A non-local quasi-linear ground state representation and criticality theory

We study energy functionals associated with quasi-linear Schrödinger operators on infinite weighted graphs, and develop a ground state representation. Using the representation, we develop a criticality theory, and show characterisations for a Hardy inequality to hold true. As an application, we show a Liouville comparison principle.


Introduction
The quasi-linear p-Laplacian on Riemannian manifolds, and especially in the Euclidean space, is one of the best studied non-linear local elliptic operators.There are many beautiful monographs related to this operator, see e.g.[ADV04; BEL15; DKN97; HKM06; Lin19], or in general metric spaces, see [BB11].
Here, we study an extended class of operators including the p-Laplacian as a special case, i.e., p-Schrödinger operators.Moreover, we study this class on locally summable weighted graphs.
Recently, the potential theory of local p-Schrödinger operators with not necessarily non-negative potential term was studied more closely, see e.g.[BEL15; DP16; HPR21; PP16; PR15; PT09; PTT08].In this theory, the ground state representation is of fundamental importance.This representation is an equivalence between functionals.It states that the p-energy functional associated with the p-Schrödinger operator is equivalent to a simplified energy functional consisting of non-negative terms only.
For non-local p-Schrödinger operators in the Euclidean space, which includes graphs as a special case, a one-sided inequality for p ≥ 2 is given in [FS08].
Here, we show a ground state representation for non-local p-Schrödinger operators for all p > 1 in terms of an equivalence between the corresponding p-energy functional and the simplified energy.We show this statement on graphs, see Theorem 3.1 and Corollary 3.2.Moreover, we briefly show the corresponding result for non-local p-Schrödinger operators on R d , see Theorem 4.2, and get as a consequence an improvement of a result in [FS08].
The ground state representation is an essential tool in criticality theory (which is sometimes also called parabolic theory).A p-energy functional is called critical if it is non-negative and the p-Hardy inequality does not hold.In the continuum, there are many characterisations of criticality known, see e.g.[HPR21;PP16].
Criticality theory on almost locally finite graphs for the standard p-Laplacian was studied in [SY93], see [Pra04] for locally finite graphs.Both show a connection between positive harmonic functions and the variational p-capacity.
In this paper, we establish a characterisation of criticality for p-Schrödinger operators in terms of null-sequences, the variational p-capacity, as well as positive harmonic functions, see Theorem 5.1.This will be achieved mainly by the aid of the ground state representation.These characterisations are the discrete counterpart to results in [HPR21;PP16].
Moreover, we show a Liouville comparison principle which is a discrete analogue to results in [PR15;PTT08].It is another application of the ground state representation and gives the criticality of an energy functional if a subharmonic function can be estimated properly by the ground state of another energy functional.
In the local and linear case, ground state representations are classical and have shown their powerfulness in many applications, see [FS08, Section 1] for a list of applications with references and more details.
This paper is organised as follows: In Section 2, we briefly introduce the basic notation and show connection between the p-Schrödinger operators and p-energy functionals via a Green's formula.Then, we turn in Section 3 to the main results of this paper, Theorem 3.1 and Corollary 3.2.After we stated the results, we discuss them in detail in Subsection 3.2.This includes a comparison with the local case in the continuum.The proofs of Theorem 3.1 and Corollary 3.2 are then divided into two parts: The proof of an elementary equivalence in Subsection 3.3, and then the application of this equivalence in Subsection 3.4.Thereafter, we show in Section 5 how this ground state representation can be used to prove some characterisations of criticality.Here, also one part of an Agmon-Allegretto-Piepenbrink-type theorem is needed, as well as a local p-Harnack inequality.This is part of Subsection 5.1.We end this paper with two simple applications of Theorem 3.1 and Theorem 5.1, one of them is a Liouville comparison principle.

Setting the Scene
In this section, we start by introducing graphs.Thereafter, we define quasi-linear Schrödinger operators on graphs.We end this section by introducing p-energy functionals and showing a connection to p-Schrödinger operators via Green's formula.
2.1.Graphs and Schrödinger Operators.Let an infinite set X equipped with the discrete topology and a symmetric function b : X × X → [0, ∞) with zero diagonal be given such that b is locally summable, i.e., the vertex degree satisfies We refer to b as a graph over X and elements of X are called vertices.Two vertices x, y are called connected with respect to the graph b if b(x, y) > 0, in terms x ∼ y.A subset V ⊆ X is called connected with respect to b, if for every two vertices x, y ∈ V there is a vertices x 0 , . . ., x n ∈ V , such that x = x 0 , y = x n and Throughout this paper we will always assume that X is connected with respect to the graph b.We now turn to functions: Let S be some arbitrary set.A function f : S → R is called non-negative, positive, or strictly positive on I ⊆ S, if f ≥ 0, f 0, f > 0 on I, respectively.If for two non-negative functions and call them equivalent on I.
The space of real valued functions on V ⊆ X is denoted by C(V ) and is a subspace of C(X) by extending the functions of C(V ) by zero on X \ V .The space of functions with compact support in V is denoted by C c (V ).
A strictly positive function m ∈ C(X) extends to a measure with full support via m The next fundamental definition is the one of the p-Laplacian.But first, we have to introduce some notation.For showing the connection to the counterpart in the continuum, we introduce the difference operator ∇ on C(X) via x, y ∈ X.
Let p ∈ [1, ∞).For V ⊆ X, let the formal space F (V ) = F b,p (V ) be given by If V = X we write F = F (X).For 1 < p < 2 we make the convention that |t| p−2 t = 0 if t = 0, i.e., 0 • ∞ = 0.Then, we can write for all p ≥ 1, Here, sgn : R → {−1, 0, 1} is the sign function, that is sgn(t) = 1 for all t > 0, sgn(t) = −1 for all t < 0, and sgn(0) = 0. We remark that F (V ) = C(X) if p = 1, by the local summability assumption on the graph.Next, we show a basic lemma, which states an alternative representation for the formal space.There, we need the following elementary inequality: we have for all p ≥ 0 that Proof.The case p = 1 is trivial.Let p > 1, and denote the set on the right-hand side by F (V ).We obviously have that C c (V ) ⊆ ℓ ∞ (V ) ⊆ F (V ).Furthermore, let f ∈ F (V ).Then, using the elementary inequality (2.1), we get for any The first sum on the right-hand side is finite by the local summability property of the graph b.The second sum is finite since Now, we are in a position to define the Laplacian: Let m be a measure on X.
Remark 2.2.Following [Mug13; Pra04; Tak03], there is the following analogy to p-Laplacians in the continuum: A vector field v is a function in C(X × X) such that v(x, y) = −v(y, x), x, y ∈ X.Moreover, define div on the space of absolutely summable vector fields in the second entry via Then, for all f ∈ F , and p ≥ 1, This shows that our Laplacian is a discrete analogue to weighted Laplace-Beltramitype operators on manifolds.
Finally, we can define Schrödinger operators as follows: Let c ∈ C(X).Then the (p-)Schrödinger operator The function c is then usually called the potential of H.If c is non-negative, then H is called p-Laplace-type operator.
A function u ∈ F (V ) is said to be harmonic, (superharmonic, subharmonic) on V ⊆ X with respect to H if If V = X we only speak of super-/sub-/harmonic functions.

Energy Functionals Associated with Graphs.
Let D = D b,c,p be given by

Then, the (p-)energy functional
If p = 2, then the energy functional is a quadratic form, and called Schrödinger form.
As in the continuum or the linear case on graphs, there exists a so-called Green's formula which shows a connection between H and h on C c (X).The Green's formula seems to be folklore in both worlds.However, for the convenience of the reader we include a proof here.A similar proof of the Green's formula for the normalised p-Laplacian, that is m = deg and c = 0, is given in [Tak03].
Then, all of the following sums converge absolutely and In particular, the formula can be applied to f ∈ C c (X), or f ∈ D, and Proof.Since ϕ ∈ C c (X), the absolute convergence follows from for any f ∈ F (V ).Applying Fubini's theorem, using the absolute convergence of the sums and the symmetry of b, we get The assertions for the Schrödinger operator H follow now easily.By Lemma 2.1, C c (X) ⊆ F (V ).Note that F (X) ⊆ F (V ).It remains to show that D ⊆ F (X).This follows from Hölder's inequality for all x ∈ X by This ends the proof.

The Ground State Representation on Graphs
In the classical linear case, ground state representations are transformations which use a superharmonic function to turn a quadratic energy form associated with a linear Schrödinger operator into a quadratic energy form associated with a linear Laplace operator, see e.g.[KPP20, Proposition 4.8] for such a statement on graphs, and e.g.[Dav89, p. 109] for a counterpart in the continuum.
In the non-linear (p = 2)-case, we do not have an equality via a transformation between functionals anymore.But instead, we achieve an equivalence between functionals, providing that a positive superharmonic function exists.The equivalent functional has the property that it consists of non-negative terms only.
Our representations in Theorem 3.1 and Corollary 3.2 can be seen as the nonlocal analogues to the local and non-linear representations in [PR15; PTT08], where p-Schrödinger operators on domains in R d are discussed.Furthermore, we briefly show the representation for weighted non-local p-Schrödinger operators in R d in the next section, Section 4.
First applications of our representations are given in Section 5.Moreover, other applications can be found in the follow-up papers [Fis22a;Fis22b].
3.1.The Statement.Let p > 1, and 0 ≤ u ∈ F (V ) for some V ⊆ X.The simplified energy (functional) h u of h with respect to u on C c (V ) be given by where we set 0 We state now the main result of this paper.
Theorem 3.1 (Ground state representation).Let p > 1 and 0 ≤ u ∈ F (V ) for some V ⊆ X.Then, we have Furthermore, the equivalence becomes an equality if p = 2.
In many applications the function u is assumed to be harmonic in V ⊆ X.In this case the representation in (3.1) reduces to A further consequence of (3.1) is, that the corresponding left-hand side is nonnegative, i.e, h(uϕ) ≥ Hu, u |ϕ| p , ϕ ∈ C c (V ).This inequality is known as Picone's inequality, see [AH98; AM16; Amg08; BF14; Fis22b; FS08; PKC09; Pic10; PTT08] for applications of this inequality in various contexts.
From the inequalities in Theorem 3.1, we get as consequences estimates between the energy associated with the Schrödinger operator and other functionals, which are usually also referred to as simplified energies (see e.g.[DP16;PTT08]).They all are called simplified, because they consist of non-negative terms only, and the difference operator ∇ applies either to u or ϕ but not to the product u • ϕ.
We set on C c (V ), and for p ≥ 2, we define on C c (V ) The following corollary is an immediate consequence of Theorem 3.1.
and if p ≥ 2 the reversed inequality in (3.2) holds true, i.e., The statements in Theorem 3.1 and Corollary 3.2 will follow mainly by pointwise inequalities without summation.Then, we will sum over X × X and use Green's formula to obtain the results.The elementary inequalities are basically given in the upcoming lemma, Lemma 3.8.
The proof does not include the case p = 1.This is because we use a quantification of the strict convexity of the mapping x → |x| p , p > 1.

Some Remarks on the Main Result.
Remark 3.3 (Comparison with the local non-linear analogue).We compare our ground state representation with results in [PTT08].Similar results associated with weighted p-Schrödinger operators can be found in [PR15].
Fix p ∈ (1, ∞) and a domain Ω ⊆ R d .Let u ∈ W 1,p loc (Ω) and ∆(u) := − div(|∇u| p−2 ∇u) be the p-Laplacian on Ω.Furthermore, let V ∈ L ∞ loc (Ω).The corresponding energy functional to the Schrödinger operator ∆ + V is given by Then, by [PTT08, Lemma 2.2], we have the following: If u is a positive harmonic function of ∆ + V in the weak sense, i.e., Ω |∇u| p−2 ∇u • ∇ϕ In particular, for p > 2, we have In the case of 1 < p < 2, we have by [PTT08, Remark 1.12] that Now, we do the comparison: In the continuum, domains of R d are considered.On graphs, we can take any subset of the graph.
Recall that u is harmonic.It is very easy to compare h u (ϕ) with the right-hand side in (3.5), see Table 1.
Table 1.Comparison of the terms in the right-hand side (RHS) of (3.5) with h u (ϕ).

RHS of (3.5)
This motivates to call the simplified energy h u the analogue to the simplified energy in the local non-linear case.Note that in the continuum, we only consider non-negative compactly supported functions ϕ, whereas on graphs, we allow ϕ to take negative values.Thus, the version in the continuum contains hidden moduli of ϕ.
Furthermore, we see that the equivalence (3.6) has the same structure as the equivalence (3.4).For a comparison of h u,1 (ϕ) + h u,2 (ϕ) with the right-hand side in (3.6) see Table 2.

RHS of (3.6)
Furthermore, we see that the estimate in (3.7) together with (3.5) has the same structure as the upper bound (3.2).
It should be mentioned that the strategy to prove the ground state representation in [PTT08] and here are similar.There, an elementary equivalence is the key ingredient and then a Picone identity is used.Here, we use different elementary equivalences and the Green's formula.However, the proof of the elementary equivalences in the discrete is technically much harder as the proof of the corresponding one in the continuum.Thus, the differences above might come from the fact that in the continuum we have a Picone identity (see [PTT08, Section 2]) which is established via the chain rule.Whereas in the discrete, we only have a one-sided Picone inequality and the missing of a chain rule in general.A general version of this one-sided Picone inequality is discussed in a follow-up paper by the author [Fis22b], see also [BF14].
Moreover, in [PTT08, Proposition 5.1] it was shown that for p > 2 both summands in the integral in (3.6) are needed in general for an upper bound.We expect that the same holds true on graphs, i.e., we expect that both h u,1 and h u,2 are needed in general for an upper bound of h.Remark 3.4 (Discussion of the constants).By comparing Theorem 3.1 with [FS08, Proposition 2.3] and Lemma 3.8 (the lemma below) with [FS08, Lemma 2.6], we see that c p in (3.3) can be stated explicitly as a minimiser, i.e., for p ≥ 2 Note that c 2 = 1/2.Moreover, we expect that the best constants in Theorem 3.1 are between 0 and 1.

Remark 3.5 (Hardy inequality)
. In [DP16] the non-linear ground state representation of [PTT08] was used to prove optimality of certain p-Hardy weights associated with p-Schrödinger operators on domains in R d .The discrete counterpart does hold as well using the here presented discrete ground state representation and are topic of a follow-up paper by the author, [Fis22a].This generalises the results of the linear case in [KPP18] to p = 2.A consequence of the ground state representation and some results in this yet unpublished paper is that the improved p-Hardy inequality on N in [FKP19] is indeed optimal.The standard (or combinatorial) p-Laplacian ∆ for real valued functions on for all functions f ∈ C(N) and n ≥ 1.The corresponding energy functional reads then as /p is a positive superharmonic function such that ∆u = wu p−1 , where w is the improved p-Hardy weight in [FKP19, Theorem 1].Let q := p/(p − 1) and Then, the equivalence (3.1) reads as follows: for all ϕ ∈ C c (N), we have If p = 2, then the equivalence is an equality and gives exactly the result of [KS21, Theorem 1].Moreover, the inequality . By (3.2), the reversed inequality holds for 1 < p ≤ 2.

Elementary Inequalities and Equivalences.
We need the following quantification of the strict convexity of the mapping x → |x| p , p > 1.In the following lemma, •, • R n denotes the standard inner product in R n .
In the previous lemma, the constant c p does not seem to be optimal.However, this is not important for our further investigations.
The next lemma is the most important tool in order to derive the ground state representations, Theorem 3.1 and Corollary 3.2.Lemma 3.8 (Fundamental inequalities and equivalences).Let a ∈ R, 0 ≤ t ≤ 1, and p > 1.Then we have where the right-hand side is understood to be zero if 1 < p < 2 and a = t = 1.

Moreover, we have
where the right-hand side is an upper bound with optimal constant c = 2, and it is a lower bound with optimal constant c = 1/2.Furthermore, if 1 < p ≤ 2, then and for p ≥ 2, the reserved inequality holds, i.e., Moreover, we have the following refinement of the elementary inequality (2.1): for all p ≥ 0, we have where the right-hand side is an upper bound with optimal constant c p = 2 1−p if 0 ≤ p ≤ 1 and c p = 1 if p ≥ 1, and it is a lower bound with optimal constant c p = 1 for 0 ≤ p ≤ 1 and c p = 2 1−p for p ≥ 1.
We do not claim that the constants we get in (3.8) are optimal.We expect that they can be improved and that the best constants should be either on the boundary of [0, 1] × R, or at (t, 0), (t, t), (t, 1), t ∈ [0, 1].Moreover, we expect that the optimal constants are between 0 and 2. Also note that the inequalities (3.8) and (3.11) show that we improved an elementary one-sided result in [FS08] for p > 2.
Moreover, in the case of 1 < p < 2, the "≥"-inequality in (3.8) was proven in [AM16, Lemma 3.3].However, the basic strategy to prove the remaining inequalities in (3.8) up to a certain point will be the similar, i.e., we start the proof with the same substitution and then use the same Taylor-Maclaurin formula (confer this also with the proof of [Lin90, Lemma 4.2]).
Furthermore, note that (3.8) is false for p = 1 as the left-hand side vanishes for a > 1 ≥ t > 0 but the right-hand side does not.A similar argument can also be made for (3.10).
Proof of Lemma 3.8.Ad (3.8):Recall that we have to show that for p > 1, The strategy of the proof is as follows: We start with some simple special cases for which the equivalence can be shown very easily.Thereafter, we do a substitution to bring the equivalence in a simpler form for the remaining cases.Then, we divide R into the three intervals [1, +∞), (t, 1), and (−∞, t] for some t ∈ [0, 1].In the two intervals [1, +∞) and (−∞, t], we then distinguish between proving lower bounds and upper bounds, as well as having p > 2 or 1 < p < 2. In the remaining interval (t, 1), we show that we can deduce the equivalence from the validity of the equivalence in [1, +∞).
1.The three cases t ∈ {0, 1}, a = t, and p = 2: If p = 2, then it is obvious that we have equality for all a ∈ R and t ∈ [0, 1].
An easy computation shows that we have indeed equality for t ∈ {0, 1}.
2. The remaining cases t ∈ (0, 1), a = t, and p = 2: We do the following substitution: Set α := (a − t)/(1 − t), then we have to show that We will do this, by considering the following three cases separately 2.1.The case α ≥ 1: The basic strategy is to use the Taylor-Maclaurin formula.Thus, let us calculate the first and the second derivatives with respect to t.Note that for α ≥ 1, we have |α where which is positive for p > 2 and negative for 1 < p < 2. Hence, g(0) = 2 is a minimum for p > 2 and a maximum for 1 < p < 2. This implies that for all t ∈ (0, 1) Now, we apply the Taylor-Maclaurin formula Since f α (0) = α p , we have (3.15)This term will be analysed in the following for upper and lower bounds and different values of p.

Upper bound for
Using this estimate in the left-hand side of (3.13), we get Since for all k ∈ 2N, 1 ≤ p ≤ 2 and α ≥ 1, we have we get for all 1 ≤ p ≤ 2 and α ≥ 1, This results in the right-hand side of (3.13) with constant 1. 2.1.4.Lower bound for p > 2 and α ≥ 1: For p ≥ 2, the function Using this estimate in the left-hand side of (3.13), we get Define for α ≥ 1, and some constant C p > 0, Thus, for p ≥ 3, Applying (3.18) and (3.19) to (3.17), results in the right-hand side of (3.13).
Moreover, this was the last puzzle stone to show (3.13) for α ≥ 1 and all 1 < p < ∞.
2.3.The case α < 0: Set β := −α.Then, substituting into (3.13),we have to show that for all β > 0 and t ∈ (0, 1), We have where Before we continue with the estimates, let us note that g t ≥ 0 and g ′ t ≥ 0. The first inequality can be seen as follows: let γ > 0. Firstly assume that γ > t.Then, Secondly, if γ ≤ t, then a similar calculation can be done to get the desired inequality (factor t out of the sum and use the binomial theorem).
Note that for all p ≥ 1, Now we continue with showing (3.20):By the first parts of the proof, i.e., the proof of (3.13), we have that for all β > 0, The strategy for the upper bound will be as follows: Clearly, (β − 1) 2 ≤ (β + 1) 2 for all β > 0. If we apply this estimate to (3.21), we are left to show that also for some positive constant C p in order to show the upper bound in (3.20).
For the lower bound, we are left to discuss the compact interval [1 − ε, 1 + ε].On this interval, we clearly have (β + 1) p ≍ 1.The equivalence (3.21) shows in particular that the corresponding left-hand side is positive.Thus, we are left to show that there exists C p,ε > 0 such that
1.The cases t ∈ {0, 1} and a = t: For t = 0, we have which is non-negative for C ≥ 2 and strictly negative for C < 2. If t = 1, then which is non-negative for C ≥ 1 and strictly negative for which is non-negative for C ≥ 2, non-positive for C ≤ 1/2 and changes sign from negative to positive as t increases in 1/2 < C < 2. Hence, it is easy to see that f •,C (•) changes sign for any 1/2 < C < 2 and an appropriate choice of t by evaluating f t,C at 0, t and 1.
Hence, f t,C has two extrema, one at a = 0 and one at a = t.If C ≥ 2, the extrema are minima, and if 0 ≤ C ≤ 1/2, the extrema are maxima.By the computations in the first case and since which is non-negative for C ≥ 2 and non-positive for 0 ≤ C ≤ 1/2, it follows that f t,C is non-negative if C ≥ 2 and non-positive if 0 ≤ C ≤ 1/2 for all t ∈ [0, 1] and we have shown that the right-hand side in (3.9) is an upper bound for every C ≥ 2, and lower bound if 0 ≤ C ≤ 1/2.Ad (3.10) and (3.11):We will show these inequalities similarly as we showed (3.8).Recall that we have to show that and Note that the inequalities basically come from the fact that for t ∈ [0, 1], we have t p/2 ≥ t for 1 < p ≤ 2, whereas t p/2 ≤ t for p ≥ 2.Here are the details: 1.The three cases t ∈ {0, 1}, a = t, and p = 2: If p = 2, then it is obvious that we have equality for all a ∈ R and t ∈ [0, 1].
An easy computation shows that we indeed have equality for t ∈ {0, 1}.If a = t, then note that t ∈ [0, 1] implies t p/2 ≥ t for 1 < p ≤ 2, and t p/2 ≤ t for p ≥ 2. This immediately yields the desired inequalities.
2. The remaining cases t ∈ (0, 1), a = t, and p = 2: We consider the cases a > t and a < t separately.
2.2.The case a < t: Note that a < t < 1.Thus, we have to show that as well as Since t ≤ t p/2 for 1 < p < 2 and t ≥ t p/2 for p > 2, we get the desired result.

Proof of the Ground State Representation Theorem.
Proof of Theorem 3.1.
By a symmetry argument, we also get for all x, y ∈ V ∪ ∂V such that u(y) ≥ u(x) > 0, Note that by Green's formula, Lemma 2.3 for the p-Laplacian L, Summing over all x, y ∈ X with respect to b and using the calculation above yields then in (3.1).Now, we can directly continue with proving Corollary 3.2.
Alternatively, one can also deduce (3.2) and (3.3) from (3.11), (3.10) and (3.8).The proof can then be mimicked from the proof of Theorem 3.1 and results in better constants.Remark 3.9 (Another Equivalence).Applying only (3.8) in the proof of Theorem 3.1 yields the following equivalence: Let 0 ≤ u ∈ F (V ), and ϕ ∈ C c (V ) for some V ⊆ X.We set for all x, y ∈ X Moreover, let the functional h u of h with respect to u on C c (V ) be given by where we again set 0 • ∞ = 0.Then, we have with equality if p = 2.

The Representation For Non-Local Operators in the Euclidean Space
The statement for graphs can be transferred to non-local p-Schrödinger operators in the flavour of [FS08].This is because the main part of the proof of the representation comes from an elementary equivalence, which does not use any knowledge of the underlying space for the corresponding energy functional.A brief comparison is given in [FS08, Remark 2.4].
To be more specific, the corresponding statement for weighted non-local p-Schrödinger operators is as follows: Fix d ≥ 1, p > 1, a non-negative symmetric and measurable function b on R d × R d and a measurable function c on R d .Set In the following, technical assumptions are needed to circumvent a regularisation of principle value type, confer [FS08, Section 2].
Assumption 4.1.Let p > 1. Assume that there exists a family of symmetric and measurable functions (b ε ) ε>0 , on R d × R d and a family of measurable functions Moreover, let u be a positive, measurable function on R d , such that the integrals are absolutely convergent for a.e.x ∈ R d , belong to L 1,loc (R d ) and the limit Hu := lim ε→0 (H ε u) exists weakly in L 1,loc (R d ).
Now we state the main result of this section, the ground state representation formula.
Theorem 4.2 (Ground state representation).Let Assumption 4.1 be fulfilled.Then, whenever E(uϕ), E u (ϕ), and u(Hu) + |ϕ| p dx are finite, we have The proof of Theorem 4.2 goes along the lines of the proof of [FS08, Proposition 2.2] doing the necessary changes.
Proof of Theorem 4.2.We can assume that ϕ is bounded.Otherwise, we replace ϕ by the function ϕ min(1, n |ϕ −1 |) and let n → ∞ using monotone convergence.Let denote by L ε the Laplacian-part of H ε .By our assumptions, we have Now, the proof for ε > 0 is exactly as the proof of Theorem 3.1.Note that the constant in the equivalence comes from a pointwise equivalence and does only depend on p (and not on ε).By the assumptions, we have weak convergence for the integral containing (H ε u) and dominated convergence for the remaining integral.Taking the limit yields (4.1).

Set
In analogy to Corollary 3.2 we have the following result.
Proof.The proof is similar to the proof of Corollary 3.2 and therefore omitted.
In [FS08], only the estimate in (4.2) in the case of p ≥ 2 is given.Note that choosing b(x, y) = |x − y| −d−ps , 0 < s < 1, x, y ∈ R d , results in the classical fractional p-Hardy inequality.
Moreover, note that for Sobolev-Bregman forms associated with the fractional p-Laplacians, another ground state alternative was found recently, see [Bog+21].

Characterisations of Criticality
In this section, we will discuss the notion of criticality.For the history of this notion see [Pin07, Remark 2.7] or [KPP20, Section 5].There it is stated that in the continuum the notion goes back to [Sim80] and was then generalised in [Mur86;Pin88].On locally summable weighted graphs, [KPP20] is the first paper discussing criticality in the context of linear Schrödinger operators.See also [KLW21, Chapter 6] (and references therein) for corresponding results for linear Laplace-type operators on graphs.
Non-negative energy functionals associated with Schrödinger operators seem to divide naturally into two categories: the ones which are strictly positive, i.e., for which a Hardy inequality holds true, and the ones which are not strictly positive, i.e., for which the Hardy inequality does not hold.In the linear (p = 2)-case, there are surprisingly many equivalent formulations to the statement that the Hardy inequality does (not) hold, for graphs confer [KPP20].For p = 2, c = 0 and m = deg, this is exactly the division of graphs into transient and recurrent graphs.
Using our recently developed ground state representation, we will see that many of the characterisations in [KPP20] remain characterisations also if p = 2.
Let h be a functional which is non-negative on C c (V ), V ⊆ X.Then, h is called subcritical in V if the Hardy inequality holds true in V , that is, there exists a positive function w ∈ C(V ) such that If such a positive w does not exist, then Other names for a subcritical functional are sometimes strictly positive, coercive, or hyperbolic.
Before we can state the main result of this section, we need the following definition: A sequence (e n ) in C c (V ), V ⊆ X, of non-negative functions is called null-sequence in V if there exists o ∈ V and α > 0 such that e n (o) = α and h(e n ) → 0.
Moreover, we define the variational capacity of h(ϕ).
Theorem 5.1 (Characterisations of criticality).Let p > 1.Furthermore, assume that there exists a positive superharmonic function in X.Then the following statements are equivalent: (i) h is critical in X.
(ii) For any o ∈ X and α > 0 there is a null-sequence (e n ) in X such that e n (o) = α, n ∈ N. (iii) cap h (x, X) = 0 for all x ∈ X. (iv) There exists a strictly positive harmonic function u in X such that h u is critical in X. (v) For all positive harmonic functions u in X, the ground state representation h u is critical in X. (vi) For any positive superharmonic function u ∈ F (X) in X and any nullsequence (e n ) in X there exists a positive constant c such that e n (x) → c u(x) for all x ∈ X as n → ∞. (vii) There exists a strictly positive harmonic function u ∈ F (X) in X and a null-sequence (e n ) in X such that e n (x) → u(x) for all x ∈ X as n → ∞.
If p ≥ 2, the sequence can be chosen such that 0 ≤ e n ≤ u for all n ∈ N. In particular, if one of the equivalent statements above is fulfilled, then there exists a unique positive superharmonic function in X (up to linear dependence) and this function is strictly positive and harmonic in X.
We remark that in the continuum, the corresponding characterisation holds true on any subdomain of R d , confer [PP16, Theorem 4.15].In Proposition 5.6, we show that h cannot be critical in V whenever V X, assumed that a positive superharmonic function on V exists.
We divide the proof of this main theorem into two subsections.In the first subsection, we show some more general auxiliary lemmata, and in the second subsection, we show the equivalences.5.1.Preliminaries.We start with a direct consequence of the ground state representation.
Lemma 5.2.Let p > 1 and V ⊆ X. Assume that there exists a function u ∈ F (V ) which is strictly positive in V and superharmonic on V .Then, we have In particular, h is non-negative on C c (V ).
Proof.By the ground state representation, Theorem 3.1, the statement follows easily since for some c p > 0, Note that the reversed statement in Lemma 5.2 is also true which is proven in a follow-up paper by the author [Fis22b].The corresponding statement is known as an Agmon-Allegretto-Piepenbrink-type theorem (see [All74;Pie74;BLS09] for a linear version in the continuum, [Dod84; KPP20] for a linear version in the discrete setting, [PP16] for a recent non-linear version in the continuum, and [LSV09] for a corresponding result on strongly local Dirichlet forms).
An immediate consequence of the definition of criticality is the following statement.
Lemma 5.3.Let p > 1. Assume that there exists a strictly positive and superharmonic function u ∈ F (X).Let h be critical in X.Then any strictly positive and superharmonic function u ∈ F (X) is harmonic on X.
Proof.Let u be such a strictly positive superharmonic function.By Lemma 5.2 we have h(ϕ) ≥ u 1−p Hu, |ϕ| p for all ϕ ∈ C c (X).Because h is critical we get that u 1−p Hu = 0 on X, and thus, u is harmonic on X.
Next we show that locally, i.e., on finite and connected sets, our graphs fulfil a so-called Harnack inequality for non-negative supersolutions.This inequality implies that non-negative supersolutions are either strictly positive or the zero function, and they do not tend to infinity in the interior of the graph.
There is a long list of proofs of various Harnack-type inequalities for the p-Laplacian, see for instance for metric spaces [BB11,Theorem 8.12] where a p-Poincaré inequality is assumed.The corresponding analogue for linear Schrödinger operators on locally summable graphs can be found in [KPP20].The basic idea of the following proof of the local Harnack inequality can also be found in [Pra04], where the standard p-Laplacian on locally finite graphs without potential (i.e., c = 0) is considered, and [HS97a], where the standard p-Laplacian on finite graphs without potential, is considered.Lemma 5.4 (Local Harnack inequality).Let p > 1, K ⊆ X be connected and finite, and f ∈ C(X).Then there exists a positive constant C K,H,f such that for any u ∈ F (K) which is non-negative on K ∪ ∂K and satisfies Hu ≥ f u p−1 on K, we have max The constant C K,H,f can chosen to be monotonous in the sense that if f ≤ g ∈ C(X) then C K,H,f ≥ C K,H,g .Specifically, for any x ∈ K we have for all y ∼ x.In particular, we have deg +c − f m ≥ 0 on K. Furthermore, if u(x) = 0 for some x ∈ K, then u(x) = 0 for all x ∈ K ∪ ∂K.Moreover, if V ⊆ X is connected, then any function which is positive on V ∪ ∂V and superharmonic on V is strictly positive on V .
Proof.Let K ⊆ X be finite and connected and let u ∈ F (K) such that u ≥ 0 on X and Hu ≥ f u p−1 on K for some f ∈ C(X).
If u(x 0 ) = 0 for some x 0 ∈ K, then we have 0 Thus, for all x ∼ x 0 , we have u(x) = 0 and since K is connected we infer by induction that u(x) = 0 for all x ∈ K ∪ ∂K.
Hence, we can assume that u > 0 on K.Because of Hu ≥ f u p−1 , we have for any x ∈ K. Furthermore, we deduce for every x ∈ K The previous calculation implies that d f ≥ 0 on K. Now assume that there is y 0 ∼ x such that u(x) ≤ u(y 0 ).Then the previous calculation also implies Hence, for all y 1 ∼ x we have Since K is finite there exists a minimum and a maximum of u in K. Let x min and x max denote the corresponding points, respectively.Moreover, let x 0 ∼ x 1 ∼ . . .∼ x n be a path in K such that x 0 = x min and x n = x max .Then we derive Again since K is finite there exists only a finite number of paths in K between x min and x max with different vertices.Hence the minimum of the product on the righthand-side exists.This minimum we denote by We still have to show the last assertion.To do so, let s ∈ F (V ) be a positive superharmonic function on V .Then there exists o ∈ X such that s(o) > 0. Since X is connected there exists a finite path from any x ∈ X to o. Denoting this path by K, we can apply the first part of this lemma and get that s(x) > 0. Since x was arbitrary, we conclude that s > 0 on V .
The following lemma is the discrete analogue of [PP16, Proposition 4.11].
Lemma 5.5.Let p > 1 and assume that there exists a positive superharmonic function u ∈ F .Furthermore, assume that (e n ) is a null-sequence in X such that e n (o) = α for some o ∈ X and α > 0.Then, e n → (α/u(o))u pointwise on X as n → ∞.In particular, for all (x, y) ∈ X × X we have ∇ x,y (e n /u) → 0 as n → ∞.
Proof.By the Harnack inequality, Lemma 5.4, any positive superharmonic function u in X is strictly positive in X.
Let o ∈ X and α > 0 be arbitrary.Set ϕ n := e n /u.Then, by the ground state representation, Theorem 3.1, Firstly, let p ≥ 2. Then the equivalence implies |∇ x,y ϕ n | → 0 for all x, y ∈ X, x ∼ y.Since X is connected, we have for any x ∈ X a k ∈ N such that x = x 1 ∼ . . .∼ x k = o.Thus, we obtain Rearranging, yields e n → (α/u(o))u pointwise on X as n → ∞.
Secondly, let 1 < p < 2. Then the equivalence implies either for each (x, y) ∈ X × X, x ∼ y.We show that (2) and (3) cannot apply: Using the triangle inequality, it is easy to see that (2) and (3) are equivalent for the pair (x, o) with x ∼ o.They are also equivalent to e Then using Hölder's inequality with p = 2/p > 1, and q = 2/(2 − p), we calculate , where c i (p), i ≤ 4, are positive constants depending only on p (and not on n).Since b(x, o), u(x), u(o) are also independent of n and strictly positive, we can rewrite the inequality above as and (e n (x)) does not converge to ∞ for all x ∼ o.Hence, (2) and (3) cannot apply for all x ∼ o, and only (1) holds true for all x ∼ o.Thus, we can continue as in the case p ≥ 2 to get that e n (x) → (α/u(o))u(x) for all x ∼ o.
Arguing similarly, we have for all y ∼ x ∼ o that p) for some positive constants C i (p), i ≤ 3. Thus, as before, (2) and (3) cannot apply for all y ∼ x ∼ o which results in e n (y) → (α/u(o))u(y).Since X is connected, we get by induction that e n (y) → (α/u(o))u(y) for all y ∈ X.This proofs the statement for 1 < p < 2.

Proof of the Characterisations of Criticality.
Here, we proof Theorem 5.1.We show the equivalences in the following order: (i) ⇐⇒ (ii) ⇐⇒ (iii), and (i) ⇐⇒ (iv) ⇐⇒ (v), and under the assumption V = X, (ii) =⇒ (vi), and ((i) & (ii) & (vi)) =⇒ (vii) =⇒ (i).From (vii), we deduce the last assertion of the theorem.The preamble ensures the existence of a positive superharmonic function u.By the the Harnack inequality, Lemma 5.4, the function u is a strictly positive superharmonic function in X.By Lemma 5.3, the criticality of h in X implies that any strictly positive superharmonic in X is a strictly positive harmonic function in X.
By (vi), any null-sequence converges to a constant multiple of u.The existence of a null-sequence is ensured by (ii).This shows the first part.
Ad (vii) =⇒ (i): By Lemma 5.2, 0 ≤ h(e n ) → 0. Hence, h is critical.Thus, we have completed the proof of the equivalences.The last statement follows immediately from (vi).This finishes the proof.Now, we show that h cannot be critical on any proper subset of X. Proposition 5.6.Let V X, and assume that there is a function u ∈ F (V ) which is positive in V ∪ ∂V and superharmonic in V .Then, h is subcritical in V .
Furthermore, in every connected component U of V where u does not vanish, there does not exists a null-sequence (e n ) in U with e n (x) = α for x ∈ ∂(X \ U) and some α > 0, and we have cap(x, U) > 0 for all x ∈ ∂(X \ U).
Proof.Since u is superharmonic in V and positive in V ∪ ∂V , there is o ∈ V ∪ ∂V such that u(o) > 0. By the Harnack inequality, Lemma 5.4, we have that u is strictly positive in the connected component U of V , where o is either on the (b) If 1 < p < 2 and h u,1 is critical in X, then h(•) − u 1−p Hu, |•| p is critical in X.
Proof.Ad (a): Recall that a ground state is harmonic, i.e., uHu = 0.Moreover, h is critical.Then, by the ground state representation, Corollary 3.2, we have that h u,1 is critical.Since 1 is a harmonic function with respect to the Laplace operator associated with h u,1 , it is a ground state.Ad (b): This is a direct consequence of the ground state representation.Then the energy functional h is critical in X with ground state ũ.
Proof.By Theorem 5.1, there exists a null-sequence (e n ) with respect to h such that e n → u pointwise as n → ∞.Denote ϕ n = e n /u, n ∈ N. From the ground state representation, Theorem 3.1, we get h(e n ) ≍ h u (ϕ n ) for all n ∈ N. Hence, using (c) and (d), we infer h u (ϕ n ) ≥ γ 1 hũ (ϕ n ) for some constant γ 1 > 0. Now, (b) implies by Lemma 5.2 that h is non-negative in C c (X).Using this fact, the calculation before, and the ground state representation, we get for some constants γ 2 , γ 3 > 0 that 0 ≤ h(ũϕ n ) ≤ γ 2 hũ (ϕ n ) ≤ γ 3 h(e n ) → 0, n → ∞.
Thus, (ũϕ n ) is a null-sequence for h and by Theorem 5.1, h is critical in X.Since ϕ n → 1, we get ũϕ n → ũ, and by Theorem 5.1, ũ is the ground state.

Example 3. 6 (
Standard p-Laplacian on N 0 ).Here, we calculate the representation for one of the simplest cases: for the graph b on N with b(n, m) = 1 if |n − m| = 1 and b(n, m) = 0 elsewhere for all n, m ∈ N.
Proof of Theorem 5.1.Ad (i) =⇒ (ii): Let w n = 1 o /n for o ∈ X and n ∈ N. Then by the criticality of h in V we have the existence of a function e n ∈ C c (X) such that h(e n ) < w n , |e n | p .By the reverse triangle inequality, we have h(|e n |) ≤ h(e n ) and thus, we can assume that e n ≥ 0. By Lemma 5.2 we have that h is non-negative in C c (X), and therefore we get0 ≤ h(e n ) < w n , |e n | p = e p n (o)m(o)/n.Hence, we can normalise e n such that e n (o) = α for any α > 0. Altogether, h(e n ) < α p m(o)/n and (e n ) is a null sequence in X.Ad (ii) =⇒ (i): Let (e n ) be a null-sequence in X with e n (o) = α > 0 for some o ∈ X.Let w ≥ 0 on X such that h(ϕ) ≥ w, |ϕ| p for ϕ ∈ C c (X). Then,0 = lim n→∞ h(e n ) ≥ lim n→∞ w, |e n | p ≥ lim n→∞ w(o)e p n (o)m(o) = w(o)α p m(o).Since o ∈ X is arbitrary, m(o) > 0 and α > 0, we get w = 0 on X.Ad (ii) ⇐⇒ (iii): This follows immediately from the definitions.Ad (i) ⇐⇒ (iv) ⇐⇒ (v): This follows from the ground state representation, Theorem 3.1.Note that the existence of such a strictly positive harmonic function is ensured by Lemma 5.3.Ad (ii) =⇒ (vi): This is Lemma 5.5.Ad ((i) & (ii) & (vi)) =⇒ (vii):

5. 3 .
Liouville Comparison Principle.Here, we show a consequence of the characterisations of criticality and the ground state representation which is usually referred to as a Liouville comparison principle, confer [Pin07, Section 11] and references therein for the linear case.For the counterpart in the continuum see [PR15, Theorem 8.1], or [PTT08, Theorem 1.9].Proposition 5.9 (Liouville comparison principle).Let p > 1.Let b and b be two graphs on X, and c, c ∈ C(X) be two potentials.Let denote h and h the energy functionals with corresponding Schrödinger operators H := H b,,c,m,X,pand H := Hb ,c,m,X,p , respectively.Assume that the following assumptions hold true:(a) The energy functional h is critical in X with ground state u.(b) There exists a positive H-superharmonic function, and also a positive Hsubharmonic function ũ.(c) There exists a constant α > 0, such that for all x, y ∈ X we have b 2/p (x, y)u(x)u(y) ≥ α b2/p (x, y)ũ(x)ũ(y).