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BY 4.0 license Open Access Published by De Gruyter Open Access November 8, 2022

Asymptotically Mean Value Harmonic Functions in Doubling Metric Measure Spaces

Tomasz Adamowicz , Antoni Kijowski EMAIL logo and Elefterios Soultanis


We consider functions with an asymptotic mean value property, known to characterize harmonicity in Riemannian manifolds and in doubling metric measure spaces. We show that the strongly amv-harmonic functions are Hölder 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 Hajłasz–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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Received: 2022-06-14
Accepted: 2022-08-16
Published Online: 2022-11-08

© 2022 Tomasz Adamowicz et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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