A MINICOURSE ON THE Lp SPECTRUM OF GRAPHS

MATTHIAS KELLER

Contents

1. Introduction 22. Graphs and Laplacians 22.1. Graphs 22.2. Generalized forms and formal Laplacians 32.3. Dirichlet forms and their generators 43. Uniformly positive measures 83.1. Liouville type theorem 83.2. Spectral inclusion 94. Intrinsic metrics 104.1. Definition of intrinsic metrics 114.2. Examples and relations to other metrics 114.3. A Hopf-Rinow theorem 144.4. Some important conditions 164.5. Construction of cut-off functions 165. Liouville type theorems 175.1. Historical notes on Yau’s and Karp’s theorem 185.2. Liouville theorems and intrinsic metrics 185.3. Green’s formula, Leibniz rules and mean value theorem 195.4. The key estimate 215.5. Caccioppoli inequality and proof of Yau’s theorem 235.6. Proof of Karp’s theorem 245.7. Domain of the generator 255.8. Recurrence 265.9. L1-Liouville theorem and counter-examples 276. Independence of the spectrum 296.1. Historical notes 296.2. The result for graphs with intrinsic metrics 296.3. Consequences of uniform subexponential growth 306.4. Lipschitz continuous functions 326.5. Kernels 326.6. Heat kernel estimates 336.7. Proof of the theorem 34References 37

Date: Bizerte, March 2014.1

2 M. KELLER

1. Introduction

The following notes are compiled for a minicourse held in BizerteTunesia in March 2014. The notes are based on joint work with FrankBauer (Harvard), Bobo Hua (Fudan) and Daniel Lenz (Jena). In par-ticular, the introduction is mainly taken from the with Daniel Lenz[28] on Dirichlet forms on graphs and the survey on intrinsic metrics[27]. The content of the first part of Section 3 is taken from [28] andthe second part from [1]. The section on Liouville theorems, Section 5,is extracted from the work with Bobo Hua [20]. Finally, the sectionon p-independence of the spectrum is taken from the paper with FrankBauer and Bobo Hua [1].

2. Graphs and Laplacians

In this section we introduce the basic notions for weighted symmetricgraphs. We mainly follow the framework developed in [28].

2.1. Graphs. Let X be a discrete and countably infinite space and ma measure of full support on X, that is a function m : X → (0,∞)which is additively extended to sets via m(A) =

∑x∈Am(x), A ⊆ X.

A graph (b, c) over X is a symmetric function b : X × X → [0,∞)with zero diagonal and∑

y∈X

b(x, y) <∞, x ∈ X,

and c : X → [0,∞). We say two vertices x, y ∈ X are neighbors ifb(x, y) > 0. In this case we write x ∼ y. The function c can describeone way edges to a virtual point at infinity or simply a potential or akilling term. If c ≡ 0, then we speak of b as a graph over X. If wealready fixed a measure, then we speak of graphs over (X,m).

We say a graph is connected if for all x, y ∈ X there are x = x0 ∼. . . ∼ xn = y. If a graph is not connected we may restrict our attentionto the connected components. Therefore, we assume in the followingthat the considered graphs are connected.

Given a pair (b, c) an important special case of a measure m is thenormalizing measure

n(x) =∑y∈X

b(x, y) + c(x), x ∈ X.

Another important special case is the counting measure m ≡ 1.We say a graph is locally finite if every vertex has only finitely many

neighbors, that is if the combinatorial vertex degree deg is finite atevery vertex

deg(x) = #{y ∈ X | x ∼ y} <∞, for all x ∈ X.

Lp SPECTRUM OF GRAPHS 3

We speak of a graph with standard weights if b : X×X → {0, 1} andc ≡ 0. In this case the normalizing measure n equals the combinato-rial vertex degree deg. Obviously, by summability of b about vertices,graphs with standard weights are locally finite.

2.2. Generalized forms and formal Laplacians. We let C(X) bethe set of complex valued functions on X and Cc(X) be the subspaceof functions in C(X) of finite support. For a graph (b, c) over X, thegeneralized quadratic form Q : C(X)→ [0,∞] is given by

Q(f) =1

2

∑x,y∈X

b(x, y)|f(x)− f(y)|2 +∑x∈X

c(x)|f(x)|2

with generalized domain

D = {f ∈ C(X) | Q(f) <∞}.

Since Q 12 is a semi norm and satisfies the parallelogram identity, by

polarization Q yields a semi scalar product on D via

Q(f, g) =1

2

∑x,y∈X

b(x, y)(f(x)− f(y))(g(x)− g(y)) +∑x∈X

c(x)f(x)g(x).

Moreover, for functions in

F = {f ∈ C(X) |∑y∈X

b(x, y)|f(y)|2 <∞ for all x ∈ X},

we define the generalized Laplacian L : F → C(X) by

Lf(x) =1

m(x)

∑y∈X

b(x, y)(f(x)− f(y)) +c(x)

m(x)f(x).

In [15, Lemma 4.7], a Green’s formula is shown for functions f ∈ Fand ϕ ∈ Cc(X)

1

2

∑x,y∈X

b(x, y)(f(x)− f(y))(g(x)− g(y)) +∑x∈X

c(x)f(x)g(x)

=∑x∈X

Lf(x)ϕ(x) =∑x∈X

f(x)Lϕ(x).

Denote

df(x, y) = f(x)− f(y), f ∈ C(X).

We call f ∈ F a solution (respectively subsolution or supersolution)for λ ∈ R if (L−λ)f = 0 (respectively (L−λ)f ≤ 0 or (L−λ)f ≥ 0). Asolution (respectively subsolution or supersolution) for λ = 0 is calleda harmonic (respectively subharmonic or superharmonic).

We say a function f ∈ C(X) is positive if f ≥ 0 and non-trivial andstrictly positive if f > 0.

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A Riesz space is a linear space equipped with a partial orderingwhich is consistent with addition, scalar multiplication and where themaximum and the minimum of two functions exist.

An important fact that is needed in the subsequent is that in orderto study existence of non-constant (respectively non-zero) solutionsfor λ ≤ 0 in a Riesz space, it suffices to study positive subharmonicfunctions.

Lemma 2.1. Let (b, c) be a connected graph over (X,m) and F0 ⊆ Fbe a Riesz space. If there are no non-constant positive subharmonicfunctions in F0, then there are no non-constant solutions for λ ≤ 0 inF0. In particular, any constant solution to λ < 0 is zero.

Proof. For a solution f to λ ≤ 0 the positive part f+ = f ∨ 0, thenegative part f− = −f ∨ 0 and the modulus |f | = f+ + f− can beseen to be non-negative subharmonic functions. Thus, the statementfollows from connectivity. The ’in particular’ is obvious. �

2.3. Dirichlet forms and their generators. The form and Lapla-cian introduced above are defined on spaces with very few properties.Next, we will consider these objects restricted to suitable Hilbert andBanach spaces.

Let `p(X,m) be the canonical complex-valued `p-spaces, p ∈ [1,∞],with norms

‖f‖p =(∑x∈X

|f(x)|pm(x)) 1p, p ∈ [1,∞),

‖f‖∞ = supx∈X|f(x)|.

As `∞(X,m) does not depend on m we also write `∞(X). For p = 2,we have a Hilbert space `2(X,m) with scalar product

〈f, g〉 =∑x∈X

f(x)g(x)m(x), f, g ∈ `2(X,m),

and we denote the norm ‖ · ‖ = ‖ · ‖2.

2.3.1. Dirichlet forms. Restricting the form Q to D∩ `2(X,m), we seeby Fatou’s lemma that this restriction is lower semi-continuous and,thus, closed. Hence, the restriction of Q to Cc(X) is closable.

The Q = Qb,c be the quadratic form given by

D(Q) = Cc(X)‖·‖Q

, where ‖ · ‖Q =(Q(·) + ‖ · ‖2

) 12

Q(f) =1

2

∑x,y∈X

b(x, y)|ϕ(x)− ϕ(y)|2 +∑x∈X

c(x)|ϕ(x)|2, f ∈ D(Q).

Lp SPECTRUM OF GRAPHS 5

It can be seen that Q is a Dirichlet form (see [8, Theorem 3.1.1] forgeneral theory and for a proof in the graph setting see [39, Proposi-tion 2.10]), that is for any f ∈ D(Q) we have 0 ∨ f ∧ 1 ∈ D(Q) and

Q(0 ∨ f ∧ 1) ≤ Q(f).

Obviously, Q is regular, that is Cc(X) ∩ D(Q) is dense in D(Q) withrespect to ‖ · ‖Q and dense in Cc(X) with respect to ‖ · ‖∞.

As it turns out, by [28, Theorem 7], all regular Dirichlet forms aregiven in this way. A fact which can be also derived directly fromthe Beurling-Deny representation formula [8, Theorem 3.2.1 and The-orem 5.2.1].

Theorem 2.2 (Theorem 7 in [28]). If q is a regular Dirichlet on`2(X,m), then there is a graph (b, c) such that q = Qb,c.

2.3.2. Markovian semigroups and their generators. By general theory(see e.g. [44, Satz 4.14]), Q yields a positive selfadjoint operator L withdomain D(L) viz

Q(f, g) = 〈L12f, L

12 g〉, f, g ∈ D(Q).

By the second Beurling-Deny criterion L gives rise to a Markoviansemigroup e−tL, t > 0, which extends consistently to all `p(X,m),p ∈ [1,∞], and is strongly continuous for p ∈ [1,∞). Markovian meansthat for functions 0 ≤ f ≤ 1, one has 0 ≤ e−tLf ≤ 1.

We denote the generators of e−tL on `p(X,m), p ∈ [1,∞), by Lp,that is

D(Lp) ={f ∈ `p(X,m) | g = lim

t→0

1

t(I − e−tL)f exists in `p(X,m)

}Lpf = g

and L∞ is defined as the dual of L1. Simultaneously, the resolventsGz = (L2 − z)−1, z ∈ {w ∈ C | <w < 0} (the open left half plane)extend consistently to `p(X,m), p ∈ [1,∞]. By the spectral theoremthe Laplace transform for the resolvent

Gzf =

∫ ∞0

eztTtfdt,

holds for f in `2. By density and duality arguments, this formulaextends to f in `p, p ∈ [1,∞), in the strong sense and to p =∞ in theweak sense. This shows that the resolvents (Lp − z)−1 are consistenton `p for z ∈ {w ∈ C | <w < 0}.

Theorem 2.3 (Theorem 9 in [28]). Let (b, c) be a graph over (X,m).Let p ∈ [1,∞] be given. Then,

Lpf = Lf for any f ∈ D(Lp).

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Proof. Let f ∈ D(Lp) be given. Then, g := (Lp + α)f exists andbelongs to `p(X,m). By the lemma below, f = (Lp + α)−1g belowsolves

(L+ α)f = g = (Lp + α)f

and we infer the statement. �

Lemma 2.4. Let (b, c) be a graph over (X,m). Let p ∈ [1,∞] be given.For any g ∈ `p(X,m), the function u := (Lp + α)−1g belongs to the setF on which L is defined and solves (L+ α)u = g.

Proof. We first consider the case p = 2. If suffices to consider thecase f ≥ 0. Choose an increasing sequence (Kn) of finite subsets ofX with

⋃Kn = X and let gn be the restriction of g to Kn. Then,

(gn) converges monotonously increasing to g in `2(X,m) and conse-quently (L + α)−1gn converges monotonously increasing to u. Thus,by monotone convergence of solutions, we can assume without loss ofgenerality that g has compact support contained in K1. By conver-

gence of resolvents, un := (L(D)Kn

+α)−1g then converges increasingly tou := (L + α)−1g. Moreover, un satisfies (L + α)un = g on Kn. Thus,the statement follows by monotone convergence of solutions.

We now turn to general p ∈ [1,∞]. Again, it suffices to considerthe case g ≥ 0. Choose an increasing sequence (Kn) of finite subsetsof X with

⋃Kn = X and let gn be the restriction of g to Kn. Then,

un := (Lp+α)−1gn converges to u. Moreover, as gn belongs to `2(X,m)consistency of the resolvents gives un = (L + α)−1gn. Now, on the`2(X,m) level we can apply the considerations for p = 2 to obtain

(L+ α)un = (L+ α)(L+ α)−1gn = gn.

Taking monotone limits now yields the statement. �

2.3.3. Graphs with standard weights. Two important special cases aregraphs with standard weights, that is b : X ×X → {0, 1} and c ≡ 0.

For the counting measure m ≡ 1, we denote the operator L on`2(X) = `2(X, 1) by ∆ which operates as

∆f(x) =∑y∼x

(f(x)− f(y)), f ∈ D(∆), x ∈ X.

We will see below in Section 2.3.4 that ∆ is bounded if and only if degis bounded.

For the normalized measure n = deg, we call the operator L on`2(X, deg) the normalized Laplacian and denoted it by ∆n. The oper-ator ∆n acts as

∆nf(x) =1

deg(x)

∑y∼x

(f(x)− f(y)), f ∈ `2(X, deg), x ∈ X,

and, as it can also be seen below, ∆n is always bounded by 2.

Lp SPECTRUM OF GRAPHS 7

2.3.4. Boundedness of the operators. We next comment on the bound-edness of the form Q and the operator L. The theorem below is takenfrom [16] and an earlier version can be found in [29, Theorem 11].

Theorem 2.5 (Theorem 9.3 in [16]). Let (b, c) be a graph over (X,m).Then the following are equivalent:

(i) X → [0,∞), x 7→ 1m(x)

(∑y∈X b(x, y) + c(x)

)is a bounded

function.(ii) Lp is bounded for some p ∈ [1,∞].

(iii) Lp is bounded for all p ∈ [1,∞].

Specifically, if the function in (i) is bounded by D <∞, then Q ≤ 2Dand ‖L‖p ≤ 2D, p ∈ [1,∞].

Proof. Note that we have seen that Lp is a restriction of L by what wehave discussed above. Thus, it suffices to consider boundedness of L.

Assume that (i) is satisfied. Then, we see that L is a bounded opera-tor on `∞. Since L is symmetric, we get, by duality, that L is boundedon `1. Applying the Riesz-Thorin theorem, we get (iii), resp. (ii).Assume, conversely, that (ii) is fulfilled. Again, by duality and sym-metry, L is also a bounded operator on `q, where 1 = 1

p+ 1

q. Using

interpolation once more, we get that L is bounded on `2. Hence, foreach x ∈ V , we have the existence of C ≥ 0 such that

〈Lδx, δx〉 ≤ Cm(x),

which gives (i). �

2.3.5. The compactly supported functions as a core. It shall be observedthat Cc(X) is in general not included in D(L). Indeed, one can give acharacterization for this situation. The proof is rather immediate andwe refer to [28, Proposition 3.3] or [14, Lemma 2.7.] for a reference.

Lemma 2.6. Let (b, c) be a graph over (X,m). Then the following areequivalent:

(i) Cc(X) ⊆ D(L).(ii) LCc(X) ⊆ `2(X,m)

(iii) The functions X → [0,∞), y 7→ 1m(y)

b(x, y) are in `2(X,m).

In particular, the above assumptions are satisfied if the graph is locallyfinite or

infy∼x

m(y) > 0, x ∈ X.

Moreover, either of the above assumptions implies `2(X,m) ⊆ F .

Proof. The equivalence of (ii) and (iii) follows from the abstract def-inition of the domain of L as D(L) = {f ∈ D(Q) | there is l ∈`2(X,m) such that 〈l, ϕ〉 = Q(f, ϕ) for all ϕ ∈ D(Q)}. The equiva-lence of (i) and (ii) is a direct calculation, see [28, Proposition 3.3] andthe ’in particular’ statements are also immediate, see [28, 14]. �

8 M. KELLER

3. Uniformly positive measures

In this section we discuss two results for graphs which seem to haveno analogue in the non-discrete setting. The condition below is about(X,m) only as a measure space. We say the measure m is uniformlypositive if

(A) infx∈X m(x) > 0.

For example this holds if m is constant as in the case of the countingmeasure or deg.

3.1. Liouville type theorem. This condition yields a result for ex-istence of certain solutions in `p(X,m) which in contrast to the metricresult above includes the case p = 1.

Theorem 3.1 (Lemma 3.2 in [28]). Let (b, c) be a graph over (X,m)with uniformly positive measure. Then any positive subharmonic func-tion in `p(X,m), p ∈ [1,∞), is zero.

Proof. Let f ∈ `p(X,m), for some p ∈ [1,∞), be positive and subhar-monic. Also, let x ∈ X such that f(x) > 0. Then, Lf(x) ≤ 0 givesusing f ≥ 0

f(x) ≤ 1∑y∈X b(x, y)

∑y∈X

b(x, y)f(y)

Thus, there must be y ∼ x such that f(x) ≤ f(y). Proceeding in-ductively there is a sequence (xn) of vertices such that 0 < f(x) ≤f(xn) ≤ f(xn+1), n ≥ 0. By (A) together with f ∈ `p(X,m), thisimplies f ≡ 0. �

Remark. Note that the theorem also holds under the weaker assumption

(A*)∑∞

n=1m(xn) =∞ for all infinite paths (xn).

Theorem 3.2 (Theorem 5 in [28]). Let (b, c) be a graph over (X,m)such that (A) holds. Then, for any p ∈ [1,∞) the operator Lp is therestriction of L to

D(Lp) = {u ∈ `p(X,m) : Lu ∈ `p(X,m)}.

Proof. Let f be positive and subharmonic. Then, Lf(x) ≤ 0 evaluatedat some x gives, using f ≥ 0,

f(x) ≤ 1∑y∈X b(x, y)

∑y∈X

b(x, y)f(y)

Thus, whenever there is x′ ∼ x with f(x′) < f(x) there must be y ∼ xsuch that f(x) < f(y). Such x′ and x exist whenever f is non-constant.Letting x0 = x, x1 = y and proceeding inductively there is a sequence(xn) of vertices such that 0 < f(x) < f(xn) < f(xn+1), n ≥ 0. Now,(A) implies that f is not in `p(X,m). On the other hand, if f isconstant, then (A) again implies f ≡ 0. �

Lp SPECTRUM OF GRAPHS 9

Theorem 3.3 (Theorem 6 in [28]). Let (b, c) be a graph over (X,m).Assume

LCc(X) ⊆ `2(X,m)

and (A). Then the restriction of L to Cc(X) is essentially selfadjointand the domain of L is given by

D(L) = {u ∈ `2(X,m) : Lu ∈ `2(X,m)}.

Proof. As LCc(X) ⊆ `2(X,m), we can define the minimal operatorLmin to be the restriction of L to

D(Lmin) := Cc(X)

and the maximal operator Lmax to be the restriction of L to

D(Lmax) := {u ∈ `2(X,m) : Lu ∈ `2(X,m)}.

The previous proposition gives

〈u, Lminv〉 = Q(u, v)

for all u, v ∈ Cc(X) which extends to all u ∈ D(Q) and v ∈ Cc(X).Thus, Lmin is a restriction of L in this case.Moreover, by Green’s formula above

〈u, Lminv〉 =∑x∈X

Lu(x)v(x)m(x)

for all v ∈ Cc(X) and u ∈ `2(X,m). Thus,

L∗min = Lmax.

Thus, essential selfadjointness of Lmin is equivalent to selfadjointness ofLmax. This in turn is equivalent to L = Lmax (as we have L ⊆ Lmax byTheorem 3.2). As (A) and Theorem 3.2 yield D(L) = {u ∈ `2(X,m) :Lu ∈ `2(X,m)}, we infer L = Lmax and essential selfadjointness of therestriction of L to Cc(X) (= Lmin) follows. �

3.2. Spectral inclusion. In this section we prove the spectral inclu-sion σ(L2) ⊆ σ(Lp) under the assumption of uniformly positive mea-sure. We notice that infx∈X m(x) > 0 implies

`p ⊆ `q, 1 ≤ p ≤ q ≤ ∞.

Moreover, by the theorem above we know the domains of the generatorsLp in this case explicitly, namely

D(Lp) = {f ∈ `p | Lf ∈ `p}.

In particular, this gives

D(Lp) ⊆ D(Lq), 1 ≤ p ≤ q ≤ ∞.

Furthermore, it can be checked directly that Cc(X) ⊆ D(Lp).

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The following theorem shows that under an assumption on the mea-sure one spectral inclusion holds without any volume growth assump-tions.

Theorem 3.4. Let (b, c) be a graph over (X,m) such that (A) holds,then, for any p ∈ [1,∞],

σ(L2) ⊆ σ(Lp).

Before we come to the proof of the theorem we have to establishconsistency of resolvents on the intersection of the resolvent sets.

Lemma 3.5. Let (b, c) be a graph over (X,m) such that (A) holds.Then, for all 1 ≤ p ≤ q ≤ ∞ and all z ∈ ρ(Lp) ∩ ρ(Lq) the resolvents(Lp − z)−1 and (Lq − z)−1 are consistent on `p = `p ∩ `q.

Proof. Let 1 ≤ p < q ≤ ∞ and z ∈ ρ(Lp) ∩ ρ(Lq). As D(Lp) ⊆ D(Lq)and Lp = Lq on D(Lp), we have for all f ∈ `p ⊆ `q

(Lq − z)(Lp − z)−1f = (Lp − z)(Lp − z)−1f = f

Hence, (Lp − z)−1 and (Lq − z)−1 are consistent on `p = `p ∩ `q. �

Proof of Theorem 3.4. Let p ∈ [1, 2]. By the lemma above the resol-vents (Lp−z)−1 and (Lq−z)−1 are consistent for z ∈ ρ(Lp) = ρ(Lq) on`p. By the Riesz-Thorin interpolation theorem (Lq − z)−1 is boundedon `2. We will show that (Lq − z)−1 is an inverse of (L2 − z) forz ∈ ρ(Lp) = ρ(Lq). So, let z ∈ ρ(Lq). As D(L2) ⊆ D(Lq), `

2 ⊆ `p∗

andL2, Lq are restrictions of L we have for f ∈ D(L2)

(Lq − z)−1(L2 − z)f = (Lq − z)−1(Lq − z)f = f.

Secondly, let f ∈ `2 and (fn) be such that fn ∈ `p and fn → f in `2. As(Lq − z)−1 is `2-bounded, (Lq − z)−1fn → (Lq − z)−1f , n→∞, in `2.By the lemma above (Lq − z)−1fn = (Lp − z)−1fn ∈ D(Lp) ⊆ D(L2),and, therefore,

(L2 − z)(Lq − z)−1fn = (Lq − z)(Lq − z)−1fn = fn → f, n→∞,

in `2. Since, L2 is closed we infer (Lq − z)−1f ∈ D(L2) and (L2 −z)(Lq − z)−1f = f . Hence, (Lq − z)−1 is an inverse of (L2 − z) and,thus, z ∈ ρ(L2). �

4. Intrinsic metrics

In this subsection we discuss the notion of intrinsic metrics for aweighted graph. Such metrics have proven to be very effective in thecontext of strongly local Dirichlet forms, see e.g. [43]. Recently, thisconcept was generalized to all regular Dirichlet forms. It was firstsystematically studied by Frank/Lenz/Wingert in [7], (see also [34] foran earlier mention of the criterion for certain non-local forms).

Lp SPECTRUM OF GRAPHS 11

4.1. Definition of intrinsic metrics. We call a symmetric map ρ :X × X → [0,∞) with zero diagonal a pseudo metric if it satisfiesthe triangle inequality. By [7, Lemma 4.7, Theorem 7.3] it can be seenthat the following definition coincides with the definition of an intrinsicmetric for general regular Dirichlet forms, [7, Definition 4.1].

Definition 4.1. A pseudo metric ρ is called an intrinsic metric withrespect to a graph (b, c) over (X,m) if for all x ∈ X∑

y∈X

b(x, y)ρ2(x, y) ≤ m(x).

Similar definitions of such metrics were also introduced in the contextof graphs or jump processes under the name adapted metrics in [5, 6,13, 21, 23, 34].

4.2. Examples and relations to other metrics.

4.2.1. The degree path metric. A first specific example of an intrinsicmetric was introduced by Huang, [21, Definition 1.6.4] and it also ap-peared in [4]. Consider the pseudo metric ρ0 : X ×X → [0,∞) givenby

ρ0(x, y) = infx=x0∼x1∼...∼xn=y

n∑i=1

(Deg(xi−1) ∨Deg(xi)

)− 12, x ∈ X,

where Deg : X 7→ (0,∞) is the weighted vertex degree given by

Deg(x) =1

m(x)

∑y∈X

b(x, y), x ∈ X.

We call such a metric that minimizes sums of weights over paths ofedges a path metric.

It can be seen directly that ρ0 defines an intrinsic metric for thegraph (b, c) over (X,m)∑

y∈X

b(x, y)ρ20(x, y) ≤∑y∈X

b(x, y)

Deg(x) ∨Deg(y)≤∑y∈X

b(x, y)

Deg(x)= m(x).

There is the following intuition behind the definition of ρ0. Considerthe Markov process (Xt)t≥0 associated to the semigroup e−tL via

e−tLf(x) = Ex(f(Xt)),

where Ex is the expected value conditioned on the process starting atx. The random walker modeled by this process jumps from a vertexx to a neighbor y with probability b(x, y)/

∑z b(x, z). Moreover, the

probability of not having left x at time t is given by

Px(Xs = x, 0 ≤ s ≤ t) = e−Deg(x)t.

Qualitatively, this indicates that the larger Deg(x), the faster the ran-dom walker leaves x. Looking at the definition of ρ0(x, y), the larger

12 M. KELLER

the degree of either x or y the closer are the two vertices. Combiningthese two observations, we see that the faster the random walker jumpsalong an edge the shorter the edge is with respect to ρ0. Of course, thejumping time along an edge connecting x to y is not symmetric anddepends on whether one jumps from x to y or from y to x as the degreesof x and y can be very different. In order to get a symmetric function,ρ0 favors the vertex with the larger degree and the faster jumping time.

There is a direct analogy to the Riemannian setting in terms of meanexit times of small balls. Consider a small ball Br of radius r on a d-dimensional Riemannian manifold, the first order term of the meanexit time of Br is r2/2d, [37]. Now, on a locally finite graph for avertex x a ’small’ ball with respect to ρ0 can be thought to have radiusr = infy∼x ρ(x, y)/2, namely this ball contains only the vertex itself.Now, computing the mean exit time of this ball gives 1/Deg(x) ≥ r2,where equality holds whenever Deg(x) = maxy∼x Deg(y).

4.2.2. The combinatorial graph distance. We call the path metric de-fined by

d(x, y) =

min #{n ∈ N | there are x0, . . . , xn with x = x0 ∼ . . . ∼ xn = y}

the combinatorial graph distance. If one considers now a graph (b, c)over (X,m) with m = n being the normalizing measure we find that∑

y∈X

b(x, y)d(x, y)2 =∑y∈X

b(x, y) ≤ n(x).

Hence, d is always an intrinsic metric for (b, c) over (X,n). In particu-lar, for c ≡ 0,

Deg(x) =1

n(x)

∑y∈X

b(x, y) = 1

for all x ∈ X which readily gives

d = ρ0

in the case of the normalizing measure n. Specifically, this is true forthe normalized Laplacian ∆n.

On the other hand, for a graph with standard weights and the count-ing measure associated to the Laplacian ∆, the combinatorial graphdistance d satisfies ∑

y∼x

b(x, y)d(x, y)2 = deg(x)

for all x ∈ X. Hence, there is an intrinsic metric equivalent to thecombinatorial graph distance if and only if the combinatorial vertex

Lp SPECTRUM OF GRAPHS 13

degree deg is bounded, and, in which case ∆ is a bounded operator.Moreover, in this case

Deg = deg

and, therefore, by connectedness

ρ0 ≤ d ≤ D12ρ0,

with D = supx∈X deg(x).

4.2.3. Comparison to the strongly local case. An important differenceto the case of strongly local Dirichlet forms is that in the graph casethere is no maximal intrinsic metric. For example for a Riemannianmanifold M the Riemannian distance dM is the maximal C1 metric ρMthat satisfies

|∇ρM(o, ·)| ≤ 1,

for all o ∈ M ,where ∇M is the Riemannian gradient. In fact, dM canbe recovered by the formula

dM(x, y) = sup{f(x)− f(y) | f ∈ C∞c (M) |∇Mf | ≤ 1}, x, y ∈ X.

Now, for discrete spaces the maximum of two intrinsic metrics isnot necessarily an intrinsic metric. In particular, consider the pseudo-metric σ

σ(x, y) = sup{f(x)− f(y) | f ∈ A}, x, y ∈ X,

where

A ={f : X → R |

∑y∈X

b(x, y)|f(x)− f(y)|2 ≤ m(x) for all x ∈ X}

As discussed for the Riemannian case above, the strongly local analoguedM of σ defines the maximal intrinsic metric in the strongly local case,but σ is in general not an intrinsic metric in the graph case.

A basic example where this can be seen immediately can be foundin [7, Example 6.2]. More generally, this can be seen for arbitrarytree graphs associated to the operator ∆. In this case σ = 1

2d and

by discussion above we already know that the combinatorial graphdistance d is typically not an intrinsic metric for ∆.

4.2.4. Resistance metrics. Another important metric appears in thecontext of resistance metrics. Let r : X ×X → [0,∞) be given by

r(x, y) = sup{f(x)− f(y) | f ∈ D, Q(f) ≤ 1}, x, y ∈ X.

Indeed, r is the square root of the resistance metric as it appears e.g.in [32], see [9, Theorem 3.20]. In [9] this metric is related to intrinsicmetrics.

14 M. KELLER

Theorem 4.2 (Theorem 3.14 in [9]). Let b be a connected graph overX. Then

r = sup{ρ | intrinsic metric for b over (X,m) with m(X) = 1}.

4.2.5. Another path metric. Colin de Verdiere/Torki-Hamza/Truc [2]studied a path pseudo metric δ which is given as

δ(x, y) = infx=x0∼...∼xn=y

n−1∑i=0

(m(x) ∧m(y)

b(x, y)

) 12, x, y ∈ X.

This metric is equivalent to the intrinsic metric ρ0 if and only if thecombinatorial vertex degree deg is bounded on the graph.

4.3. A Hopf-Rinow theorem. We shall stress that in general anintrinsic metric ρ (and in particular ρ0) is not a metric and (X, ρ)might not even be locally compact, as can be seen from examples in[24, Example A.5]. However, for locally finite graphs and path metricssuch as ρ0 the situation is much tamer. For example one can show aHopf-Rinow type theorem. Note that we do not need that the metricsare intrinsic for this section.

Theorem 4.3 (Theorem A1 in [16]). Let (b, c) be a connected graphover (X,m) and ρ be a path metric. Then, the following are equivalent:

(i) (X, ρ) is complete as a metric space.(ii) (X, ρ) is geodesically complete, that is any infinite path (xn) of

vertex that realizes the distance has infinite length.(iii) Every distance ball is finite.(iv) Every bounded and closed set is compact.

Remark. (a) Anytime a path pseudo metric d induces the discrete topol-ogy on X the following implications hold: (iii)⇔(iv)⇒(i)⇒(ii). Thisis the case if and only if infy∼x σ(x, y) > 0 for all x ∈ X. In fact,(iv)⇒(i) holds for general metric spaces. The stronger assumption oflocal finiteness is needed for the implications (ii)⇒(i), (i)⇒(iii) (or (iv))and (ii)⇒(iii) (or (iv)).

(b) A similar statement as (i)⇒(iii) can also be found in [36].

We prove the theorem in several steps through the following lemmas.

Lemma 4.4. Let (b, c) be a connected locally finite graph over (X,m)and ρ be a path metric. Then, the following hold:

(a) (X, ρ) is a discrete metric space. In particular, (X, ρ) is locallycompact.

(b) A set is compact in (X, ρ) if and only if it is finite.

Proof. Local finiteness implies that for all x ∈ X there is an r > 0 suchthat ρ(x, y) > r for all y ∈ X with y ∼ x. First, by the definition ofρ, we have that for all x, z ∈ X there is y ∼ x with ρ(x, y) ≤ ρ(x, z).Thus, ρ(x, y) = 0 implies x = y, therefore, d is a metric. Second, it

Lp SPECTRUM OF GRAPHS 15

yields that Br(x) = {x} and {x} is an open set which shows (a). Fromthis we conclude that for any infinite set U the cover {{x} | x ∈ U}has no finite subcover. The other direction of (b) is clear. �

Lemma 4.5. Let (b, c) be a connected locally finite graph over (X,m)and ρ be a path metric. Then, there exists an infinite geodesic ofbounded length.

Proof. Let o ∈ X be the center of the infinite ball Br of radius r andlet d be the combinatorial graph distance. Let Pn, n ≥ 0, be the set offinite paths (x0, . . . , xk) such that x0 = o, xi 6= xj for i 6= j, ρ(xk, o) = nand d(xj, o) ≤ n for j = 0, . . . , k. Let

l((x0, . . . , xk)) =k−1∑i=0

ρ(xi, xi+1).

Claim: Γn = {γ ∈ Pn | γ geodesic with respect to ρ, l(γ) ≤ r} 6= ∅for all n ≥ 0.Proof of the claim: The set Pn is finite by local finiteness of the graphand, thus, contains a minimal element γ = (x0, . . . , xK) with respectto the length l, i.e., for all γ′ ∈ Pn we have l(γ′) ≥ l(γ). Then, γ is ageodesic: for every path (x′0, . . . , x

′M) with x′0 = o and x′M = xK , we let

m ∈ {n, . . . ,M} be such that (x′0, . . . , x′m) ∈ Pn. By the minimality of

γ we infer

l((x′0, . . . , x′M)) ≥ l((x′0, . . . , x

′m)) ≥ l(γ).

It follows that γ is a geodesic. Clearly, l(γ) ≤ r, as otherwise Br ⊆{y ∈ X | d(y, o) ≤ n − 1} which would imply the finiteness of Br bythe local finiteness of the path space. Thus, γ ∈ Γn which proves theclaim.

We inductively construct an infinite geodesic (xk) with boundedlength: We set x0 = o. Since Γn 6= ∅, there is a geodesic in Γn for everyn ≥ 0 such that x0 is a subgeodesic. Suppose we have constructed a ge-odesic (x1, . . . , xm) such that for all n ≥ m there is a geodesic in Γn thathas (x1, . . . , xm) as a subgeodesic. By local finiteness xm has finitelymany neighbors. Thus, there must be a neighbor xm+1 of xm such thatfor infinitely many n the path (x0, . . . , xm, xm+1) is a subpath of a geo-desic in Γn. However, a subpath of geodesic is a geodesic. Thus, there isan infinite geodesic γ = (xk)k≥0 with l(γ) = limn→∞ l((x0, . . . , xn)) ≤ ras (x0, . . . , xn) ∈ Γn for all n ≥ 0. �

Proof of Theorem 4.3. The fact that (X, ρ) is a discrete metric spacefollows from Lemma 4.4. We now turn to the proof of the equiva-lences. We start with (i)⇒(ii). If there is a bounded geodesic, then itis a Cauchy sequence. Since a geodesic is a path it is not eventuallyconstant, thus it does not converge by discreteness. Hence, (X, d) isnot metrically complete. To prove (ii)⇒(iii) suppose that there is adistance ball that is infinite. By Lemma 4.5 there is a bounded infinite

16 M. KELLER

geodesic and (X, d) is not geodesically complete. From Lemma 4.4 (b)we deduce (iii)⇔(iv). Finally, we consider the direction (iv)⇒(i). Ifevery bounded and closed set is compact, then every closed distanceball is compact. Then, by Lemma 4.4 (b) every distance ball is finiteand it follows that (X, d) is metrically complete. �

4.4. Some important conditions. In the general situation when oneis not necessarily locally finite or has a path metric, we will often makeassumptions. These assumptions are discussed next.

We say a pseudo metric ρ admits finite balls if (iii) in the theoremabove is satisfied for ρ, i.e.,

(B) The distance balls Br(x) = {y ∈ X | ρ(x, y) ≤ r} are finite forall x ∈ X, r ≥ 0.

A somewhat weaker assumption is that the degree is bounded onballs :

(D) The restriction of Deg to Br(x) is bounded for all x ∈ X, r ≥ 0.

Clearly, (B) implies (D). Moreover, (D) is equivalent to the fact thatL restricted to the `2 space of a distance ball is a bounded operator.

The assumptions (B) and (D) can be understood as bounding ρ in acertain sense from below. Next, we come to an assumption which maybe understood as an upper bound.

We say a pseudo-metric ρ has finite jump size if

(J) The jump size s = sup{ρ(x, y) | x, y ∈ X, x ∼ y} is finite.

The assumptions (B) and (J) combined have strong geometric con-sequences.

Lemma 4.6. Let (b, c) be a graph over (X,m) and ρ be an pseudometric. If ρ satisfies (B) and (J), then the graph is locally finite.

Proof. If there was a vertex with infinitely many neighbors, then therewould be a distance ball containing all of them by finite jump size.However, this is impossible by (B). �

4.5. Construction of cut-off functions. From an analyst’s point ofview a major interest in metrics is to construct cut-off functions withdesirable properties. Let us illustrate in which sense intrinsic metricsserve this purpose.

Given a intrinsic metric ρ, a subset A ⊆ X and R > 0, the mostbasic cut-off function is defined by η = ηA,R : X → [0,∞)

η(x) =(

1− ρ(x,A)

R

)∧ 0, x ∈ X.

Such test functions are often used to approximate a solution f ∈C(X) by ηBr,Rf , where Br is a ball with respect to ρ about somevertex.

If (B) holds, then ηBr,Rf is in Cc(X) which is for example sufficientto apply Green’s formula.

Lp SPECTRUM OF GRAPHS 17

Secondly, if (J) holds, i.e., s < ∞. Then, the function dη on theedges given by

dη(x, y) = η(x)− η(y)

is supported on BR+s \Br−s.Let us collect some basic properties.

Lemma 4.7. Let η = ηBr,R, 0 < r < R, be given as above. Then,

(a) η|Br ≡ 1 and η|X\BR ≡ 0.(b) For x ∈ X,∑

y∈X

b(x, y)|η(x)− η(y)|2 ≤ 1

(R− r)21BR+s\Br−s(x)m(x).

Proof. (a) is obvious from the definition of η and (b) follows directlyfrom∑y∈X

b(x, y)(η(x)−η(y))2

=∑

y∈BR+s\Br−s

b(x, y)(η(x)− η(y))2

≤ 1

(R− r)2∑

y∈BR+s\Br−s

b(x, y)(ρ(x,Br)− ρ(y,Br))2

≤ 1

(R− r)2∑

y∈BR+s\Br−s

b(x, y)ρ2(x, y)

≤ 1

(R− r)2m(x),

for x ∈ BR+s \Br−s and the term is 0 otherwise. �

5. Liouville type theorems

The classical Liouville theorem states that if a harmonic function isbounded from below, then the function is constant. Here, we look intoboundedness assumptions such as `p growth bounds. First, we presentYau’s Lp Liouville theorem and Karp’s improved bound in the case ofmanifolds. Secondly, we discuss the case of the normalized Laplacianfor graphs and the results that have been proven for this operator.Thirdly, we present theorems that recover Yau’s and Karp’s resultsfor weighted graphs using intrinsic metrics. This yields a sufficientcriterion for recurrence. Finally, we round off the section by a result ofa type which seems to hold exclusively for discrete spaces.

18 M. KELLER

5.1. Historical notes on Yau’s and Karp’s theorem. We considera connected Riemannian manifold M , together with its Laplace Bel-trami operator ∆M . A twice continuously differentiable function f onM is called harmonic (respectively subharmonic) if ∆Mf = 0 (respec-tively ∆Mf ≤ 0).

In 1976 Yau [47] proved that on a complete Riemannian manifold Many harmonic function or positive subharmonic function in Lp(M) isalready constant.

This result was later strengthened by Karp in 1982. Namely, anyharmonic function or positive subharmonic function f that satisfies

infr0>0

∫ ∞r0

1

‖f1Br‖p

p

dr =∞,

is already constant, where 1Br is the characteristic function of the geo-desic ball Br about some arbitrary point in the manifold. Karp’s resulthas Yau’s theorem as a direct consequence.

Later in 1994 Sturm generalized Karp’s theorem to strongly localDirichlet forms, where balls are taken with respect to the intrinsicmetric. The underlying assumption on the metric is that it generatesthe original topology and all balls are relatively compact.

For graphs b over (X,m) the first results in this direction were ob-tained for the normalizing measure m = n. In this case, the operatorL is bounded, see Section 2.5, and the combinatorial graph distanced is an intrinsic metric, see Section 4.2.2. (Of course, the fact thata function is harmonic depends only on the graph b and not on themeasure m, but being in an `p space does.)

Starting 1997 with Holopainen/Soardi [18], Rigoli/Salvatori/Vignati[38], Masamune [33], eventually in 2013 Hua/Jost [19] showed that if aharmonic or positive subharmonic function f satisfies

lim infr→∞

1

r2‖f1Br(x)‖pp <∞,

for some p ∈ (1,∞) and x ∈ X, then f must be constant. Here,the balls are taken with respect to the natural graph distance. Thisdirectly implies Yau’s theorem for p ∈ (1,∞). Moreover, Hua/Jost [19]also show Yau’s theorem for p = 1.

5.2. Liouville theorems and intrinsic metrics. For graphs b over ageneral discrete measure space one can not expect such results to holdwithout any further conditions. Since the property of being harmonicdoes not depend on the measure, for any harmonic function f there isa measure m such that f is in `p(X,m), p ∈ (1,∞).

The following theorem for p ∈ (1,∞) is found in [20, Corollary 1.2]with the additional assumption of finite jump size (J). Below we sketchhow to omit (J) an idea which was communicated by Huang [22]. For

Lp SPECTRUM OF GRAPHS 19

p = 2, the theorem (without the assumption (J)) is found in [14] basedon ideas developed in [20, 24] and [36].

Theorem 5.1 (Corollary 1.2 in [20] and [22]). Let (b, c) be a connectedgraph over (X,m) and ρ be an intrinsic metric with bounded degreeon balls (D). If f ∈ `p(X,m), p ∈ (1,∞), is a positive subharmonicfunction then f is constant.

Refining the Caccioppoli inequality from the proof of Theorem 5.1above, see [20, Lemma 3.1], and applying it recursively, we obtain adiscrete version of Karp’s theorem. However, for the recursive schemewe need the finite jump size assumption (J).

Theorem 5.2 (Theorem 1.1 in [20]). Let b be a connected graph over(X,m) and ρ be an intrinsic metric with bounded degree on balls (D)and finite jump size (J). If f is a positive subharmonic function suchthat for some p ∈ (1,∞) and x ∈ X

infr0>0

∫ ∞r0

1

‖f1Br(x)‖p

p

dr =∞,

then f is constant, where 1B is the characteristic function of a setB ⊆ X.

In particular, the theorem above implies the result of Hua/Jost [19]and even implies that a harmonic function f satisfying

lim supr→∞

1

r2 log r‖f1Br(x)‖pp <∞,

for some p ∈ (1,∞), is constant.

5.3. Green’s formula, Leibniz rules and mean value theorem.We first prove a Green’s formula which is an Lp version of the one in[24, Lemma3.3]. The improvement is an idea due to Huang [22]. For aset U let n(U) be the combinatorial neighborhood of U , that is

n(U) = {y ∈ X | there is x ∈ U x ∼ y} ∪ U.

Lemma 5.3 (Green’s formula). Let (b, c) be a connected graph over(X,m) and ρ an intrinsic metric. Let p, q ∈ [1,∞) such that p−1 +q−1 = 1, ε > 0 and assume Deg is bounded on Bε(U). Then for all fwith f1n(U) ∈ `p(X,m) ∩ F and g ∈ `q(X,m) with supp g ⊆ U∑

x∈X

(Lf)(x)g(x)m(x) =1

2

∑x,y∈U

b(x, y)(f(x)− f(y))(g(x)− g(y)).

Proof. We let b′ = b−b1X\U×X\U and c′ = c1U . Denote by L′ the formalLaplacian for the graph (b′, c′) and Deg′ the corresponding weighteddegree. We claim that Deg′ is bounded on n(U). Clearly, Deg′ = Deg

20 M. KELLER

on U and Deg′ ≤ Deg on Bε(U), therefore, it is bounded by assumption.For x ∈ n(U) \Bε(U) note that by the property of ρ

ε2Deg′(x) ≤ ε2

m(x)

∑y∈X

b′(x, y) =ε2

m(x)

∑y∈U

b(x, y)

≤ 1

m(x)

∑y∈U

b(x, y)ρ2(x, y) ≤ 1

and, hence, Deg′(x) ≤ ε−2 for x ∈ n(U) \ Bε(U). Now, as Deg′ isbounded on n(U) we get by Holder’s inequality∣∣∣ ∑x,y∈n(U)

b′(x, y)f(x)g(y)∣∣∣ ≤ (∑

n(U)

|f |pDeg′m) 1p(∑n(U)

|g|qDeg′m) 1q<∞,

where the finiteness follows from the boundedness of Deg′, f1n(U) ∈`p(X,m) and g ∈ `q(X,m). Simultaneously,∣∣∣ ∑x,y∈n(U)

b′(x, y)f(x)g(x)∣∣∣ ≤ (∑

n(U)

|f |pDeg′m) 1p(∑n(U)

|g|qDeg′m) 1q<∞.

Now, we proceed as in the proof of [24, Lemma3.3]. Note that absoluteconvergence of every term is ensured by the estimates above∑x∈X

(Lf)(x)g(x)m(x)

=∑x∈U

(L′f)(x)g(x)m(x)

=∑

x∈U,y∈n(U)

b′(x, y)df(x, y)g(x) +∑x∈U

c′(x)f(x)g(x)

=∑

x,y∈n(U)

b′(x, y)df(x, y)g(x) +∑

x∈n(U)

c′(x)f(x)g(x)

=1

2

∑x,y∈n(U)

b′(x, y)df(x, y)dg(x, y) +∑

x∈n(U)

c′(x)f(x)g(x)

=1

2

∑x,y∈X

b(x, y)df(x, y)dg(x, y) +∑x∈X

c(x)f(x)g(x).

�

The following Leibniz rules follow by direct computations.

Lemma 5.4 (Leibniz rules). For all x, y ∈ X, x ∼ y and f, g : X → R

d(fg)(x, y) =f(y)dg(x, y) + g(x)df(x, y)

=f(y)dg(x, y) + g(y)df(x, y) + df(x, y)dg(x, y).

Lp SPECTRUM OF GRAPHS 21

A fundamental difference of Laplacians on graphs and on manifoldsis the absence of a chain rule in the graph case. In particular, exis-tence of a chain rule can be used as a characterization for a regularDirichlet form to be strongly local. We circumvent this problem byusing the mean value theorem from calculus instead. In particular, fora continuously differentiable function ϕ : R → R and f : X → R, wehave

d(ϕ ◦ f)(x, y) = ϕ′(ζ)(f(x)− f(y)),

for some ζ ∈ [f(x) ∧ f(y), f(x) ∨ f(y)]. In this paper we will applythis formula to get estimates for the function ϕ : t 7→ tp−1, p ∈ (1,∞).However, we need a refined inequality as it was already used in the proofof [18, Theorem 2.1]. For the convenience of the reader, we include ashort proof here.

Lemma 5.5 (Mean value inequalities). For all f : X → R and x ∼ ywith (f(x)− f(y)) ≥ 0,

(a) (f(x)− f(y))p−1 ≥ 12(fp−2(x) + fp−2(y))(f(x)− f(y)), for p ∈

[2,∞),(b) (f(x) − f(y))p−1 ≥ C(f(x) ∨ f(y))p−2(f(x) − f(y)), for p ∈

(1,∞), where C = (p− 1) ∧ 1.

Proof. (a) Denote a = f(y), b = f(x). As it is the only non-trivialcase, we assume 0 < a < b. Note that for p 6= 1

bp−1 − ap−1 = (b− a)(bp−2 + ap−2) + ab(bp−3 − ap−3).

Thus, the statement is immediate for p ≥ 3 since the second term onthe right side is non-negative in this case. Let 2 ≤ p < 3 and noteap−3 > bp−3. The function t 7→ t2−p is convex on (0,∞) and, thus, itsimage lies below the line segment connecting (b−1, bp−2) and (a−1, ap−2).Therefore,

ap−3 − bp−3 ≤ ap−3 − bp−3

(3− p)

=

∫ a−1

b−1

t2−pdt ≤ (a−1 − b−1)((bp−2 − ap−2)

2+ ap−2

)=

1

2ab(b− a)(ap−2 + bp−2).

From the equality in the beginning of the proof we now deduce theassertion in the case 2 ≤ p < 3.(b) The case p ≥ 2 follows from (a). The case 1 < p ≤ 2 in (b) followsdirectly from the mean value theorem. �

5.4. The key estimate. The lemma below is vital for the proof ofYau’s and Karp’s theorem.

22 M. KELLER

Lemma 5.6 (Lemma 3.1 in [20]). Let p ∈ (1,∞), 0 ≤ ϕ ∈ `∞(X),ε > 0 and U = suppϕ. Assume Deg is bounded on Bε(U). Then, forevery non-negative subharmonic function f with f1n(U) ∈ `p(X,m),∑

x,y∈X

b(x, y)(f(x) ∨ f(y))p−2ϕ(y)2|df(x, y)|2

≤ C∑

x,y∈n(U)

b(x, y)fp−1(x)ϕ(y)|df(x, y)dϕ(x, y)|,

where C = 4/((p− 1) ∧ 1).

Proof. From the assumptions f1U ∈ `p(X,m) and ϕ ∈ `∞(X), we inferϕ2fp−1 ∈ `q(X,m) with q = p/(p−1), i.e., p−1+q−1 = 1. Thus, bound-edness of Deg on U implies applicability of Green’s formula with f andg = ϕ2fp−1. We start by using non-negativity and subharmonicity off then applying Green’s formula (Lemma 5.3) and the first and secondLeibniz rule (Lemma 5.4)

0 ≥∑x∈X

(Lf)(x)(ϕ2fp−1)(x)m(x)

=1

2

∑x,y∈X

b(x, y)df(x, y)d(ϕ2fp−1)(x, y)

=1

2

∑x,y∈X

b(x, y)df(x, y)(ϕ2(y)dfp−1(x, y) + fp−1(x)dϕ2(x, y)

)=

1

2

∑x,y∈X

b(x, y)df(x, y)

·(ϕ2(y)dfp−1(x, y) + 2fp−1(x)ϕ(y)dϕ(x, y) + fp−1(x)|dϕ(x, y)|2

)We apply the mean value inequality, Lemma 5.5 (b), to the first term,that is

ϕ2(y)df(x, y)dfp−1(x, y) ≥ Cϕ2(y)(fp−1(x) ∨ fp−1(y))p−2|df(x, y)|2

and for third term we notice that∑x,y∈X

b(x, y)fp−1(x)df(x, y)|dϕ(x, y)|2

=1

2

∑x,y∈X

b(x, y)dfp−1(x, y)df(x, y)|dϕ(x, y)|2 ≥ 0

Hence, we obtain

0 ≥C2

( ∑x,y∈X

b(x, y)(ϕ2(y)(fp−1(x) ∨ fp−1(y))p−2|df(x, y)|2

+ 2∑x,y∈X

b(x, y)fp−1(x)ϕ(y)df(x, y)dϕ(x, y))

Lp SPECTRUM OF GRAPHS 23

Absolute convergence of the two terms in the last line can be checkedusing Holder’s inequality and the assumptions f1U ∈ `p(X,m), ϕ ∈`∞(X) and boundedness of Deg on U . Hence, we obtain the statementof the lemma. �

5.5. Caccioppoli inequality and proof of Yau’s theorem.

Theorem 5.7 (Caccioppoli-type inequality, Theorem 1.8 in [20]). Let(b, c) be a connected graph over (X,m) and ρ be an intrinsic metric withbounded degree on balls (D). Let p ∈ (1,∞). Then, there is C > 0 suchthat for every non-negative subharmonic function f and all 0 < r < R∑x,y∈Br

b(x, y)(f(x) ∨ f(y))p−2|(f(x)− f(y))|2 ≤ C

(R− r)2‖f1BR+s\Br−s‖pp,

where s is the jump size of the intrinsic metric.

Proof. Using Lemma 5.6 and the inequality ab ≤ εa2 + 14εb2, ε > 0, we

estimate ∑x,y∈Br

b(x, y)(f(x) ∨ f(y))p−2ϕ(y)2|df(x, y)|2

≤C∑

x,y∈n(Br)

b(x, y)fp−1(x)ϕ(y)|df(x, y)dϕ(x, y)|

≤1

2

∑x,y∈X)

b(x, y)(f(x) ∨ f(y))p−2ϕ(y)2|df(x, y)|2

+ C∑x,y∈X

b(x, y)(f(x) ∨ f(y))p|dϕ(x, y)|2.

Letting ϕ = η = ηBr,R with 0 < r < R and

η(x) =(

1− ρ(x,Br)

R

)∧ 0, x ∈ X.

and using the cut-off function lemma, Lemma 4.7, we arrive at∑x,y∈Br

b(x, y)(f(x) ∨ f(y))p−2η(y)2|df(x, y)|2

≤C∑x,y∈Br

b(x, y)(f(x) ∨ f(y))p|dϕ(x, y)|2 ≤ C

(R− r)2‖f1BR+s\Br−s‖pp.

�

Proof of Theorem 5.1. Letting first R → ∞ and then r → ∞ in theCaccioppoli inequality, we see as f ∈ `p(X,m), for all x

(f(x) ∨ f(y))p−2|(f(x)− f(y))|2 = 0

for all x ∼ y which immediately implies that f is constant. �

24 M. KELLER

5.6. Proof of Karp’s theorem.

Proof of Theorem 5.2. Let p ∈ (1,∞) and let f be a non-negative sub-harmonic function. Assume f1Br ∈ `p(X,m) for all r ≥ 0 since oth-erwise infr0

∫∞r0r/‖f1Br‖ppdr = 0. We assume finite jump size s < ∞.

Let η = ηBr+s,R−s with 0 < r < R − 3s and Letting ϕ = η = ηr+s,R−swith 0 < r < R− 3s and

η(x) =(

1− ρ(x,Br+s)

R− s

)∧ 0, x ∈ X.

Then by Lemma 5.6 (applied with ϕ = η) we obtain∑x,y∈BR

b(x, y)(f(x) ∨ f(y))p−2η(y)2|df(x, y)|2

=∑

x,y∈Br+2s

b(x, y)(f(x) ∨ f(y))p−2η(y)2|df(x, y)|2

≤ C∑

x,y∈n(Br+2s)

b(x, y)fp−1(x)ϕ(y)|df(x, y)dϕ(x, y)|

= C∑

x,y∈BR\

b(x, y)fp−1(x)ϕ(y)|df(x, y)dϕ(x, y)|

where we use r < R − 3s and dη(x, y) = 0, x, y ∈ Br in the last line.Now, the Cauchy-Schwarz inequality and the cut-off function lemma,Lemma 4.7, yield( ∑x,y∈BR

b(x, y)(f(x) ∨ f(y))p−2η(y)2|df(x, y)|2)2

≤ C( ∑x,y∈BR\Br

b(x, y)fp(x)|dϕ(x, y)|2)( ∑

x,y∈BR\Br

b(x, y)fp−2(x)ϕ(y)2|df(x, y)|2)

≤ C

(R− r)2‖f1BR\Br‖pp

(( ∑x,y∈Br

−∑x,y∈Br

)b(x, y)fp−2(x)ϕ(y)2|df(x, y)|2

)Let R0 ≥ 3s be such that f1BR0

6= 0 and denote

v(r) = ‖f1Br‖pp, r ≥ 0.

Moreover, for j ≥ 0, let Rj = 2jR0, ϕj = ηRj+s,Rj+1−s and

Qj+1 =∑

x,y∈BRj+1

b(x, y)fp−2(x)ϕ2j(y)|df(x, y)|2.

As ϕj−1 ≤ ϕj, we get Qj ≤ Qj+1 and together with the estimate abovethis implies

QjQj+1 ≤ Q2j+1 ≤ C

v(Rj+1)

(Rj+1 −Rj)2(Qj+1 −Qj), j ≥ 0.

Lp SPECTRUM OF GRAPHS 25

Since Rj+1 = 2Rj, dividing the above inequality byv(Rj+1)

R2j+1

QjQj+1 and

adding C/Qj+1 yield

R2j+1

v(Rj+1)+

C

Qj+1

≤ C

Qj

and, thus,

1

C

∞∑j=1

R2j+1

v(Rj+1)≤ 1

Q1

.

Now, the assumption∫∞R0r/v(r)dr = ∞ implies

∑∞j=0

R2j

v(Rj)= ∞.

Therefore, Q1 = 0. As this is true for all R0 large enough, we have

(f(x) ∨ f(y))p−2|df(x, y)|2 = 0,

for all edges x ∼ y. For p ≥ 2, connectedness clearly implies that f isconstant. On the other hand, for p ∈ (1, 2], we always have fp−2(e+) >0 and, thus, f is constant. �

5.7. Domain of the generator. In Lemma 2.6 we demonstrated thatfor a graph we may not have LCc(X) ⊆ `2(X,m). Hence, in general Lis not a symmetric operator on the subspace Cc(X) of `2(X,m). Never-theless, we can still specify the domain of the operator. The followingresult is found in [24] for graph Laplacians and in [14] for magneticSchrodinger operators. We only sketch the idea of the proof whichfollows essentially from Theorem 5.1 above and standard argumentsfound in [28, Proof of Theorem 5 and 6]. One may find a version of theTheorem below in [20, Corollary 1.4].

Theorem 5.8. Let (b, c) be a graph over (X,m) and ρ be an intrinsicmetric with bounded degree on balls (D). Then,

D(Lp) = {f ∈ `p(X,m) | Lf ∈ `p(X,m)}, for all p ∈ (1,∞).

In particular, if additionally LCc(X) ⊆ `2(X,m), then L|Cc(X) is es-sentially selfadjoint on `2(X,m).

Idea of the proof for p = 2. For two extensions L′ and L′′ one sees byGreen’s formula that both are restrictions of L. Then u = (L′−λ)−1f−(L′′ − λ)−1f is a solution to (L − λ)u = 0 in `p(X,m) for any f ∈`2(X,m). By Theorem 5.1 and Lemma 2.1 we know u ≡ 0. Hence,(L′ − λ)−1 = (L′′ − λ)−1 and, therefore, L′ = L′′. �

Proof of Theorem 5.8 (Domain of the `p generators). Let f ∈ `p(X,m)∩F be such that (L+ 1)f = 0. Since the positive and negative part f+,f− of f are non-negative, subharmonic and in `p(X,m), they must beconstant by Yau’s theorem, Theorem 5.1. This implies f± ≡ 0 and,thus, f ≡ 0. Now, the proof of the corollary works literally line by lineas the proof of Theorem 3.2. �

26 M. KELLER

Combining this with the Hopf-Rinow type theorem, Theorem 4.3,above, we obtain an analogue to a classical result in Riemannian geom-etry which as discussed above is often referred to as Gaffney’s theorem.

Corollary 5.9 (Theorem 2 in [24]). Let b be a locally finite graphover (X,m) and ρ be an intrinsic path metric. If (X, ρ) is metricallycomplete, then L|Cc(X) is essentially selfadjoint on `2(X,m).

5.8. Recurrence. As a direct consequence of Karp’s theorem we geta sufficient criterion for recurrence of a graph. A connected graph iscalled recurrent if for all m and some (all) x, y ∈ X, we have∫ ∞

0

e−tL1{x}(y)dt =∞

which is equivalent to absence of non-constant bounded subharmonicfunctions. For a collection of various equivalent statements of recur-rence, see [20, Proposition 3.3] and references therein.

Similar analogous results to the criterion below are due to [25, The-orem 3.5] and [43, Theorem 3] which generalizes for example [3, The-orem 2.2], [38, Corollary B], [45, Lemma 3.12], [12, Corollary 1.4], [35,Theorem 1.2] on graphs.

Theorem 5.10 (Corollary 1.6 in [20]). Let b be a connected graph over(X,m) and ρ be an intrinsic metric with bounded degree on balls (D)and finite jump size (J). If for some x ∈ X∫ ∞

1

r

m(Br(x))dr =∞,

then the graph is recurrent.

For the proof of the theorem above we recall the following well knownequivalent conditions for recurrence.

Proposition 5.11 (Characterization of recurrence). Let a connectedgraph X be given. Then the following are equivalent.

(i) For the transition matrix P with Px,y = b(x, y)/∑

z∈X µxz,

x, y ∈ X, we have∑∞

n=0 P(n)(x, y) =∞ for some (all) x, y ∈ X,

where P (n) denotes the n-th power of P .(ii) For m ≡ 1 and some (all) x, y ∈ X, we have

∫∞0e−tLδx(y)dt =

∞, where δx(y) = 1 if x = y and zero otherwise.(iii) For all m and some (all) x, y ∈ X, we have

∫∞0e−tLδx(y)dt =

∞.(iv) Every bounded superharmonic (or subharmonic) function is con-

stant.(v) Every non-negative superharmonic function is constant.(vi) Every superharmonic (or subharmonic) function of finite energy

is constant.

Lp SPECTRUM OF GRAPHS 27

(vii) cap(x) := inf{E(f)|f ∈ Cc(X), f(x) = 1} = 0 for some (all)x ∈ X

A graph is called recurrent if one of the equivalent statements ofProposition 5.11 is satisfied.

Proof. The equivalence (i)⇔(ii) is shown in [39, Theorem 6]. Theequivalences (ii)⇔(vi)⇔(iii) are in [39, Theorem 2 and Theorem 9](confer [41, Theorem 3.34]). The equivalences (i)⇔(v)⇔(vii) are foundin [45, Theorem 1.16, Theorem 2.12]. The equivalence (iv)⇔(v) followssince every non-negative superharmonic function f can be approxi-mated by the bounded superharmonic functions f ∧ n, n ≥ 1. �

Proof of Theorem 5.10 (Recurrence). Theorem 5.2 implies that any non-negative bounded subharmonic function f is constant provided infr0

∫∞r0r/m(Br)dr =

∞ since ‖f1Br‖pp ≤ ‖f‖p∞m(Br), r ≥ 0. By Proposition 5.11 the graphis recurrent. �

5.9. L1-Liouville theorem and counter-examples. In this sectionwe deal with the borderline case of the `p Liouville theorem p = 1. Wefirst prove Theorem 5.12 which deals with the stochastic complete caseand then give two examples which show that there is no `1 Liouvilletheorem for non-negative subharmonic functions in the general case.

A graph is called stochastically complete if e−tL1 = 1, where 1 denotesthe function that is constantly one on X. For the relevance of theconcept see [11, 28, 46]. The proof of Theorem 5.12 follows along thelines of the proof of [11, Theorem 13.2].

Theorem 5.12 (Grigor’yan’s L1 theorem). Assume a connected graphis stochastically complete. Then, every non-negative superharmonicfunction in `1(X,m) is constant.

Proof of Theorem 5.12. If the graph is recurrent, then there are no non-constant non-negative superharmonic functions by Proposition 5.11. Soassume the graph is not recurrent which impliesG(x, y) =

∫∞0e−tLδx(y)dt <

∞, x, y ∈ X, again by Proposition 5.11. Let Kn, n ≥ 0, be an sequenceof finite sets exhausting X and Gn(x, y) =

∫∞0e−tLnδx(y)dt, where

Ln are the finite dimensional operators arising from the restriction ofthe form Q to Cc(Kn). By domain monotonicity, [28, Proposition 2.6and 2.7] the semigroups e−tLn converge monotonously increasing to e−tL

and, hence, Gn(x, y) ≤ G(x, y) for x, y ∈ Kn, and Gn ↗ G, n → ∞,pointwise. By direct calculation for any x ∈ Kn

LnGn(x, y) =

∫ ∞0

Lne−tLnδx(y)dt =

∫ ∞0

∂te−tLnδx(y)dt

= [e−tLnδx(y)]∞0 = δx(y)

and, hence, Gn(x, ·) are harmonic on Kn \ {x}, n ≥ 0.

28 M. KELLER

Let u be a non-trivial non-negative superharmonic function which isstrictly positive by the minimum principle [28, Theorem 8]. Let U ⊆ Xbe finite with o ∈ U ⊆ Kn, n ≥ 0 and C > 0 be such that Cu ≥ G(o, ·)on U . By the minimum principle Cu ≥ Gn(o, ·) on Kn \ {o} and,hence, Cu ≥ G(o, ·) on X by the discussion above. If the graph isstochastically complete, then we get by Fubini’s theorem

C‖u‖1 ≥ ‖G(o, ·)‖1 =

∫ ∞0

∑x∈X

e−tLδo(x)m(x)dt =

∫ ∞0

e−tL1(o)dt

=

∫ ∞0

dt =∞.

Hence, u is not in `1(X,m). �

In the proof we show that in the non-recurrent case there are nonontrivial superharmonic functions in `1. This is explained since inthe case of finite measure recurrence and stochastic completeness areequivalent [39, Theorem 12].

Next, we show that in general there is no `p Liouville theorem forp ∈ (0, 1]. This is analogous to the situation in Riemannian geometry,where counter-examples were given by Chung and Li/Schoen. Our firstexample is a graph of finite volume and the second is of infinite volume.

Example 5.13 (Finite volume). Let G = (X,µ,m) be an infinite linegraph, i.e., X = Z and x ∼ y iff |x − y| = 1 for x, y ∈ Z. Definethe edge weight by b(x, y) = 21−(|x|∨|y|) for x ∼ y and the measure mby m(x) = (|x| + 1)−22−|x|, x ∈ Z, which implies m(X) < ∞. Thevertex degree path metric ρ0 is compatible as it satisfies δ(x, x + 1) ≥C(|x|+1)−1 and, thus,

∑∞x=−∞ δ(x, x+1) =∞. However, the function

f defined as

f(x) = sign(x)(2|x| − 1), x ∈ Z,

is harmonic and, clearly, f ∈ `p(X,m), p ∈ (0, 1].

Example 5.14 (Infinite volume). We can extend the example aboveto the infinite volume case. Let G be the graph from above and G′ be alocally finite graph of infinite volume which allows for a compatible pathmetric. We glue G′ to the vertex x = 0 of the graph G by identifyinga vertex in G′ with x = 0. Next, we extend the path metrics in thenatural way and obtain (by renormalizing the edge weights of the metricat the edges around x = 0 if necessary) again a compatible intrinsicmetric and the graph has infinite volume. Moreover, we extend f onG from above by zero to G′ and obtain a harmonic function which isin `p, p ∈ (0, 1].

Lp SPECTRUM OF GRAPHS 29

6. Independence of the spectrum

In the beginning of the 80’s Simon [40] asked a famous questionwhether the spectra of certain Schrodinger operators on Rd are inde-pendent on which Lp space they are considered. Hempel/Voigt [17]gave an affirmative answer in 1986. Here, we consider a geometricanalogue of this question. First, we discuss a theorem by Sturm onRiemannian manifolds and secondly, we present a result for weightedgraphs involving intrinsic metrics.

6.1. Historical notes. In 1993 Sturm proved a theorem for uniformlyelliptic operators on a complete Riemannian manifold M whose Riccicurvature is bounded below. We assume that M grows uniformlysubexponentially if for any ε > 0 there is C > 0 such that for allr > 0 and all x ∈M

vol(Br(x)) ≤ Ceεrvol(B1(x)).

Then the spectrum of a uniformly elliptic operator on this manifold isindependent of the space Lp(M), p ∈ [1,∞] on which it is considered.

6.2. The result for graphs with intrinsic metrics. A graph (b, c)over (X,m) with an intrinsic metric ρ is said to have uniform subexpo-nential growth if for any ε > 0 there is C > 0 such that for all r > 0and all x ∈M

m(Br(x)) ≤ Ceεrm(x).

The proof of the following theorem follows closely the strategy of Sturmin [42].

Theorem 6.1 (Theorem 1 in [1]). Let b be a connected graph over(X,m) and ρ be an intrinsic metric such that the balls are finite (B),which has finite jump size (J) and the graph grows uniformly subexpo-nentially. Then,

σ(Lp) = σ(L2), p ∈ [1,∞].

The statement of the theorem is in general wrong if one drops thegrowth assumption. In particular, if λ0(L2) > 0, then the graph growsexponentially, i.e., µ > 0 by the section above. On the other hand,if the graph is stochastically complete, then 1 is an eigenfunction ofL∞ to the eigenvalue 0 and, therefore, λ0(L1) = 0 by duality. Hence,λ0(L1) < λ0(L2).

On the other hand, it is an open question what happens for graphsthat are subexponentially growing, i.e., µ = 0, but not uniformly subex-ponentially growing.

30 M. KELLER

6.3. Consequences of uniform subexponential growth.

Lemma 6.2. Assume the graph has uniform subexponential growth.Then, for all ε > 0 there is C(ε) > 0 such that

(a) m(x) ≤ Ceεd(x,y)m(y), for all x, y ∈ X,(b) #Br(x) ≤ Ceεr, for all r ≥ 0, where #Br(x) denotes the num-

ber of vertices in Br(x).(c)

∑y∈X e

−εd(x,y) ≤ C, for all x ∈ X.

Proof. To prove (a) let x, y ∈ X. Using the uniform subexponentialgrowth assumption and x ∈ Bd(x,y)(y) yields m(x) ≤ m(Bd(x,y)(y)) ≤Ceεd(x,y)m(y). Turning to (b), let x ∈ X and r ≥ 0. We obtain using(a) and the uniform subexponential growth assumption

#Br(x) =∑

y∈Br(x)

m(y)/m(y) ≤ Ceεrm(Br(x))

m(x)≤ C2e2εr.

The final statement (c) follows also by direct calculation using (b)(with ε1)∑y∈X

e−εd(x,y) =∞∑r=1

∑y∈Br(x)\Br−1(x)

e−εd(x,y) ≤∞∑r=1

e−ε(r−1)#Br(x) ≤ C∞∑r=1

e(ε1−ε)r

Hence, choosing ε1 = ε/2 yields the statement. �

Remark. (a) Lemma 6.2 (b) implies finiteness of distance balls. Onthe other hand, finite jump size s implies that for each vertex x allneighbors of x are contained in Bs(x). Hence, graphs with uniformsubexponential growth and finite jump size are locally finite.

(b) Finiteness of distance balls has strong consequences on the unique-ness of selfadjoint extensions. In particular, by [24, Corollary 1] impliesthatQ is the maximal form on `2 and that the restriction of L2 to Cc(X)(whenever Cc(X) ⊆ D(L2)) is essentially selfadjoint.

In the following we discuss examples to clarify the relation betweenuniform subexponential growth and bounded geometry in the discretesetting.

Recall that we speak of bounded geometry if n/m is a bounded func-tion which is a natural adaption to the situation of weighted graphs.In Examples 6.3 and 6.4 below, we show that there are graphs withuniform subexponential growth and unbounded geometry. For com-pleteness we also give a example of bounded geometry and exponentialgrowth which is certainly well-known.

Example 6.3 (Uniform subexponential growth and unbounded geom-etry). Let X = N, m ≡ 1, c ≡ 0 and consider b such that b(x, y) = 0for |x − y| 6= 1, b(x, x + 1) = x for x ∈ 4N and b(x, x + 1) = 1 oth-erwise. Clearly, (n/m)(x) = n(x) = x + 1 for x ∈ 4N and, thus, thegraph has unbounded geometry. In particular, Lp is unbounded for all

Lp SPECTRUM OF GRAPHS 31

p ∈ [1,∞]. Moreover, let ρ be the path metric induced by the edge

weights w(x, x + 1) = (n(x) ∨ n(x + 1))−12 . Then, ρ is intrinsic and

we obtain that ρ(x, y) ≥ (|x − y| − 3)/(4√

2) for all x, y ∈ X. Hence,m(Br(x)) = #Br(x) ≤

√2(8r+ 6) which implies uniform subexponen-

tial growth.

Example 6.4 (Exponential growth and bounded geometry). Take aregular tree, c ≡ 0, set b to be one on the edges and zero otherwiseand let m ≡ 1. This graph has bounded geometry but is clearly ofexponential growth.

It is apparent that the graph in Example 6.3 (a) above has boundedcombinatorial vertex degree and the unbounded geometry is inducedby the edge weights. So, one might wonder whether there are alsoexamples of uniform subexponential growth but with unbounded com-binatorial vertex degree which is the criterion for unbounded geometryin the classical setting. The proposition below shows that this is im-possible under the assumptions of uniform subexponential growth andfinite jump size. Recall that by the remark below Lemma 6.2 we al-ready know that the graph must be locally finite.

The combinatorial vertex degree deg is the function that assigns toeach vertex the number of neighbors, that is deg(x) = #{y ∈ X |b(x, y) > 0}, x ∈ X.

Proposition 6.5. If the graph has uniform subexponential growth withrespect to a metric with finite jump size s, then the combinatorial vertexdegree is bounded.

Proof. Suppose the graph has unbounded vertex degree, i.e., there is asequence of vertices (xn) such that deg(xn) ≥ n2 for all n ≥ 1. We showthat there is a sequence of vertices zn such that m(Bs(zn))/m(zn) isunbounded and thus the graph does not have uniform subexponentialgrowth.

If, for n ≥ 1, there is a neighbor yn of xn such that m(yn) ≤m(xn)/

√deg(xn), then we estimate using xn ∈ Bs(yn)

m(Bs(yn))

m(yn)≥ m(xn)

m(yn)≥√

deg(xn) ≥ n

We set zn = yn in this case. If, on the other hand, m(y) ≥ m(xn)/√

deg(xn)for all neighbors y of xn, then

m(Bs(xn))

m(xn)≥ 1

m(xn)deg(xn)

m(xn)√deg(xn)

≥√

deg(xn) ≥ n

and set zn = xn in this case. Hence, we have proven the claim. �

Corollary 6.6. Assume there is D > 0 such that b ≤ D and m ≥ 1/D.Then, uniform subexponential growth with respect to a metric with finitejump size implies bounded geometry.

32 M. KELLER

Proof. One simply observes that n/m ≤ D2 deg and the statementfollows from the proposition above. �

The lemma and the corollary above mean for the standard Laplacians

∆pϕ(x) =∑

y∼x(ϕ(x)−ϕ(y)) on `p(X, 1) and ∆(n)p ϕ(x) = 1

deg(x)

∑y∼x(ϕ(x)−

ϕ(y)) on `p(X, deg), that uniform subexponential growth implies boundedgeometry, both in the sense of bounded n/m and also in the sense ofbounded deg.

6.4. Lipschitz continuous functions. We denote by CLip the realvalued bounded Lipschitz continuous functions with Lipschitz constantε > 0 with respect to an intrinsic metric ρ, i.e.,

CLip := {ψ : X → R | ψ(x)− ψ(y) ≤ ερ(x, y), x, y ∈ X} ∩ `∞(X,m).

Lemma 6.7. Let (b, c) be a connected graph over (X,m) and ρ be anintrinsic metric which has finite jump size (J). Let ε > 0 and let s bethe jump size of ρ. Then, for all ψ ∈ CLip,

(a) eψ is a bounded Lipschitz continuous function, in particular,eψD(Q) = D(Q).

(b) |1− eψ(x)−ψ(y)| ≤ εeεsρ(x, y), for x ∼ y.(c) |(e−ψ(x) − e−ψ(y))(eψ(x) − eψ(y))| ≤ 2ε2eεsρ(x, y)2, for x ∼ y.

Proof. The first statement of (a) follows from mean value theorem, thatis for any x, y ∈ X we have

|eψ(x) − eψ(y)| ≤ |ψ(x)− ψ(y)|e‖ψ‖∞ ≤ ερ(x, y)e‖ψ‖∞ .

Now, eψD(Q) ⊆ D(Q) is a consequence of [13, Lemma 3.5]. The otherinclusion follows since e−ψ is also bounded and Lipschitz continuous.Similarly, we get (b) using the Taylor expansion of the exponentialfunction

|1− eψ(x)−ψ(y)| =∑k≥1

(ψ(x)− ψ(y))k

k!≤ ερ(x, y)

∑k≥1

(εs)k−1

k!≤ ερ(x, y)eεs,

and, similarly, using |(e−ψ(x) − e−ψ(y))(eψ(x) − eψ(y))| = 2∑

k∈2N(ψ(x)−ψ(y))k

k!we get (c). �

6.5. Kernels. Let A : D(A) ⊆ `p → `q, p, q ∈ [1,∞] be a denselydefined linear operator. We denote by ‖A‖p,q the operator norm of A,i.e.

‖A‖p,q = supf∈D(A), ‖f‖p=1

‖Af‖q.

Note that any such operator A : D(A) ⊆ `p → `q, p < ∞, withCc(X) ⊆ D(A) admits a kernel kA : X ×X → C such that

Af(x) =∑x∈X

kA(x, y)f(y)m(y)

Lp SPECTRUM OF GRAPHS 33

for all f ∈ D(A), x ∈ X, which can be obtained by

kA(x, y) =1

m(x)m(y)〈A1y, 1x〉,

where 1v(w) = 1 if w = v and 1v(w) = 0 otherwise.We recall the following well known lemma which shows that the

operator norm of A : `p → `q can be estimated by its integral kernel.

Lemma 6.8. Let p, p∗ ∈ [1,∞), p−1 +p∗−1 = 1, and let A be a denselydefined linear operator with Cc(X) ⊆ D(A) ⊆ `p. Then,

(a) ‖A‖p,q ≤(∑

y ‖kA(·, y)‖p∗q m(y)) 1p∗

for q < ∞ and p ∈ (1,∞),

and‖A‖1,q ≤ supy ‖kA(·, y)‖q for q <∞.

(b) ‖A‖p,∞ ≤ supx ‖kA(x, ·)‖p∗ and equality holds if p = 1.

Proof. Statement (a) follows from the fact ‖f‖p = supg∈`p∗ , ‖g‖p∗=1〈g, f〉and twofold application of Holder inequality. The first part of (b)follows simply from Holder inequality. For the second part note that‖A‖1,∞ ≥ supx,y∈X |A1y(x)|/m(y). �

6.6. Heat kernel estimates. We denote the kernel of the semigroupTt, t ≥ 0 by pt. As the semigroups are consistent on `p, p ∈ [1,∞], i.e.,they agree on their common domains, the kernel pt does not dependon p.

The following heat kernel estimate will be the key to p-independenceof spectra of Lp. It is proven in [4], for locally finite graphs and c ≡ 0.However, on the one hand, local finiteness is not used in [4] for thisresult and, on the other hand, the remark below Lemma 6.2 showsthat we are in the local finite situation anyway whenever we assumeuniform subexponential growth. We conclude the statement for c ≥ 0by a Feynman-Kac formula.

Lemma 6.9. Let (b, c) be a connected graph over (X,m) and ρ be anintrinsic metric such that the balls are finite (B) and which has finitejump size (J). We have for all t ≥ 0 and x, y ∈ X

pt(x, y) ≤ (m(x)m(y))−12 e−ρ(x,y) log

ρ(x,y)2et .

Proof. Denote the semigroup of the graph (b, 0) by T(0)t and the kernel

by p(0)t and correspondingly for (b, c) by Tt and pt.

For c ≡ 0 the estimate above is found in [4, Theorem 2.1] for p(0)t .

Now, by a Feynman-Kac formula, see e.g. [14], we have

pt(x, y) = Ttδy(x) = Ex[e−

∫ t0cm(Xs)dsδy(Xt)

]≤ Ex

[δy(Xt)

]= T

(0)t δy(x) = p

(0)t (x, y),

where δy = 1y/m(y). This proves the claim. �

By basic calculus, we obtain the following heat kernel estimate.

34 M. KELLER

Lemma 6.10. Let (b, c) be a connected graph over (X,m) and ρ bean intrinsic metric such that (B) and (J). For all β > 0 there exists aconstant C(β) such that for all t ≥ 0, x, y ∈ X

pt(x, y) ≤ (m(x)m(y))−12 e−βρ(x,y)+C(β)t.

Proof. Let β > 0 and r > 0 let f(r) = −r log(r/2e) + βr. Directcalculation shows that the function f assumes its maximum on thedomain (0,∞) at the point r0 = 2eβ. In particular, setting C(β) = 2eβ

yields

−ρ(x, y)

tlog

ρ(x, y)

2et≤ −βρ(x, y)

t+ C(β).

for all t > 0 and x, y ∈ X. The statement follows now from the lemmaabove. �

6.7. Proof of the theorem. In this section we prove Theorem 6.1following the strategy of [42]. The proof is divided into several lemmasand as always we assume that d is an intrinsic metric with finite jumpsize s.

Lemma 6.11. Let (b, c) be a connected graph over (X,m) and ρ be anintrinsic metric such (B) and (J). For every compact set K ⊆ ρ(L2)there is ε > 0 and C <∞ such that for all z ∈ K and all ψ ∈ CLip

‖e−ψ(L2 − z)−1eψ‖2,2 ≤ C.

Proof. Let ε > 0 and ψ ∈ CLip. By Lemma 6.7 we have e−ψD(Q) =eψD(Q) = D(Q). Let Qψ be the (not necessarily symmetric) form withdomain D(Qψ) = D(Q) acting as

Qψ(f, g) := Q(e−ψf, eψg)−Q(f, g).

Application of Leibniz rule yields, for f ∈ D(Qψ),

|Qψ(f, f)| =1

2

∑x,y∈X

b(x, y)|f(y)|2(e−ψ(x) − e−ψ(y))(eψ(x) − eψ(y))

+1

2

∑x,y∈X

b(x, y)f(y)(f(x)− f(y))(1− eψ(y)−ψ(x))

+1

2

∑x,y∈X

b(x, y)f(y)(1− eψ(x)−ψ(y))(f(x)− f(y)).

Applying Cauchy-Schwarz inequality, Lemma 6.7 (b) and (c) and theintrinsic metric property, gives

. . . ≤ε2C∑x,y∈X

|f(x)|2b(x, y)ρ(x, y)2 + 2Cε( ∑x,y∈X

|f(x)|2b(x, y)ρ(x, y)2) 1

2Q(f)

12

≤ Cε2‖f‖22 + 2Cε‖f‖2Q(f)12 .

Lp SPECTRUM OF GRAPHS 35

Hence, the basic inequality 2ab ≤ (1/δ)a2 + δb2 for δ > 0 and a, b ≥ 0

(applied with a = Cε‖f‖22 and b = Q(f)12 ) yields

|Qψ(f, f)| ≤ Cε2(1 + 1δ)‖f‖22 + δQ(f).

This shows that Qψ is Q bounded with bound 0. According to [26, The-orem VI.3.9] this implies that the form Qψ +Q is closed and sectorial.It can be checked directly that the corresponding operator is eψL2e

−ψ

with domain Dψ = eψD(L2). Moreover, for K ⊆ ρ(L2) compact, wecan choose ε, δ > 0 that 2

∥∥(C(1 + 1/δ)ε2e2s + δL2

)· (L2− z)−1

∥∥2,2< 1

for all z ∈ K since C is a universal constant. Therefore, again by [26,Theorem VI.3.9] this implies existence of C = C(K, ε) such that

‖e−ψ(L2 − z)−1eψ‖2,2 = ‖(eψL2e−ψ − z)−1‖2,2 ≤ C

for all z ∈ K and ψ ∈ CLip. �

Let us recall some well known facts about consistency of semigroupsand resolvents. The semigroups Tt are consistent on `p. By the spectraltheorem the Laplace transform for the resolvent Gz = (L2 − z)−1

Gzf =

∫ ∞0

eztTtfdt,

holds for f in `2 and z ∈ {w ∈ C | <w < 0} (the open left halfplane). By density and duality arguments, this formula extends tof in `p, p ∈ [1,∞), in the strong sense and to p = ∞ in the weaksense. This shows that the resolvents (Lp − z)−1 are consistent on `p

for z ∈ {w ∈ C | <w < 0}.We denote by gα the kernel of the resolvent Gα = (Lp − α)−1 which

is independent of p ∈ [1,∞] for α < 0.

Lemma 6.12. Let (b, c) be a connected graph over (X,m) and ρ be anintrinsic metric such that (B), (J) and assume the graph has uniformsubexponential volume growth. For any ε > 0 there exists α < 0 andC <∞ such that

(a) |gα(x, y)| ≤ C(m(x)m(y))−12 e−ερ(x,y) for all x, y ∈ X,

(b) ‖eψGαe−ψm

12‖1,2 ≤ C for all ψ ∈ CLip,

(c) ‖m 12 eψGαe

−ψ‖2,∞ ≤ C for all ψ ∈ CLip.

Proof. (a) By the Laplace transform of the resolvent and Lemma 6.10,we get

gα(x, y) =

∫ ∞0

eαtpt(x, y)dt ≤ (m(x)m(y))−12 e−ερ(x,y)

∫ ∞0

e(α+C)tdt,

which yields the statement for α < −C.(b) By Lemma 6.8 (a), ψ ∈ CLip, part (a) above (with ε1 ≥ 2ε) and

36 M. KELLER

Lemma 6.2 (c)

‖eψGαe−ψm

12‖21,2 ≤ sup

y∈X‖gα(·, y)eψ(·)−ψ(y)m(y)

12‖22

≤ C supy∈X

∑x∈X

e(2ε−2ε1)ρ(x,y) <∞.

The proof of (c) works similarly using Lemma 6.8 (b). �

Lemma 6.13. Let (b, c) be a connected graph over (X,m) and ρ be anintrinsic metric such that (B), (J) and assume the graph has uniformsubexponential volume growth. Then, (L2 − z)−2 extends to a boundedoperator on `p for all z ∈ ρ(L2) and p ∈ [1,∞]. Moreover, for allcompact K ⊆ ρ(L2) there is C < ∞ such that for all z ∈ K andp ∈ [1,∞]

‖(L2 − z)−2‖p,p ≤ C.

Proof. For z ∈ ρ(L2) denote by g(2)z the kernel of the squared resolvent

(Gz)2 = (L2 − z)−2. Applying the resolvent identity twice yields

(Gz)2 = (Gα + (z − α)GαGz)(Gα + (z − α)GzGα) = Gα(I + (z − α)Gz)

2Gα,

for all α < 0. Therefore,

m12 eψ(Gz)

2e−ψm12 =

(m

12 eψGαe

−ψ)(I + (z − α)eψ/2Gze

−ψ/2)2(

eψGαe−ψm

12

),

for all ε > 0 and ψ ∈ CLip. Taking the norm ‖ · ‖1,∞ and factorizing

‖ . . . ‖1,∞ ≤ ‖(. . .)‖2,∞‖(. . .)‖2,2‖(. . .)‖1,2 yields that U := m12 eψG2

ze−ψm

12

is a bounded operator `1 → `∞ by Lemma 6.11 and Lemma 6.12 withappropriate choice of α < 0, ε > 0 and all ψ ∈ CLip. Hence, the

operator U admits a kernel kU(x, y) = (m(x)m(y))12 eψ(x)−ψ(y)g

(2)z (x, y),

x, y ∈ X, and we conclude from Lemma 6.8 (b) that

|g(2)z (x, y)| ≤ C(m(x)m(y))−12 eψ(y)−ψ(x).

For chosen ε > 0 and any fixed x, y ∈ X let ψ : u 7→ ε(ρ(u, y)∧ρ(x, y))and we obtain from Lemma 6.2 (a) (with ε)

|g(2)z (x, y)| ≤ C(m(x)m(y))−12 e−ερ(x,y) ≤ Cm(x)−1e−

ε2ρ(x,y).

Thus, using Lemma 6.8 (a) and Lemma 6.2 (c), we obtain

‖G2z‖1,1 ≤ sup

y∈X

∑x∈X

|g(2)z (x, y)|m(x) ≤ C supy∈X

∑x∈X

e−ε2ρ(x,y) <∞.

As G2z is bounded for p = 1 and p = 2, it follows from the Riesz-Thorin

interpolation theorem that it is bounded for p ∈ [1, 2] and by dualityfor p ∈ [1,∞]. �

Lemma 6.14. Let (b, c) be a connected graph over (X,m). If σ(Lp) ⊆[0,∞) for all p ∈ [1,∞], then σ(L2) ⊆ σ(Lp).

Lp SPECTRUM OF GRAPHS 37

Proof. The operators (Lp − z)−1 and (Lq − z)−1 are consistent forz ∈ {w ∈ C | <w < 0} ⊆ ρ(Lp) = ρ(Lp∗) by the discussion aboveLemma 6.12 for all p ∈ [1,∞]. By the assumption σ(Lp) ⊆ [0,∞)the resolvent sets are connected which yields by [17, Corollary 1.4]that (Lp − z)−1 and (Lq − z)−1 are consistent for z ∈ ρ(Lp) ∩ ρ(Lq)for p, q ∈ [1,∞]. Moreover, by the standard theory (Lp − z)−1 and(Lq−z)−1 are analytic on ρ(Lp) = ρ(Lp∗). By the Riesz-Thorin theoremthese resolvents can be consistently extended to analytic `2-boundedoperators. That is, as a `2-bounded operator-valued function (Lp−z)−1

is analytic on ρ(Lp) which is consistent with (L2−z)−1 on ρ(Lp)∩ρ(L2).Note that, (L2 − z)−1 is analytic on ρ(L2) which is also the maximaldomain of analyticity. Thus, the statement follows by unique continu-ation. �

Proof of Theorem 6.1. We start by showing σ(Lp) ⊆ σ(L2). For the

kernel g(2)z of (L2 − z)−2 and fixed x, y ∈ X the function ρ(L2) →

C, z 7→ g(2)z (x, y) is analytic. By Lemma 6.13 we know that for any

compact K ⊆ ρ(L2) the operators (L2 − z)−2, z ∈ K are boundedon `p, p ∈ [1,∞]. Therefore (L2 − z)−2 is analytic as a family of`p-bounded operators for z ∈ ρ(L2). On the other hand, (Lp − z)−2 isanalytic as a family of `p-bounded operators with domain of analyticityρ(Lp) by [17, Lemma 3.2]. Since (L2 − z)−2 and (Lp − z)−2 agree on{w ∈ C | <w < 0}, by unique continuation they agree as analytic `p

operator-valued functions on ρ(L2). As the domain of analyticity of(Lp − z)−2 is ρ(Lp), this implies ρ(L2) ⊆ ρ(Lp).

On the other hand, since σ(Lp) ⊆ σ(L2) ⊆ [0,∞) the statementσ(L2) ⊆ σ(Lp) follows from Lemma 6.14. �

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Mathematisches Institut, Friedrich Schiller Universitat Jena, 07743Jena, Germany

E-mail address: m.keller@uni-jena.de