International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Issue 8, Pages 92176, 9 p.
doi:10.1155/IJMMS/2006/92176

Riesz-Martin representation for positive super-polyharmonic functions in a Riemannian manifold

Abstract

Let u be a super-biharmonic function, that is, Δ2u≥0, on the unit disc D in the complex plane, satisfying certain conditions. Then it has been shown that u has a representation analogous to the Poisson-Jensen representation for subharmonic functions on D. In the same vein, it is shown here that a function u on any Green domain Ω in a Riemannian manifold satisfying the conditions (−Δ)iu≥0 for 0≤i≤m has a representation analogous to the Riesz-Martin representation for positive superharmonic functions on Ω.