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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.

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    $\begingroup$ Yes, that works, and in fact the proof given in the wikipedia article generalizes with no changes whatsoever (there is the issue of ensuring enough regularity of $w$, obtained as a solution of the Poisson equation, but I don't think that will be a real problem): en.wikipedia.org/wiki/Harmonic_function#The_mean_value_property $\endgroup$
    – Christian Remling
    Commented Jun 9, 2017 at 0:30
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    $\begingroup$ Probably not exactly what you want, but the use of "neighborhood" in your title made me think of the limit of small balls and hence of the Lebesgue differentiation theorem: if $f\in L^1$, then one has that for almost all $x$, $f(x)=\lim_{B\to x}|B|^{-1}\int_Bf(x)\,dx$. According to the link, this works for $B$ ranging over balls centered at $x$ in any norm. $\endgroup$
    – Francois Ziegler
    Commented Jun 9, 2017 at 0:57
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    $\begingroup$ As an explicit example, I took the harmonic function $f(x,y) = e^x \cos(y)$ and averaged it over the square $[-1,1]^2$, which is the ball of the $\ell^\infty$ norm, and I didn't get $1$. $\endgroup$
    – Nate Eldredge
    Commented Jun 9, 2017 at 1:49
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    $\begingroup$ There may be results of the form "for each norm $\|\cdot\|$ satisfying certain conditions there exists a second-order elliptic operator $L$ such that solutions $u$ of $Lu=0$ satisfy the averaging property with respect to the balls of $\|\cdot\|$". Is that the kind of thing you want? $\endgroup$
    – Nate Eldredge
    Commented Jun 9, 2017 at 2:01
  • $\begingroup$ @ChristianRemling Thank you! Please note that the harmonic functions need not have the mean value property w.r.t. non-Euclidean norms (see an example given by Nate Eldredge), so it would be an overstatement to claim that the proof in the Wikipedia article on the harmonic functions can be generalized "with no changes whatsoever". Still, the approach used in the proof looks promising, and I will look if it could be appropriately modified to address my question. $\endgroup$
    – Grove
    Commented Jun 9, 2017 at 4:27

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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.

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    $\begingroup$ Thank you very much for providing the reference! This publication (and other papers referenced there) seems to address the following situation: if a function has the mean value property w.r.t some metrics/measure, what regularity and Dirichlet-problem results hold for it. What I was looking for (and I should have been more accurate in stating my question) are questions like this: for harmonic functions, is it true that their mean values over non-Euclidean balls (e.g. corresponding to the norms I listed above) are equal to the value at the center of the "ball"? Are there publications on this? $\endgroup$
    – Grove
    Commented Jun 8, 2017 at 23:02

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