Evans potentials and the Riesz decomposition

Abstract

A superharmonic function $u$ on a parabolic Riemannian manifold $M$ is shown to admit the Riesz decomposition $u=h+(1/c_{d})\int_{M}e(\cdot,y)d\mu(y)$ on $M$ into the harmonic function $h$ on $M$ and the Evans potential of an Evans kernel $e(x,y)$ on $M$ and of the Borel measure $\mu:=-\Delta u\geqq 0$ on $M$ multiplied by a certain constant $1/c_{d}$ if and only if $m(t^{2},u)-2m(t,u)={\cal O}(1)\ (t\rightarrow+\infty)$, where $m(t,u)$ is the spherical mean over the sphere of radius $t$ all induced by the above chosen Evans kernel $e(x,y)$ on $M$.