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arXiv · 2005.13902

Asymptotically mean value harmonic functions in doubling metric measure spaces

Tomasz Adamowicz·Antoni Kijowski·Elefterios Soultanis
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Abstract

We consider functions with an asymptotic mean value property, known to characterize harmonicity in Riemannian manifolds, in doubling metric measure spaces. We show that the strongly amv-harmonic functions are H\"older continuous for any exponent below one. More generally, we define the class of functions with finite amv-norm and show that functions in this class belong to a fractional Hajlasz-Sobolev space and their blow-ups satisfy the mean-value property. Furthermore, in the weighted Euclidean setting we find an elliptic PDE satisfied by amv-harmonic functions.

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Tomasz Adamowicz, Antoni Kijowski, Elefterios Soultanis. 2020-05-28. Asymptotically mean value harmonic functions in doubling metric measure spaces. https://arxiv.org/abs/2005.13902

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