AI 中文总结
研究通过迭代均值刻画多重调和函数，引入一族均值公式，证明其可刻画有限阶实值多重调和函数，还得出均值性质强逆命题及正则性结果，即满足该族均值性质的局部可积函数是多重调和且光滑的。
AI 中文摘要
我们引入了一族由迭代均值的线性组合给出的均值公式（精确的和渐近的）。我们证明了该族中的均值公式刻画了有限阶实值多重调和函数，并且一个简单的代数条件根据多重调和性的阶将该族划分成等价类。我们的关键结果包括均值性质的强逆命题——满足该族中均值性质的局部可积函数是多重调和的——以及一个正则性结果——满足该族中均值性质（精确的或渐近的）的局部可积函数是光滑的。
英文摘要
We introduce an infinite family of mean-value formulas (exact and asymptotic) given in terms of linear combinations of iterated means. We prove that the mean-value formulas in this family characterize real-valued polyharmonic functions of finite order, and that a simple algebraic condition partitions the family into equivalence classes according to the order of polyharmonicity. Our key results include strong converses to the mean-value properties -- locally integrable functions satisfying a mean-value property in the family are polyharmonic -- and a regularity result -- locally integrable functions satisfying a mean-value property in the family, whether exact or asymptotic, are smooth.
