A Liouville theorem for polyharmonic functions (Q1348160)

Suppose that \(u: \mathbb{R}^d\to \mathbb{R}\) satisfies \(\Delta^p u\equiv 0\), where \(p\in\mathbb{N}\) and \(\Delta^p\) denotes the iterated Laplacian. Further, let \(M(f,r)\) denote the mean value of a function \(f\) over the sphere of centre \(0\) and radius \(r\) in \(\mathbb{R}^d\). This note establishes that, if \(\liminf_{r\to\infty} r^{-s}M(u^+, r)= 0\) for some \(s> 2p-2\), then \(u\) is a polynomial of degree less than \(s\). This generalizes work of several authors, and the proof is notably brief. Reviewer's remark: A similar result was obtained independently by \textit{T. Futamura}, \textit{K. Kishi} and \textit{Y. Mizuta} [J. Math. Soc. Japan 53, 113-118 (2001; Zbl 0977.31005)].