Asymptotic mean value Laplacian on equiregular sub-Riemannian manifolds
Abstract
Let (M,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 Ahf(x)=1/(h2μ(B(x,h)) )∫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.
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BibTeX
@misc{baudoin2026asymptotic,
title = {{Asymptotic mean value Laplacian on equiregular sub-Riemannian manifolds}},
author = {Fabrice Baudoin and Jonathan Junné and David Tewodrose},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02710},
eprint = {2609.02710},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02710},
}
RIS
TY - UNPB
TI - Asymptotic mean value Laplacian on equiregular sub-Riemannian manifolds
AU - Fabrice Baudoin
AU - Jonathan Junné
AU - David Tewodrose
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02710
UR - https://arxiv.org/abs/2609.02710
AB - Let (M,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)=1/(h^2μ(B(x,h)) )∫_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.
ER -