Completely superharmonic polyharmonic functions on a Riemannian manifold (Q2580226)

Let \(\Omega\) be a domain in a Riemannian manifold \(R\). An \(m\)-tuple \(h=(h_m,h_{m-1}, \ldots,h_2,h_1)\) of real functions on \(\Omega\) is called \(m\)-harmonic function, if and only if \(h_1\) is harmonic on \(\Omega\) and \((-\Delta )h_{i+1}=h_i\) for \(1\leq i\leq m-1\). Here, \(- \Delta :=d\delta +\delta d\) denotes the Laplace-Beltrami operator (\(\Delta =\sum_{k=1} ^n\frac{\partial^2}{\partial x_k^2}\) for \(R= {\mathbb R}^n\)). An \(m\)-harmonic function \((h_i) _{m\geq i\geq 1}\) is called completely superharmonic if \(h_i\geq 0\) for \(1\leq i\leq m\). This definition generalizes the notion of a completely superharmonic function as defined by \textit{M. Nicolesco} on a domain in \({\mathbb R}^n\) [Les fonctions polyharmoniques (Hermann, Paris) (1936; Zbl 0016.02505)]. The author proves the following two major results for a hyperbolic Riemannian manifold \(R\) (i.e. there exists the Green function on \(R\)): -- Let \(G\) be the Green function on \(R\). If \[ \int G(x,y_m)G(y_m,y_{m-1})\dots G(y_3,y_2)\, dy_2dy_3\ldots dy_m \] is finite for some \(x\in R\), then there exists an \(m\)-harmonic function on \(R\) that is completely superharmonic. -- If there exists a completely superharmonic \(m\)-harmonic function on \(R\), then there exists an \(m\)-harmonic Green function on \(R\) (i.e. a positive symmetric function \(g(x,y)=g_y(x)\) which, considered as a function of \(x\), is a potential satisfying the condition \((-\Delta) ^m g_y(x)=\delta_y(x)\), \(\delta_y\) being the Dirac measure at \(y\)).