Optimal Hypercontractivity and Log--Sobolev inequalities on Cyclic Groups $\mathbb{Z}_{m\cdot 2^k}$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917121688076288 |
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| author | Yao, Gan |
| author_facet | Yao, Gan |
| contents | For $1<p\le q<\infty$ and $n\in\{3\cdot 2^{k},2^{k}\}$ with $k\ge 1$, 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 $t\ge \tfrac{1}{2}\log\big(\tfrac{q-1}{p-1}\big)$. We establish sharp Log--Sobolev inequalities with the optimal constant $2$, by performing a KKT analysis, and lifting from the base cases $\mathbb{Z}_6$ and $\mathbb{Z}_4$ via a Cooley--Tukey $n\mapsto 2n$ comparison of Dirichlet forms. The general case for arbitrary $n$ remains open. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_03489 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal Hypercontractivity and Log--Sobolev inequalities on Cyclic Groups $\mathbb{Z}_{m\cdot 2^k}$ Yao, Gan Classical Analysis and ODEs Functional Analysis 47D03, 47D07, 42A16 For $1<p\le q<\infty$ and $n\in\{3\cdot 2^{k},2^{k}\}$ with $k\ge 1$, 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 $t\ge \tfrac{1}{2}\log\big(\tfrac{q-1}{p-1}\big)$. We establish sharp Log--Sobolev inequalities with the optimal constant $2$, by performing a KKT analysis, and lifting from the base cases $\mathbb{Z}_6$ and $\mathbb{Z}_4$ via a Cooley--Tukey $n\mapsto 2n$ comparison of Dirichlet forms. The general case for arbitrary $n$ remains open. |
| title | Optimal Hypercontractivity and Log--Sobolev inequalities on Cyclic Groups $\mathbb{Z}_{m\cdot 2^k}$ |
| topic | Classical Analysis and ODEs Functional Analysis 47D03, 47D07, 42A16 |
| url | https://arxiv.org/abs/2512.03489 |