平均截算子的不动点
Fixed Points of Mean Section Operators
摘要：
利用旋转等变有界线性算子的核函数的球拉普拉斯算子在小极冠上的质量分布刻画了从$C(\mathbb{S}^{n-1})$到$C^2(\mathbb{S}^{n-1})$的旋转等变有界线性算子.利用这个特征，我们表明，每一个连续的，齐次的，平移不变的，旋转等变的Minkowski值$Φ$是弱单调的映射空间的凸体与$C^2$支撑函数到自己。作为应用，证明了若$Φ$是加法偶算子或平均截算子，则欧氏球是它在单位球的某个$C^2$邻域中的唯一不动点.我们的方法统一并扩展了Ivaki从2017年开始的先前结果以及第二作者与Schuster从2021年开始的先前结果。
英文摘要：
We characterize rotation equivariant bounded linear operators from $C(\mathbb{S}^{n-1})$ to $C^2(\mathbb{S}^{n-1})$ by the mass distribution of the spherical Laplacian of their kernel function on small polar caps. Using this characterization, we show that every continuous, homogeneous, translation invariant, and rotation equivariant Minkowski valuation $Φ$ that is weakly monotone maps the space of convex bodies with a $C^2$ support function into itself. As an application, we prove that if $Φ$ is in addition even or a mean section operator, then Euclidean balls are its only fixed points in some $C^2$ neighborhood of the unit ball. Our approach unifies and extends previous results by Ivaki from 2017 and the second author together with Schuster from 2021.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
