发表机构
Institute of Applied Mathematics, Shenzhen Polytechnic University; Faculty of Natural Sciences and Mathematics, University of Montenegro（深圳职业技术学院应用数学研究所; 黑山大学自然科学与数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该文证明单调带状算子的锐利平均宽度不等式，并导出单位球上调和自映射的全局与微分界，回答了Koh与Kovalev的面积及高维体积问题。
AI 中文摘要
我们证明了单调带状算子的锐利平均宽度不等式，并导出了单位球上欧几里得与双曲调和自映射的全局与微分界。设 $k:[0,\pi]\to\mathbb R$ 为连续且非增函数，$T_k$ 为相应的带状积分算子，\\[(T_kF)(\xi)=\int_{\mathbb S^{n-1}} k\\!\left(\arccos\langle\xi,\eta\rangle\right) F(\eta)\\,d\sigma(\eta),\\]其中 $\sigma$ 为归一化曲面测度。对每个可测函数 $F:\mathbb S^{n-1}\to\overline{\mathbb B^n}$，我们证明 \\[w\\!\left(\operatorname{co}T_kF(\mathbb S^{n-1})\right) \le2\lambda_1(k),\\]其中 $w$ 表示平均宽度，归一化为 $w(\overline{\mathbb B^n})=2$，$\lambda_1(k)$ 为 $T_k$ 在一次球谐函数空间上的特征值。对于严格递减核，等号成立当且仅当 $F(\eta)=Q\eta$ 几乎处处成立，其中 $Q\in O(n)$。对于普通泊松核，乘子为 $r$，从而得到锐利的平均宽度、内蕴体积与像体积收缩。特别地，$|f(r\mathbb B^n)|\le\omega_n r^n$ 无需单射性假设，回答了Koh与Kovalev的面积及高维体积问题。
英文摘要
We prove a sharp mean-width inequality for monotone zonal operators and derive global and differential bounds for Euclidean and hyperbolic-harmonic self-maps of the unit ball. Let $k:[0,π]\to\mathbb R$ be continuous and nonincreasing, and let $T_k$ be the associated zonal integral operator, \[ (T_kF)(ξ)=\int_{\mathbb S^{n-1}} k\!\left(\arccos\langleξ,η\rangle\right) F(η)\,dσ(η), \] where $σ$ is normalized surface measure. For every measurable $F:\mathbb S^{n-1}\to\overline{\mathbb B^n}$, we prove \[ w\!\left(\operatorname{co}T_kF(\mathbb S^{n-1})\right) \le2λ_1(k), \] where $w$ denotes mean width, normalized by $w(\overline{\mathbb B^n})=2$, and $λ_1(k)$ is the eigenvalue of $T_k$ on the space of spherical harmonics of degree one. For strictly decreasing kernels, equality holds exactly for $F(η)=Qη$ almost everywhere, with $Q\in O(n)$. For the ordinary Poisson kernel, the multiplier is $r$, yielding sharp mean-width, intrinsic-volume, and image-volume contraction. In particular, $|f(r\mathbb B^n)|\leω_n r^n$ without injectivity assumptions, answering the area and higher-dimensional volume question of Koh and Kovalev.
Comments36 pages
