AI 中文总结
该研究在单位球面上建立分形雷梅兹不等式，基于此得到球面上热方程的精确可观测性不等式并改进已有结果，还证明了带超二次势的热方程的低维可观测性不等式。
AI 中文摘要
本文研究雷梅兹型不等式及其在可观测性不等式中的应用，目标有两个：第一，在单位球面$\boldsymbol{\reals}^{n-1}$上建立如下分形雷梅兹不等式：$\boldsymbol{\reals}^{n-1}$上的球多项式$p$（次数至多为正整数$N$）的上确界，不超过常数$C(M,N,n,\boldsymbol{\reals})$乘以$p$在分形集$M$上的上确界，其中$M$是$\boldsymbol{\reals}^{n-1}$（$n\boldsymbol{\reals}$）中具有正$(n-2+\boldsymbol{\reals})$豪斯多夫内容的分形集，$\boldsymbol{\reals}\boldsymbol{\reals}$；第二，基于该分形框架，在球面上建立对所有$\boldsymbol{\reals}\boldsymbol{\reals}$都成立的热方程的精确可观测性不等式，在球面情形下改进了Burq与Moyano[《欧洲数学会杂志》（JEMS），25卷4期（2023）]的结果；此外，作为额外应用，还证明了整个空间$\boldsymbol{\reals}^n$上带有超二次势$V(x)=|x|^{2m}$（正整数$m\boldsymbol{\reals}$）的热方程的低维可观测性不等式。
英文摘要
This paper is concerned with Remez-type inequalities and their applications in observability inequality. Our aim is twofold. First, we establish the following fractal Remez's inequality on the unit sphere $\mathbb{S}^{n-1}$ \begin{align*} \sup_{\mathbb{S}^{n-1}} |p|\le C(M,N,n,δ)\sup_{M} |p|, \end{align*} where $M \subset \mathbb{S}^{n-1}$ ($n \ge 2$) is a fractal set of positive $(n-2+δ)$-Hausdorff content for arbitrary $δ\in (0,1)$, and $p$ is a spherical polynomial of degree at most $N\in \mathbb{Z}^+$. Second, building upon this fractal framework, we establish sharp observability inequalities for the heat equation on the sphere, again valid for all $δ\in (0, 1)$, which improve the result of Burq and Moyano [J. Eur. Math. Soc. (JEMS), 25 (4) (2023)] in the spherical setting. Furthermore, as an additional application, we prove a lower-dimensional observability inequality for the heat equation with super-quadratic potentials $V(x) = |x|^{2m}$ ($m \in \mathbb{Z}^+, m\ge 2$) on the whole space $\mathbb{R}^n$.
Comments26 pages
