Question:

When the value of a function in a point is equal to its integral average over the point's neighborhood?

Carter: 2 days ago

It is well-known that the harmonic functions have this remarkable Averaging Property: if $f$ is harmonic in a domain $U \subset R^n$, then, for any point $x \in U$, $f(x)$ is equal to the integral average of $f$ over any ball centered at $x$ and lying in $U$.

Here a ball is understood with respect to the Euclidean distance -- generated by the norm $\sqrt{\sum_{i=1}^{n} x_i^2}.$ But one can consider the "balls" with respect to other "classical" distances in $R^n$ -- generated by such norms as $\max(|x_1|,|x_2|,...,|x_n| )$ or $\sum_{i=1}^{n} |x_i|$. I have done some research investigating functions satisfying this Averaging Property with respect to the "balls"'corresponding to the distances generated by other norms in $R^n$ (such as the two norms listed above). However I suspect that some studies on this topic have been done already. I would be most grateful if somebody could inform me about publications on this topic.

Answer:
Owen: 2 days ago

You are essentially asking for functions which have the mean value property but for balls with respect to different metrics, and possibly using different measures. This seems amenable (no pun intended) to googling. For example, I googled "harmonic functions on metric measure spaces" and found this paper: https://arxiv.org/pdf/1601.03919.pdf , which contains some regularity and Dirichlet-problem results for such functions in some general metric measure spaces.

There is also a very large literature related to (discrete) harmonic functions on graphs, which may be relevant to you, but I know little about this, so whatever you find by googling will be better.

Hope it's helpful.