Let $f$ be a continuous real-valued function on $\mathbb{R}^n$. It is well known that the following are equivalent:
- $f$ is harmonic.
$f$ satisfies the ball mean value property $$ f(x)=\frac{1}{|B(x,r)|}\int_{B(x,r)}fdV $$ for all $x\in \mathbb{R}^n$ and all $r>0$.
$f$ satisfies the sphere mean value property $$ f(x)=\frac{1}{|\partial B(x,r)|}\int_{\partial B(x,r)}fdV $$ for all $x\in \mathbb{R}^n$ and all $r>0$.
QUESTION: Does the above hold if we restrict that the mean value properties hold for some specific $r$ (which may be a function of $x$)?
In my search for an answer, I have stumbled upon the following hints:
This M.SE post gives an elementary counter example to the case of $\mathbb{R}$. A non-harmonic continuous function $f:\mathbb{R} \to \mathbb{R}$ that satisfies both the MVP's with $r=1$ is constructed.
In this M.OF post there is a counter example to the case of $\mathbb{R}^2$, altought strictly speaking the function constructed there attains complex values.
In a series of papers by W. Hansen and N. Nadirashvili, most recent of which is "A Liouville property for spherical averages in the plane" (dating 1999), some partial positive results are given: in $\mathbb{R}^2$, and for continuous bounded functions, the above three are indeed equivalent (note that bounded harmonic on $\mathbb{R}^d$ are just constants by the Liouville property).
So is this question still open for $\mathbb{R}^d$, $d\geq 3$? what about for other cases, say of Lie groups?
As a general note, the motivation for my question comes from the theory of random walks on groups, where harmonic function are typically defined as function that satisfy this weak form of MVP: Given a probability measure $\mu$ on a group $G$, $f$ is called $\mu$-harmonic if $f(g)=\int_G f(gs)d\mu(s)$ for all $g\in G$.
Thanks!!
