EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5761 ISSN 1307-5543 – ejpam.com Published by New York Business Global Subordinate Average Structures on Random Walks M. Surya Priya1, N. Nathiya1,∗ 1 Department of Mathematics, Vellore Institute of Technology Chennai, Tamilnadu, India Abstract. In a random walk {N, p(x, y)} where N is an infinite graph and {p(x, y)} is a set of transition probabilities, if {p′ (x, y)} is another set subordinate to the set {p(x, y)} such that p ′ (x, y) ≤ p(x, y) for all pairs (x, y) and p ′ (x, y) < p(x, y) for atleast one pair (x, y), then {N, p ′ (x, y)} can be identified as a Schrödinger network. In general, we consider the random walk {N,P ′} which is subordinate to {N,P} and discuss the relation between the classes of super- average functions defined by the transition probabilities sets {p(x, y)} and {p′ (x, y)}. Moreover, we define {N,P ′} as parahyperbolic if 0 is the only bounded P ′ -average function on N and study various potential-theoretic properties of parahyperbolic networks. We also give equivalent condi- tions for a random walk to be parahyperbolic. Finally, we discuss the relation between bounded P ′ and P -average functions. 2020 Mathematics Subject Classifications: 31C20, 31C05, 60J45 Key Words and Phrases: Superaverage functions, subordinate structure, parahyperbolic, P ′ - Green’s potential, bounded P and P ′ functions 1. Introduction In the state space N = {0, 1, 2, ....} with the set P = {p(x, y)} of transition probabili- ties given by p(n, n + 1) = αn, p(n, n − 1) = βn, for n ≥ 1, αn, βn > 0, αn + βn ≤ 1 and 0 ≤ p(0, 1) ≤ 1, the transience, the recurrence, the hitting time etc. of the random walk {N,P} depend on P . For example if p(n, n+1) = p(n, n−1) = 1 2 for n ≥ 1 and p(0, 1) = 1 then {N,P} is recurrent and any function u(x) on N such that u(n) = 1 2u(n+1)+ 1 2u(n−1) for n ≥ 1 and u(0) = u(1) is constant. This example is the motivation for the consider- ation of the following problem: Let {N,P} be a random walk ([2] and [12]) where N is an infinite graph which is connected and P = {p(x, y)} is a set of transition probabilities, p(x, y) > 0 if and only if x and y are neighbours; p(x, y) and p(y, x) may have different values. Suppose P ′ = {p′ (x, y)} is another set of transition probabilities on N such that p ′ (x, y) ≤ p(x, y) for every pair x, y. The problem is to study how the properties of tran- sience, recurrence and other probabilistic results in {N,P} get transformed in the random walk {N,P ′}. We refer to P ′ as a transition probability structure on N subordinate to P . ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5761 Email addresses: surya.smiley20@gmail.com (M. Surya Priya), nadhiyan@gmail.com (N. Nathiya) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Surya Priya, N. Nathiya / Eur. J. Pure Appl. Math, 18 (1) (2025), 5761 2 of 11 There is an analogy in the context of general infinite networks ([1] and [10]) {X, t(x, y)}. Here potential-theoretic properties of functions on X are studied using the Laplace op- erator ∆u(x) = ∑ t(x, y)[u(y) − u(x)]. If q ≥ 0 is a function on X, there is another interesting Schrödinger operator ([4] and [7]) ∆qu(x) = ∆u(x) − q(x)u(x). If we write t ′ (x, y) = t(x,y) q(x)+ ∑ t(x,y) than {X, t ′ (x, y)} is another network where t ′ (x, y) ≤ t(x, y) which provides a convenient base for the study of Schrödinger potentials with a comparable study of Laplace potentials. In this article, we discuss the potential-theoretic aspects of functions on N determined by the original structure P and another structure P ′ that is subordinate to P . For related results, we have referred [6] and [11]. The classification of connected and locally finite in- finite network into parabolic and hyperbolic of order p is investigated in [13], where as the same is done for non-locally finite networks in [3]. The concept of recurrent random walks on countable infinite state spaces is explored by V. R. Manivannan and M. Venkataraman [5]. In [8] M. Surya priya and N. Nathiya establishes the existence of Green’s function on non-reversible random walks through the application of potential theoretic techniques. With reference to the aforementioned, we have looked into the concepts of P ′ -Green’s function as well as parahyperbolic and bounded hyperbolic subordinate structures. S. Sivan and M. Venkataraman [9], says that an infinite network is parahyperbolic if and only if constant 1 is a potential. They have given neccessary and sufficient condition for a network to be parahyperbolic. V. Anandam [1] has studied the Schrödinger operators and subordinate structures on infinite networks. Where as in the present article, we delve into the potential theory associated to a structure subordinate to a non-locally finite random walk. We define parahyperbolic random walk and its subordinate structure. Finally a section is devoted to investigate the relation between bounded P -average functions and bounded P ′ -average functions. 2. Preliminaries Definition 1. Random walk: Let {N,P} be a random walk with a countable infinite number of states N and P = {p(x, y)} is the probability transition matrix, where p(x, y) denotes the transition probability from state x to state y. We assume {N,P} is connected (i.e, for any two distinct states there exists a path connecting them) and without self loops. As usual, we shall take N as an infinite graph by defining [x, y] as an edge if and only if p(x, y) > 0. We say two states x and y are neighbours if there exists an edge between them and it is denoted by x ∼ y and p(x) = ∑ y∼x p(x, y) = 1 for every x ∈ N . Note: We do not place the condition that the number of neighbours of any state is finite. Hence we consider only those real-valued functions s on N for which ∑ y∼x p(x, y)|s(y)| < ∞ for any x ∈ N . Write As(x) = ∑ y p(x, y)s(y). Definition 2. Interior and Boundary of a set: We say a state x is an interior state of M. Surya Priya, N. Nathiya / Eur. J. Pure Appl. Math, 18 (1) (2025), 5761 3 of 11 a subset K if and only if x and all its neighbours are in a subset K of N . The set of all interior states of K is denoted by K̊ and the boundary of K by ∂K = K\K̊. Definition 3. Laplacian(∆): Let s(x) be a real valued function defined on N . For x ∈ K̊, K ⊂ N , the Laplacian (∆) of s at x is defined as ∆s(x) = ∑ y∼x p(x, y)[s(y)− s(x)] = (A− I)s(x) Definition 4. A function u defined on a subset K is said to be P -superaverage (re- spectively, P -subaverage and P -average) on K if and only if s(x) ≥ As(x) (respectively s(x) ≤ As(x) and s(x) = As(x)) for every x ∈ K̊. Definition 5. If p ≥ 0 is a P -superaverage function such that any P -subaverage function majorized by p is non-positive, then p is called a P -potential. Definition 6. If s is a P -superaverage function on N and if K is a subset of N such that ∆s(x) = 0 for each x in N\K, then K is said to be the P -average support of s in N . If there is a perturbation on Laplace operator indicated by the operator ∆qu(x) = ∆u(x) − q(x)u(x), q ≥ 0, (the operator ∆q is commonly referred to as a Schrödinger operator on N) we have ∆qu(x) = ∑ y p(x, y)u(y)− [1 + q(x)]u(x) =[1 + q(x)][A ′ − I]u(x) where A ′ u(x) = ∑ p ′ (x, y)u(y), p ′ (x, y) = p(x,y) 1+q(x) ≤ p(x, y). Then the potential theory associated with the Schrödinger operator ∆q depends on A ′ , just as the potential theory associated with the Laplace operator ∆ depends on the operator A. Since A ′ φ(x) ≤ Aφ(x) for any functions φ ≥ 0 on N , the relation between the Schrödinger potentials and the Laplace potentials is exhibited by the relation between P ′ = {p′ (x, y)} and P = {p(x, y)}. Here p ′ (x, y) ≤ p(x, y) for any pair x, y and we say that P ′ is subordinate to P . In the following sections we investigate this subordinate structure in an abstract setting. 3. Subordinate structure Definition 7. Let {p′ (x, y)} be a set of transition indices on N such that p(x, y) ≥ p ′ (x, y) ≥ 0 for any pair of states x and y and p ′ (x, y) < p(x, y) for atleast one pair of x and y. Then we say that P ′ = {p′ (x, y)} defines a submarkov average structure on N that is subordinate to the average structure defined by P = {p(x, y)}; or simply that P ′ is subordinate to P on N . M. Surya Priya, N. Nathiya / Eur. J. Pure Appl. Math, 18 (1) (2025), 5761 4 of 11 Remark 1. A Schrödinger operator ∆q defines a subordinate structure on N , when q > 0 and p ′ (x, y) = p(x,y) 1+q(x) . But for a subordinate structure {N,P ′} to define a Schrödinger operator on N , it is necessary that for every x ∈ N , the quantity p(x,y) p′ (x,y) is independent of y, for any y ∼ x. Definition 8. A real valued function u on a subset K of N is said to be P ′ -superaverage ( respectively P ′ -subaverage ) on K if and only if ∆ ′ u(x) ≤ 0 (∆ ′ u(x) ≥ 0 respectively) or u(x) ≥ ∑ y p ′ (x, y)u(y) for every x in K̊. (Here ∆ ′ u(x) = [ ∑ y p ′ (x, y)u(y)] − u(x) = (A ′ − I)u(x)). Definition 9. A real valued function u on a subset K of N is said to be P ′ -average on K if and only if ∆ ′ u(x) = 0 for every x in K̊. Proposition 1. If P ′ is subordinate to P then for any u ≥ 0 on a subset K of N ,∑ p(x, y)u(y) ≥ ∑ P ′ (x, y)u(y), for every x ∈ K̊. Hence (i) If u ≥ 0 is P -superaverage on K then u is P ′ -superaverage on K. (ii) If u ≥ 0 is P ′ -subaverage on K then u is P -subaverage on K. (iii) If u = 0 is P ′ -average on K then u is P ′ -superaverage on K. Proof. For a P -superaverage function u on K ⊂ N , we have ∑ p(x, y)u(y) ≤ u(x). Since P ′ is subordinate to P ,∑ p(x, y)u(y) ≥ ∑ p ′ (x, y)u(y) u(x) ≥ ∑ p(x, y)u(y) ≥ ∑ p ′ (x, y)u(y) u(x) ≥ ∑ p(x, y) ′ u(y) Thus, if u is P -superaverage on a subset K of N then u is P ′ -superaverage on K ⊂ N . Similarly the proof follows for (ii) and (iii). 3.1. Properties of P ′ -Superaverage Functions (i) If s1 and s2 are P ′ -superaverage on a subset K and if α1, α2 are two non-negative numbers, then α1s1 + α2s2 and inf(s1, s2) are P ′ -superaverage on K. (ii) If {si} is a lower directed family of P ′ -superaverage functions on K, then s(x) = infi ui(x) and then s is P ′ -superaverage onK. (A lower directed family F of functions means that if f, g ∈ F then inf(f, g) is also in F) (iii) Greatest P ′ -average minorant (g.P ′ -a.m): Suppose u(x) ≥ v(x) on N where u(x) is P ′ -superaverage and v(x) is P ′ -subaverage on N . Then there exists a P ′ - average function h(x) on N , u(x) ≥ h(x) ≥ v(x) and if h1 is any other P ′ -average function on N between u(x) and v(x), then h(x) ≥ h1(x) on N (Section 3.1, [1]). M. Surya Priya, N. Nathiya / Eur. J. Pure Appl. Math, 18 (1) (2025), 5761 5 of 11 Proof. Let {Kn} be an exhaustion of N by finite sets; that is Kn ⊂ K̊n+1 ⊂ Kn+1 and X = ∪Kn. let Dnu denote the P ′ -superaverage function on N , equal to the Dirichlet solution on Kn with boundary values u and extended by u outside Kn. Then {Dnu} is a decreasing sequence of P ′ -superaverage functions, each Dnu ≥ v on N . Hence D[u] = limnDnu is P ′ -superaverage function. Now for any z in N , z ∈ K̊m for some m. Hence Dn(x) is P ′ -average at x = z for all n ≥ m. Consequently, D[u](x) is P ′ -average at x = z. This shows that D[u] is P ′ -average on N . Thus u ≥ D[u] ≥ v. Moreover if h1 is P -average, u ≥ h1 ≥ v, then Dnu ≥ h1 for any n so that D[u] ≥ h1. we term D[u] as the greatest P ′ -average minorant of u on N . (iv) Riesz representation theorem: Any non-negative P ′ -superaverage function s on a subsetK can be written as the sum of a P ′ -potential and a non-negative P ′ -average function on K and this representation is unique. Proposition 2. If s is a P -potential then it is a P ′ -potential. Proof. First note that s is P ′ -superaverage on N . Let u ≥ 0 be a P ′ -subaverage function such that u ≤ s on N . Note that u is P -subaverage function on N and s is a P-potential, therefore u = 0. Consequently s is a P ′ -potential on N . 3.2. P ′ -Green’s Potential Though positive P ′ -potentials always exist on N , positive P -potentials may exist (hy- perbolic random walk) or may not exist (parabolic random walk) on N . Thus on a hyperbolic random walk N , for a fixed state e, we have the P -Green’s potential Ge(x), −∆[Ge(x)] = δe(x) and P ′ -Green’s potential G ′ e(x) = −∆ ′ [G ′ e(x)] = δe(x). The following theorem indicates a relation between them. Lemma 1. Let s ≥ 0 be a P ′ -superaverage function and p be a P ′ -potential on N . If (−∆ ′ )s ≥ (−∆ ′ )p, then s ≥ p on N . Proof. By hypothesis, s = p+u where u is P ′ -superaverage on N . Since s ≥ 0,−u ≤ p on N . Hence −u ≤ 0 so that s ≥ p on N . Theorem 1. Let (N,P ) be a hyperbolic network. Let Ge(x) be the P -Green’s function on N with average support {e}. Then Ge ′ (x) ≤ Ge(x) for every x ∈ N . Proof. Since any positive P -superaverage function on N is a P ′ -superaverage function on N , Ge(x) is a P ′ -superaverage function on N . If x ̸= e, (−∆ ′ )Ge(x) ≥ 0 and (−∆ ′ )G ′ e(x) = 0 when x = e, (−∆ ′ )Ge(e) =Ge(e)− ∑ y p ′ (e, y)Ge(y) M. Surya Priya, N. Nathiya / Eur. J. Pure Appl. Math, 18 (1) (2025), 5761 6 of 11 ≥Ge(e)− ∑ y p(e, y)Ge(y) =(−∆)Ge(e) =1 =(−∆ ′ )G ′ e(e) Since for all x ∈ N , (−∆ ′ )Ge(x) ≥ (−∆ ′ )G ′ e(x) By the above Lemma 1, Ge ′ (x) ≤ Ge(x) for every x ∈ N . Lemma 2. If pn is a sequence of P ′ -potentials and if p(x) = ∑ n pn(x) is finite at one state, then p is a P ′ -potential. Proof. The P ′ -superaverage function ( actually a P ′ -potential ) sm = ∑m 1 pn intro- duces a sequence {sm} of increasing P ′ -superaverage functions so that s = lim m sm is a P ′ -superaverage function if s is finite at one state. Hence p(x) = ∑ n pn(x) is a P ′ - superaverage function. To show p(x) is a P ′ -potential: Let h(x) be a non-negative P ′ -average and h ≤ p. Then h − ∞∑ 2 pn ≤ p1. Here the left side is P ′ -subaverage and the right side a P ′ -potential, so that h− ∞∑ 2 pn ≤ 0. Continuing this process we find h(x) ≤ ∞∑ m pn(x) for any m. For any z in N , since ∞∑ 1 pn(z) is convergent h(z) ≤ ∞∑ m pn(z) ≤ ϵ for sufficiently large m. This leads to h(z) = 0, hence h = 0 and consequently p = ∞∑ 1 pn is a P ′ -potential on N . Recall that for any P ′ -superaverage function s ≥ 0 we write by Riesz representation, s = p+D[s], where D[s] is the greatest P ′ -verage minorant of s. Theorem 2. Any P ′ -superaverage function s ≥ 0 has a unique representation s(x) =∑ y [−∆ ′ s(y)]G ′ y(x) +D[s](x). Proof. LetK be a finite set and uk(x) = s(x)− ∑ y∈k [−∆ ′ s(y)]G ′ y(x). Then −∆ ′ [uk(x)] = 0 if x ∈ k and −∆ ′ [uk(x)] ≥ 0 if x ∈ N\K. Then uk(x) is a P ′ -superaverage function on N and −uk(x) ≤ ∑ y∈K [−∆ ′ s(y)]G ′ y(x). Since the left side is P ′ -subaverage and the right side is a P ′ -potential; −uk(x) ≤ 0 on N . That is ∑ y∈K [−∆ ′ s(y)]G ′ y(x) ≤ s(x). Allowing K to grow into N , s(x) ≥ ∑ y∈N [−∆ ′ s(y)]G ′ y(x). Note that the right side is a P ′ -potential by Lemma 2. Write h(x) = s(x)− ∑ y∈N [−∆ ′ s(y)]G ′ y(x). Note −∆ ′ h = 0 so that h is a P ′ -average function on N . By the uniqueness of Riesz representation, h(x) = D[s](x). M. Surya Priya, N. Nathiya / Eur. J. Pure Appl. Math, 18 (1) (2025), 5761 7 of 11 In the particular case, when s = 1 is the constant function then −∆ ′ s(x) = s(x) −∑ y p ′ (x, y)s(y) = 1− p ′ (x) when p ′ (x) = ∑ y p ′ (x, y). Hence obtain the following result. Corollary 1. For any x ∈ N , 1 = ∑ y[1− p ′ (y)]G ′ y(x) +D[1](x). 4. Parahyperbolic subordinate structures Since A ′ u(x) ≤ Au(x) for any non-negative functions u(x), then any non-negative P - superaverage functions is a P ′ -superaverage function. In particular, the constant function 1 is a P ′ -superaverage function so that 1 = s + h where s > 0 is P ′ -superaverage and h ≥ 0 is a P ′ -average function. Since h > 0 or h ≡ 0, the constant 1 is a P ′ -potential or just a positive P ′ -superaverage function that is not a P ′ -potential. This opens up two possibilities in the study of P ′ -superaverage functions on N , as shown in this section. In a random walk (N,P ) the constant 1 is P -average. It is possible that any positive P -superaverage function is constant, hence there may not be any positive P -potential on N . On the other hand, the constant 1 is P ′ -superaverage but not P ′ -average. Hence there are always P ′ -potentials on N . Let P ′ be a subordinate structure to P . Then the constant 1 is a P ′ -superaverage function, write 1 = v + h where v is a P ′ -potential and h ≥ 0 is a P ′ -average function. (i) It is possible that h ̸= 0. It means that there are bounded positive P ′ -average functions N . (ii) If h = 0, then 1 is a P ′ -potential, hence there is no bounded positive P ′ -average functions on N . Definition 10. If the constant 1 is a P ′ -potential, then (N,P ′ ) is referred to as parahy- perbolic. Otherwise (N,P ′ ) is termed bounded hyperbolic. Proposition 3. (Maximum Principle:) The following are equivalent( Theorem 4.3.7, [1]): (i) (N,P ′ ) is parahyperbolic. (ii) In an arbitrary subset F of N , if u is an upper bounded subaverage function such that u ≤ 0 on ∂F , then u ≤ 0 on F . Definition 11. (Perron family:) Let F be the family of all P ′ -subaverage functions u on N such that for a P ′ -superaverage function v on N , u ≤ v on N . If u1, u2 ∈ F, then sup(u1, u2) ∈ F, hence is an upper directed family of P ′ -subaverage functions . Fix a state z and choose any u ∈ F. Then the function uz(x) = { u(x), if x ̸= z∑ p ′ (z, y)u(y), if x = z (Known as the Poisson modification of u(x) at x = z) also is in F. Note uz ≥ u and uz(x) is P ′ -average at x = z. Consequently, h(x) = sup u∈F u(x) is P ′ -subaverage on M. Surya Priya, N. Nathiya / Eur. J. Pure Appl. Math, 18 (1) (2025), 5761 8 of 11 N and at x = z, h(z) = sup u∈F uz(z) is P ′ -average. Since z is arbitrary, we conclude that h(x) = sup u∈F u(x) is P ′ -harmonic on N . We refer to F as the Perron family of P ′ -subaverage functions. Theorem 3. The following are equivalent: (i) Any bounded P ′ -superaverage function u defined outside a finite set is of the form u = p− q where p and q are bounded P ′ -potentials on N . (ii) Any bounded P ′ -superaverage function in N is a P ′ -potential. (iii) 0 is the only bounded P ′ -average function in N . (iv) The constant function 1 is a P ′ -potential on N , that is N is parahyperbolic. Proof. (i) implies (ii). Let s be a bounded P ′ -superaverage function in N . Then by (i), s = p − q outside a finite set A. Hence |s| ≤ p + q on N/A. Since A is a finite set, s is bounded on A and we select a large constant α > 1 such that |s| ≤ α(p + q) on A. Consequently, |s| ≤ α(p + q) on N . Since −s ≤ α(p + q), we see that −s ≤ 0, then 0 ≤ s ≤ α(p+ q) so that s is a P ′ -potential on N . (ii) implies (iii) if h ̸= 0 is a bounded P ′ -average function on N , then by (ii) it is a P ′ -potential. (iii) implies (iv) Since 1 is P ′ -superaverage on N , the greatest P ′ -average minorant of 1 is 0. Hence 1 is a P ′ -potential, thus {N,P ′} is parahyperbolic. (iv) implies i) Let u = p − q outside a finite set in N . Since u is bounded by hypothesis and q is bounded, it is clear that p is bounded on N . Since 1 is a P ′ -potential by (iv) the bounded P ′ -superaverage function p is a P ′ -potential. Theorem 4. If (N,P ) is parabolic, then (N,P ′ ) is parahyperbolic. Proof. For let h ′ be a P ′ -average function on N such that |h′ | ≤ M , where M is a constant. Then, |h′ | is P ′ -subaverage on N and hence P -subaverage. Since, by assumption there is no positive P -potential on N , |h| must be a constant thus |h| = c. If c ̸= 0, in |h| = c, |h| is P ′ -subaverage and c is P ′ -superaverage which is a contradiction. Hence c = 0 that is h = 0. Thus 0 is the only bounded P ′ -average function on N . Hence the constant function 1 is a P ′ -potential on N by the Theorem 3. Theorem 5. If (N,P ′ ) is parahyperbolic, then any lower bounded P ′ -superaverage func- tion is non-negative. Conversely, if any lower bounded P ′ -average function is non-negative, then (N,P ′ ) is parahyperbolic. Proof. Let (N,P ′ ) be parahyperbolic. Suppose s is a P ′ -superaverage function on N such that s ≥ −M for some M > 0. Since M is P ′ -potential by assumption , −s ≤ M implies that s ≥ 0. Conversely, suppose any lower bounded P ′ -average function on N is non-negative. If (N,P ′ ) is not parahyperbolic, then by Theorem 4 there exists a P ′ -average function h on N , 0 < h < 1. Since −h is lower bounded, −h ≥ 0 a contradiction. M. Surya Priya, N. Nathiya / Eur. J. Pure Appl. Math, 18 (1) (2025), 5761 9 of 11 Corollary 2. Suppose h is a P ′ -average function bounded on one side in N . If h takes both positive and negative values in N , then there exists a bounded P ′ -average function H, 0 < H < 1, on N , hence N is bounded hyperbolic. 5. Relation between bounded P ′ and P -average functions In a random walk the constant function 1 is P -average on N . The question is: what can we say about the existence of bounded or just positive P -average functions on N that are not constants? We have examples of {N,P} on which there are no non-constant bounded or just positive P -average functions. In this section we try to assert the existence of such functions on {X,P} if similar functions exist on {X,P ′} where P ′ is subordinate to P . If there are non-zero bounded P ′ -average functions on N , then the constant 1 is not a P ′ -potential, hence there are bounded positive P ′ -average functions on N . In this section we investigate the relation between bounded P ′ -average functions and bounded P -average functions on N . Theorem 6. Let (N,P ) be hyperbolic with its Green’s potential Gy(x) satisfying the con- dition supz∈NGz(z) ≤ M . If ∑ x [1 − p ′ (x)] < ∞, then N has bounded positive P ′ -average functions on N . Proof. If 0 is the only bounded positive P ′ -average function on N , then constant 1 is a P ′ -potential in N and 1 = ∑ y [1− p ′ (y)]G ′ y(x) for x ∈ N ≤ ∑ y [1− p ′ (y)]Gy(x) ≤ ∑ y [1− p ′ (y)]Gy(y) ≤M ∑ y [1− p ′ (y)] <∞. Hence u(x) = ∑ y[1− p ′ (y)]Gy(x) should be a P -potential. But this is not possible since u(x) maximizes the P -average function 1. Theorem 7. Let B (respectively B ′ ) be the set of all bounded non-negative P -(respectively P ′ -) average functions in N . Then there is an injective map S : B ′ → B such that S(α1h1 + α2h2) = α1S(h1) + α2S(h2) where α1, α2 are non-negative constants and h1, h2 are in B ′ . M. Surya Priya, N. Nathiya / Eur. J. Pure Appl. Math, 18 (1) (2025), 5761 10 of 11 Proof. Let h ∈ B ′ . Then h is a bounded P -subaverage function. Let S(h) be the least P -average majorant of h. Then S(α1h1+α2h2) = α1Sh1+α2Sh2. Suppose S(h1) = S(h2). Note that for h ∈ B ′ , S(h) − h is a P -potential and hence a P ′ -potential. Consequently, if S(h1) = S(h2), then |h1 − h2| = |[S(h1)− h1]− [S(h2)− h2]| ≤ p1 + p2 where p1 and p2 are P ′ -potential on N . since |h1 − h2| is P ′ -subaverage function on N , h1 = h2. Corollary 3. If there are non-proportional bounded non-negative P ′ -average functions in N , then there is atleast one non-constant bounded P -average function in N . Proof. If h1 and h2 are non-proportional in B ′ , then S(h1) and S(h2) are non- proportional bounded P -average functions in N . Hence atleast one of them is non- constant. Lemma 3. Let h be a P ′ -average function in N , such that |h| ≤ s where s is P ′ - superaverage on N . Then h = h1 − h2 where h1 and h2 are non-negative P ′ -average functions such that h1 − h+ and h2 − h− are P ′ -potentials. This decomposition is unique. Proof. Let h1 be the least P ′ -average majorant of h+ and h2 be the least P ′ -average majorant of h−. Then p1 = h1 − h+ and p2 = h2 − h− are P ′ -potentials on N . Hence h = h+ − h− = (h1 − h2) − (p1 − p2). Then by the uniqueness of Riesz decomposition, h = h1 − h2 on N . Suppose h = u1 − u2 is another such decomposition. Since u1 − h+ and h1 − h+ are potentials, so u1 = h1 and then u2 = h2. Theorem 8. If there exists a bounded P ′ -average function on N that takes both positive and negative values, then there is atleast one bounded non-constant P -average function on N . Proof. 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