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Let $f$ be a continuous function defined on $\mathbb{R^n}$. It is well known that both the spherical mean value property (MVP) of $f$, i.e. $$f(x)=\frac{1}{|\partial B(x,r)|}\int_{\partial B(x,r)}f,\ \forall x\in\mathbb{R^n}, r>0$$ and the ball MVP, i.e. $$f(x)=\frac{1}{|B(x,r)|}\int_{B(x,r)}f,\ \forall x\in\mathbb{R^n},r>0$$ imply that $f$ is harmonic.

Note that in the definitions we require the redius $r$ to run over all the positive numbers. Out of curiosity I tried to find non-harmonic functions which satysfy the MVPs only for $r=1$. I did some search and found a remarkable fact called Delsarte's two-radius theorem saying that the spherical MVP with two fixed radii is enough to imply harmonicity of $f$. But for the $1$-radius MVP I haven't found any statement.

In the case $n=1$ examples have been found nicely in this M.SE post. But it is still unclear to me how to construct similar examples in higher dimensions. Any comments would be appreciated!

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    $\begingroup$ I constructed examples of non-harmonic functions satisfying this property in a previous answer. mathoverflow.net/questions/70922/… $\endgroup$
    – George Lowther
    Commented Mar 5, 2012 at 0:37
  • $\begingroup$ My linked example is for the spherical MVP in $\mathbb{R}^2$ although I've no doubt that it generalizes to any $n$ and to the ball MVP. $\endgroup$
    – George Lowther
    Commented Mar 5, 2012 at 0:44
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    $\begingroup$ Ok, after seeing Igor's answer, I do now doubt that it generalizes to any $n$ and the ball MVP! $\endgroup$
    – George Lowther
    Commented Mar 5, 2012 at 0:52

1 Answer 1

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It is a theorem of Hansen and Nadirashvili: W. Hansen, N. Nadirashvili, "Littlewood's one circle problem" J. London Math. Soc. , 50 (1994) pp. 349–360 that one radius (which is allowed to be a function of the point) is not enough for the spherical MVP for functions defined in domains in $\mathbb{C}$ -- this is still open in dimension > 2, while it IS enough for the ball MVP (W. Hansen, N. Nadirashvili, "A converse to the mean value theorem for harmonic functions" Acta Math. , 171 (1993) pp. 139–163), in every dimension. If instead of a domain we take all of $\mathbb{C}$ the answer is YES, and the proof is elementary, see: W. Hansen, A strong version of Liouville’s theorem, Am. Math. Mon. 115 (7) (2008) 583–595.

EDIT

To answer @George's comment: I was actually misled by Hansen's statement in a later paper. What he actually seems to show is that a positive bounded function satisfying the mean-value property for a single $r(x)$ at every $x,$ where $r(x) < | x| + M,$ for some $M$ is constant.

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  • $\begingroup$ "If instead of a domain we take all of ℂ the answer is YES". To which question is the answer yes? $\endgroup$
    – George Lowther
    Commented Mar 5, 2012 at 1:07
  • $\begingroup$ @Igor Awesome! Just a minor correction: the implication "ball MVP ⟹ harmonicity" does not hold in one dimension. $\endgroup$
    – Syang Chen
    Commented Mar 5, 2012 at 18:26

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