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