发表机构
Aarhus University; INRIA Rennes, Univ Rennes & Institut de Recherche Mathématiques de Rennes, CNRS UMR 6625; Vrije Universiteit Brussel（奥胡斯大学; 雷恩INRIA、雷恩大学与雷恩数学研究所，法国国家科学研究中心UMR 6625; 布鲁塞尔自由大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究等正则次黎曼流形上度量球均值算子的小尺度极限，给出其收敛判据，在步长≤2的流形上验证该判据，计算李群上的极限并证明体积加权算子的无条件分布收敛性。
AI 中文摘要
设$(M,\u0394,g)$为配备光滑正测度$\u03bc$的光滑等正则次黎曼流形，我们研究度量球均值算子$A_hf(x)=\frac{1}{h^{2}\u03bc(B(x,h)) }\u222b_{B(x,h)}(f(q)-f(x)) d\u03bc(q)$的小尺度极限。精确齐次性在Carnot群上给出逐点极限；在一般等正则流形上，对每个光滑测试函数的收敛性等价于第一类特权坐标下度量球的重标水平一阶矩的收敛，该判据与$\u03bc$无关。当该判据成立时，主符号由切单位球的归一化二阶矩张量确定，漂移项满足显式测度变换公式。在步长至多为2的情形中，我们结合实解析有限喷管约化与tame积分验证了该判据，排除了归一化矩的振荡；还计算了李群上的极限，并证明了体积加权算子在所有等正则流形上的无条件分布收敛性。
英文摘要
Let $(M,\mathcal{D},g)$ be a smooth equiregular sub-Riemannian manifold equipped with a smooth positive measure $μ$. We study the small-scale limit of the metric-ball mean-value operator \[ A_hf(x)=\frac{1}{h^{2}μ(B(x,h)) }\int_{B(x,h)}(f(q)-f(x))\, dμ(q). \] Exact homogeneity yields the pointwise limit on Carnot groups. On a general equiregular manifold, convergence for every smooth test function is equivalent to convergence of the rescaled horizontal first moments of metric balls in first-kind privileged coordinates. This criterion is independent of $μ$; when it holds, the principal symbol is determined by the normalized second-moment tensor of the tangent unit ball, and the drift satisfies an explicit change-of-measure formula. In step at most two, we verify the criterion by combining a real-analytic finite-jet reduction with tame integration, which rules out oscillation of the normalized moments. We also compute the limit on Lie groups and prove unconditional distributional convergence of the volume-weighted operators on every equiregular manifold.
