Asymptotically mean value harmonic functions in sub-Riemannian and RCD settings

Tomasz Adamowicz; Antoni Kijowski; Elefterios Soultanis

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arXiv · 2301.06513

Asymptotically mean value harmonic functions in sub-Riemannian and RCD settings

Tomasz Adamowicz·Antoni Kijowski·Elefterios Soultanis

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Abstract

We consider weakly and strongly asymptotically mean value harmonic (amv-harmonic) functions on subriemannian and RCD settings. We demonstrate that, in non-collapsed RCD-spaces with vanishing metric measure boundary, Cheeger harmonic functions are weakly amv-harmonic and that, in Carnot groups, weak amv-harmonicity equivalently characterizes harmonicity in the sense of the sub-Laplacian. In homogeneous Carnot groups of step $2$, we prove a Blaschke-Privaloff-Zaremba type theorem. Similar results are discussed in the settings of Riemannian manifolds and for Alexandrov surfaces.

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Tomasz Adamowicz, Antoni Kijowski, Elefterios Soultanis. 2023-01-16. Asymptotically mean value harmonic functions in sub-Riemannian and RCD settings. https://arxiv.org/abs/2301.06513

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