有限环上对数Sobolev不等式的尖锐一致稳定性及三次Sobolev不等式的稳定性

中文摘要 AI 辅助

我们确定了有限环上对数Sobolev不等式的四次稳定性估计中的尖锐一致常数。我们赋予$C_{n}=\mathbb{Z}/n\mathbb{Z}$均匀概率测度，并令$\mathcal{E}_{n}$为$C_{n}$上简单随机游走的Dirichlet形式，归一化使其谱间隙为1。Frank和Ivanisvili最近证明了尖锐对数Sobolev不等式$2\mathcal{E}_{n}(u)\geq \mathrm{Ent}(u^{2})$对所有$n\geq 4$成立。我们证明$2\mathcal{E}_{n}(u)-\mathrm{Ent}(u^{2})\geq \frac{1}{3}\Vert u-1\Vert _{2}^{4}$对所有$n\geq 4$和所有满足$\Vert u\Vert _{2}=1$的非负$u$成立。这是Xie和Zhang的四次稳定性估计的尖锐形式，在$n$上一致成立，他们得到了常数$\frac{1}{12}$。同样的估计在圆上和$C_{n}$上的词长能量上也成立。在这些模型中，四环是唯一极端的：在$C_{4}$上常数$\frac{1}{3}$是尖锐的但未达到，而在$C_{n}$（$n\geq 5$）和圆上尖锐常数至少为$\frac{1}{3}+c$，其中$c>0$不依赖于$n$。我们还给出了这些尖锐常数的显式上界。我们的第二个主要结果是Frank和Ivanisvili的三次Sobolev不等式的稳定性估计，这是他们证明中的主要步骤。它给出了Frank和Ivanisvili所期望的该不等式的定量版本，具有$\Vert u-1\Vert _{2}$的最优幂次：在$C_{n}$（$n\geq 5$）和圆上为四次，在$C_{4}$上为六次。证明的两个关键新要素是：一个关于$u$的第一傅里叶系数以其均值为界的估计，该估计因$u$非负而成立；以及Gross两点对数Sobolev不等式的尖锐稳定性估计，在距离的四次幂和六次幂中都具有尖锐常数。

英文摘要

We determine the sharp uniform constant in the quartic stability estimate for the logarithmic Sobolev inequality on finite cycles. We equip $C_{n}=\mathbb{Z}/n\mathbb{Z}$ with the uniform probability measure, and we let $\mathcal{E}_{n}$ be the Dirichlet form of the simple random walk on $C_{n}$, normalized so that its spectral gap is one. Frank and Ivanisvili recently proved the sharp logarithmic Sobolev inequality $2\mathcal{E}_{n}(u)\geq \mathrm{Ent}(u^{2})$ for all $n\geq 4$. We prove that $2\mathcal{E}_{n}(u)-\mathrm{Ent}(u^{2})\geq \frac{1}{3}\Vert u-1\Vert _{2}^{4}$ for all $n\geq 4$ and all nonnegative $u$ with $\Vert u\Vert _{2}=1$. This is the sharp form, uniformly in $n$, of the quartic stability estimate of Xie and Zhang, who obtained the constant $\frac{1}{12}$. The same estimate holds on the circle and for the word-length energy on $C_{n}$. Among these models, the four-cycle is the only extremal one: on $C_{4}$ the constant $\frac{1}{3}$ is sharp but not attained, while on $C_{n}$ with $n\geq 5$ and on the circle the sharp constant is at least $\frac{1}{3}+c$, where $c>0$ does not depend on $n$. We also give explicit upper bounds for these sharp constants. Our second main result is a stability estimate for the cubic Sobolev inequality of Frank and Ivanisvili, which is the main step in their proof. It gives the quantitative version of this inequality expected by Frank and Ivanisvili, with the optimal powers of $\Vert u-1\Vert _{2}$: four on $C_{n}$ with $n\geq 5$ and on the circle, and six on $C_{4}$. Two key new ingredients of the proofs are a bound for the first Fourier coefficient of $u$ in terms of its mean, which holds because $u$ is nonnegative, and a sharp stability estimate for Gross's two-point logarithmic Sobolev inequality, with sharp constants in both the fourth and the sixth powers of the distance.

发表机构

• Key Laboratory of Algebraic Lie Theory and Analysis of Ministry of Education, School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院教育部代数Lie理论与分析重点实验室）
• School of Science and the Environment, Grenfell Campus, Memorial University of Newfoundland（纪念大学纽芬兰分校格伦费尔校区科学与环境学院）
• Department of Mathematics, University of Connecticut（康涅狄格大学数学系）

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