Mean values of positive superharmonic functions on half-spaces (Q1103068)

Let w denote a non-negative superharmonic function on the open half-space \(D=R^ n\times]0,\infty[\), \(n\geq 1\), and let \(\Phi\) be a non-decreasing function \(\Phi: [0,\infty[\to [0,\infty[,\) not necessarily concave. The present paper is devoted to the study of mean values, \[ M(\Phi(w),t)=\int_{R^ n}\Phi (w(x,t))dx, \] and extends results of the first author [J. Lond. Math. Soc., II. Ser. 23, 129-136 (1981; Zbl 0427.31002)] and of the second author [Mathematika 29, 55-57 (1982; Zbl 0508.31004)]. For instance, under the stated conditions, there exists \(C>0\) (possibly depending on w) such that \[ \liminf_{t\to \infty}M(\Phi(w),t)/(t^ n\Phi(Ct^{-n}))>0. \] Stronger results are obtained under further conditions on the function \(\Phi\). Conditions on \(M(\Phi(w),t)\) are given that imply \(w\equiv 0\) on D.