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.


Introduction
Let u be a locally Lebesgue integrable function defined on the unit disc D in the complex plane.u is called a super-biharmonic function if Δ 2 u ≥ 0 in the sense of distributions.Abkar and Hedenmalm [1] consider a super-biharmonic function u on D, satisfying two conditions which regulate the growth of u near the boundary ∂D.These conditions are used to split u into its biharmonic Green potential part and its biharmonic part.Using this decomposition, they show that u can be represented by three measures, one on D and two on the boundary ∂D.This comes out as a generalization of the Riesz-Poisson integrals to the super-biharmonic functions on D. However, an extension of this representation in the case of the unit ball in R n , n > 2 (or to the case of Δ m u ≥ 0 with suitable restrictions on u in the unit disc itself) seems complicated.
In this paper, we consider a set of two other conditions on a function u satisfying Δ 2 u ≥ 0, namely, u ≥ 0 and Δu ≤ 0. These conditions are more appropriate as a generalization of the positive superharmonic functions.For, suppose u is a locally Lebesgue integrable function on a bounded domain Ω in R n , n ≥ 2, such that u ≥ 0, Δu ≤ 0, and Δ 2 u ≥ 0. Then u can be represented by three positive measures, one on Ω and two on the Martin boundary of Ω.Interestingly, the method of proof is general enough to be used in the case of (−Δ) i u ≥ 0, 0 ≤ i ≤ m, for any integer m ≥ 2, and any domain Ω in R n on which the Green function is defined (in particular on any bounded domain Ω in R n , n ≥ 2); actually, it goes through in the case of a Riemannian manifold also.Accordingly, we prove this result in the context of a Riemannian manifold.

Preliminaries
Let R be an oriented Riemannian manifold of dimension ≥ 2, with local coordinates x = (x 1 ,...,x n ) and a C ∞ -metric tensor g i j such that g i j x i x j is positive definite.Denote the volume element by dx = det(g i j )dx 1 ,...,dx n .Let Δ be the Laplace-Beltrami operator which, acting on a C 2 -function f , gives Δ f = div grad f .However, we will assume that Δ is taken in the sense of distributions.Thus, a locally dx-integrable function f on an open set ω in R is said to be superharmonic (resp., harmonic) if Δ f ≤ 0 (resp., Δ f = 0) on ω; a positive superharmonic function u on ω is called a potential if and only if the greatest harmonic minorant of u on ω is 0, (i.e., if h is harmonic on ω and h ≤ u, then h should be negative).
For each open set ω in R, let H(ω) denote the class of C 2 -functions u on ω such that Δu = 0.If ω is a domain, H(ω) has the Harnack property, namely, if h n is an increasing sequence in H(ω) and if h = suph n , then h ∈ H(ω) or h ≡ ∞.We can also solve the Dirichlet problem on any parametric ball.This means that the set of harmonic functions H(ω) satisfies the axioms 1, 2, 3 of Brelot [7, pages 13-14].Consequently, we can use the results and the terminology of the Brelot axiomatic potential theory in the context of the Riemannian manifold R.
A domain Ω in R is called a Green domain if the Green function G(x, y) is well defined on Ω.On a Green domain Ω in R, we can construct the Martin compactification Ω of Ω as in [8, pages 111-115].Some of the important points to remember here are the following: fix a point y 0 in a Green domain Ω.If G(x, y) is the Green function on Ω, write k y (x) = k(x, y) = G(x, y)/G(x, y 0 ) with the convention k(y 0 , y 0 ) = 1.Then there exists only one (metrizable) compactification Ω up to homeomorphism such that (i) Ω is dense open in the compact space Ω; (ii) k y (x), y ∈ Ω, extends as a continuous function of x on Ω; (iii) the family of these extended continuous functions on Ω separates the points x ∈ Δ = Ω\Ω.Ω is called the Martin compactification of Ω and Δ = Ω\Ω is called the Martin boundary.A positive harmonic function u > 0 is called minimal if and only if for any harmonic function v, 0 ≤ v ≤ u, we should have v = αu for a constant α, 0 ≤ α ≤ 1.It can be proved that every minimal harmonic function u(y) on Ω is of the form u(y 0 )k(x, y) for some x ∈ Δ, and the points x ∈ Δ corresponding to these minimal harmonic functions are called the minimal points of Δ, and the set of minimal points of Δ is denoted by Δ 1 , called the minimal boundary.
With these remarks, we can state the Martin representation theorem: for any harmonic function u ≥ 0 on Ω, there exists a unique Radon measure μ ≥ 0 on Δ with support in the minimal boundary In the particular case of R = R n , n ≥ 2, and Ω = B(0,1) the unit ball, taking the fixed point y 0 as the centre 0, we have the following: the Martin boundary Δ = Ω\Ω is homeomorphic to the unit sphere S and k(x, y) is the Poisson kernel; also Δ 1 = Δ = S. Then the Martin representation gives the familiar result (see, e.g., Axler et al. [4, page 105]): if u is positive and harmonic on B, then there exists a unique positive Borel measure on S such that u(x) = S p(x, y)dμ(y), where p(x, y), x ∈ B, y ∈ S, is the Poisson kernel.

Riesz-Martin representation for positive super-biharmonic functions
Let Ω be a Green domain in a Riemannian manifold R, with the Green function G(x, y) which is a symmetric function and for fixed y, G y (x) = G(x, y) is a potential on Ω; we have also ΔG y (x) = −δ y (x), after a normalization.
The above definition is given in Sario [10] when Ω = R, a hyperbolic manifold.On an arbitrary hyperbolic Riemannian manifold R, the biharmonic Green function may or may not exist.It is shown in [2, Theorem 3.2] that the biharmonic Green function G 2 (x, y) can be defined on a hyperbolic Riemannian manifold R if and only if there exist two positive potentials p and q on R such that Δq = −p.
Consequently, any relatively compact domain Ω in a Riemannian manifold R is a biharmonic Green domain, whether R is hyperbolic or parabolic.Note that if Ω is a biharmonic Green domain in R, then u(x) = G 2 (x, y) is a potential on Ω, for fixed y; and Δu(x Hence p(z)= Ω G(z,y)dμ(y) ≡∞, so that p(z) is a potential on Ω, and q(x)= Ω G(x,z)p(z)dz, which shows that q(x) is a potential on Ω and Δq(x) = −p(x) = − Ω G(x, y)dμ(y).
For the construction of s in R n , we refer to Brelot [6].A similar method, with the use of an approximation property given in Bagby and Blanchet [5, Theorem 3.10], proves the result in a Riemannian manifold.(For a more general discussion of this result, see [3, Section 2].)By Definition 3.1, a Green domain Ω in a Riemannian manifold R (whether hyperbolic or parabolic) is a biharmonic Green domain if and only if G 2 (x, y) ≡ ∞ on Ω.Note that u(x) = G 2 y (x) > 0, Δu(x) = G y (x) < 0, and Δ 2 u(x) = δ y (x) ≥ 0 on Ω. Hence on a biharmonic Green domain Ω, functions v of the type v > 0, Δv ≤ 0, and Δ 2 v ≥ 0 exist.The following theorem gives an integral representation for such functions.
Theorem 3.4.Let Ω be a biharmonic Green domain in a Riemannian manifold R (whether R is hyperbolic or parabolic) and let v be a locally dx-integrable function on Ω.Then the following are equivalent.
(ii) Let , where h 1 (x) is a positive harmonic function on Ω, so that Δu 2 ≤ 0 and Then u 3 ≥ 0 is harmonic on Ω, so that Δu 3 ≡ 0 and where μ is a positive Radon measure on Ω.Since Δ(Δv) = μ, Δv is a subharmonic function on Ω.Since Δv ≤ 0 by hypothesis, −Δv is a positive superharmonic function on Ω. Hence by the Riesz representation theorem, where h(x) is a positive harmonic function on Ω.
Hence q(x) has the greatest harmonic minorant h 2 (x) on Ω, and by the Riesz representation theorem, (3.7) Similarly, dealing with the superharmonic function H(x) and its greatest harmonic minorant h 3 (x) on Ω, we can write by using the Martin representation for the positive harmonic function h on Ω.Note that v 1 ∈ ∧ 1 and is uniquely determined.Consequently, (3.9) Now, using (3.6), (3.7), and (3.9), we write where h 0 = h 1 + h 2 + h 3 is harmonic on Ω.Now by hypothesis v ≥ 0, so that 6 Riesz-Martin representation Now the two terms on the right side are potentials on Ω and hence their sum also is a potential on Ω.This means that the harmonic function −h 0 is majorized by a potential on Ω, so that −h 0 ≤ 0. Thus h 0 is a positive harmonic function Ω. Use the Martin representation to conclude that there exists a unique measure v 0 on the Martin boundary with support in Δ 1 , such that Thus, from (3.10) and (3.12), we finally arrive at the representation for v(x) on Ω: where (μ,v 1 ,v 0 ) ∈ π 2 × ∧ 1 × ∧ 0 is uniquely determined.

Representation for positive super-polyharmonic functions
By induction, we can extend Theorem 3.4 to obtain the Riesz-Martin representation for positive super-polyharmonic functions.
Let Ω be a Green domain in a Riemannian manifold R, with G(x, y) as the Green function of Ω.For an integer m ≥ 2, we will denote and say that a positive Radon measure μ on Ω is in π m if u(x) = Ω G m (x, y)dμ(y) ≡ ∞ on Ω, in which case u(x) is a potential on Ω and (−Δ) m u = μ; also (−Δ) j u ≥ 0 for 0 ≤ j ≤ m.When such a function u(x) exists on Ω, we say that Ω is an m-harmonic Green domain in R, whether R is hyperbolic or parabolic.
Let Ω be the Martin compactification of Ω and let k(x, y) be the Martin kernel.For any i, 1 ≤ i ≤ m − 1, let ∧ i denote the set of positive Radon measures v i on Δ = Ω\Ω with support in the minimal boundary Δ 1 , such that is well defined on Ω with the above properties.As before, let ∧ 0 denote the set of positive Radon measures v on Δ, with support in Δ 1 .
Then, the proof of Theorem 3.4 can be extended by using the method of induction to arrive at the following result.

Integral representations in a Riemann surface
We are not in a position to say that the above integral representation theorems in a Riemannian manifold R are automatically valid in a Riemann surface S. For, we have used the Laplace-Beltrami operator Δ on R to define polyharmonic-superharmonic functions on R and also to obtain some of their properties.But the Laplacian is not invariant under a parametric change in an abstract Riemann surface S. Hence there is a problem.We indicate in this section how to get over this difficulty.
Let S be a Riemann surface.Let μ ≥ 0 be a Radon measure defined on an open set ω in S.Then, using an approximation theorem of Pfluger [9, page 192], we can show that there exists a superhamonic function s on ω with associated measure μ in a local Riesz representation as explained in Lemma 3.3 (see [3,Theorem 2.3]).Let us symbolically denote this relation between s and μ by Ls = −μ on ω.
Let now dσ denote the surface measure on S.Then, given any locally dσ-integrable function f on an open set ω, let λ be the signed measure on ω defined by dλ = f dσ.Construct as above two superharmonic functions s 1 and s 2 on ω, such that Ls 1 = −λ + and Ls 2 = −λ − .Let us denote this relation between the δ-superharmonic function s = s 1 − s 2 and the locally dσ-integrable function f by Ls = − f .
We will say that s = (s m ,s m−1 ,...,s 1 ) is a polyharmonic-superharmonic function of order m in an open set ω, if s 1 is superharmonic on ω and Ls i = −s i−1 for 2 ≤ i ≤ m.We will say that s ≥ 0 if each s i ≥ 0. If there exists a polyharmonic-superharmonic function s = (s m ,s m−1 ,...,s 1 ) ≥ 0, s i ≡ 0 for any i, on a domain Ω in S, we say that Ω is an mharmonic Green domain in S.
Let now Ω be a Green domain in a Riemann surface S. As before, let Ω be the Martin compactfication of Ω, let Δ = Ω\Ω be the Martin boundary, and let Δ 1 be the minimal boundary.Then, with the notations as in Section 4, we can prove the following.

Lemma 3 . 3 .
Let μ ≥ 0 be a Radon measure on an open set ω in a Riemannian manifold R, hyperbolic or parabolic.Then there exists a superharmonic function s on ω with μ as the associated measure in a local Riesz representation.

Theorem 4 . 1 .
Let Ω be an m-harmonic Green domain in a Riemannian manifold R and let v be a locally dx-integrable function on Ω.Let m ≥ 1 be an integer.Then the following V. Anandam and S. I. Othman 7 are equivalent.(a)(−Δ) i v ≥ 0 on Ω for 0 ≤ i ≤ m.(b) There exist unique measures μ ∈ π m and v