As is well known, harmonic functions satisfy the mean value property, i.e. the average of such a function over a ball is equal to its value at the center. This fact naturally raises the question on whether this is a feature characterizing only balls, namely, is a set, for which all harmonic functions satisfy the mean value property, necessarily a ball?

This question was investigated by several authors, including Bernard Epstein (1962), Bernard Epstein and Schiffer (1965), Myron Goldstein and Wellington (1971), who obtained a positive answer to this question under suitable additional assumptions.

The problem was finally elegantly, completely and positively settled by Ülkü Kuran (1972), with an artful use of elementary techniques.

This classical problem has been recently fleshed out by Giovanni Cupini, et al. (in press) who proved a quantitative stability result for the mean value formula, showing that a suitable “mean value gap” (measuring the normalized difference between the average of harmonic functions on a given set and their pointwise value) is bounded from below by the Lebesgue measure of the “gap” between the set and the ball (and, consequently, by the Fraenkel asymmetry of the set). That is, if a domain “almost” satisfies the mean value property for all harmonic functions, then that domain is “almost” a ball.

The goal of this note is to investigate some nonlocal counterparts of these results. Some of our arguments rely on fractional potential theory, others on purely nonlocal properties, with no classical counterpart, such as the fact that “all functions are locally fractional harmonic up to a small error”.

众所周知，谐波函数满足平均值的性质，即该函数在球上的平均值等于其在中心的值。这个事实自然引发了一个问题，即这是否是仅表征球的特征，即是否为一组，所有谐波函数都满足均值特性，必然是球？

包括伯纳德·爱泼斯坦（1962），伯纳德·爱泼斯坦和席弗（1965），迈伦·戈德斯坦和惠灵顿（1971）在内的多位作者对此问题进行了调查，他们在适当的附加假设下获得了该问题的肯定答案。

这个问题最终由ÜlküKuran（1972）巧妙地运用了基本技术巧妙地解决了。

Giovanni Cupini等人最近已经充实了这个经典问题。（印刷中）证明了均值公式的定量稳定性结果，表明一个合适的“均值间隙”（测量给定集合上的谐波函数的平均值与它们的逐点值之间的归一化差）由下面限定Lebesgue度量集合和球之间的“间隙”（因此，通过集合的Fraenkel不对称性）。也就是说，如果一个域“几乎”满足所有谐波函数的平均值属性，则该域“几乎”就是一个球。

本说明的目的是调查这些结果中的一些非本地对应项。我们的某些论点依赖于分数势理论，另一些论点则依赖于纯粹的非局部性质，而没有经典的对应关系，例如“所有函数都是局部分数谐波，直至出现很小的误差”这一事实。