+}$ on $\mathbb{Z}_n$, associated with the word length $ψ_n(k)=\min(k,n-k)$, is hypercontractive from $L_p$ to $L_q$ if and only if $t\ge \tfrac{1}{2}\log\big(\tfrac{q-1}{p-1}\big)$. To this end, we establish the corresponding sharp Log--Sobolev inequalities with the optimal constant $2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_30867 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Complete Solution of Optimal Hypercontractivity and Log--Sobolev inequalities on Cyclic Groups $\mathbb{Z}_{n}$ for $n\geq 4$ Yao, Gan Classical Analysis and ODEs Functional Analysis For $1<p\le q<\infty$ and $n\geq 4$, we prove that the Poisson-like semigroup $(P_t)_{t\in \mathbb{R}_+}$ on $\mathbb{Z}_n$, associated with the word length $ψ_n(k)=\min(k,n-k)$, is hypercontractive from $L_p$ to $L_q$ if and only if $t\ge \tfrac{1}{2}\log\big(\tfrac{q-1}{p-1}\big)$. To this end, we establish the corresponding sharp Log--Sobolev inequalities with the optimal constant $2$. |
| title | A Complete Solution of Optimal Hypercontractivity and Log--Sobolev inequalities on Cyclic Groups $\mathbb{Z}_{n}$ for $n\geq 4$ |
| topic | Classical Analysis and ODEs Functional Analysis |
| url | https://arxiv.org/abs/2605.30867 |