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arXiv · 2607.19684

On the characterization of polyharmonic functions through iterated means

Fernando Charro·Catherine Lebiedzik·Md Nurul Raihen
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Abstract

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.

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Fernando Charro, Catherine Lebiedzik, Md Nurul Raihen. 2026-07-22. On the characterization of polyharmonic functions through iterated means. https://arxiv.org/abs/2607.19684

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