International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 92176, 9 pages
doi:10.1155/IJMMS/2006/92176
Riesz-Martin representation for positive
super-polyharmonic functions in A Riemannian manifold
Department of Mathematics, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
Received 19 December 2005; Revised 28 February 2006; Accepted 4 April 2006
Copyright © 2006 V. Anandam and S. I. Othman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.