On Riesz decomposition for super-polyharmonic functions in \({\mathbb R}^{n}\)
The Riesz decomposition theorem tells us that a superharmonic function \(u\) on \(\mathbb{R}^{n}\), \(n\geq 3\), can be written as the sum of a Newtonian potential and a harmonic function if and only if the mean values of \(u\) over the spheres \(\{|x| =r\}\), \(r>0\), are bounded below. The author proves an analogous decomposition theorem for functions \(u\) that are \(m\)-superharmonic, which means roughly that the distribution \((-\Delta )^{m}u\) can be identified with a positive measure. The particular case where \(m=2\) had previously been investigated by \textit{K. Kitaura} and \textit{Y. Mizuta} [J. Math. Soc. Japan 58, No. 2, 521--533 (2006; Zbl 1103.31004)].
scientific article; zbMATH DE number 6536134