Universal inequalities for Dirichlet eigenvalues on discrete groups

We prove universal inequalities for Laplacian eigenvalues with Dirichlet boundary conditions on subsets of certain discrete groups. The study of universal inequalities on Riemannian manifolds was initiated by Weyl, Polya, Yau, and others. Here we focus on a version by Cheng and Yang. Specifically, we prove Yang-type universal inequalities for Cayley graphs of finitely generated amenable groups, as well as for the d-regular tree (simple random walk on the free group).


Introduction
The spectral theory of Laplace-Beltrami operators on Riemannian manifolds was extensively studied in the literature, see e.g.[CH53,Cha84,SY94,Li12].For a bounded domain Ω in a Riemannian manifold, we denote by the spectrum of the Laplace-Beltrami operator with Dirichlet boundary condition on Ω, counting the multiplicity of eigenvalues.
where ω n is the volume of the unit ball in R n and vol(Ω) is the volume of Ω.It was conjectured by Pólya [P 61] that (ω n vol(Ω)) Li and Yau [LY83] proved that Payne, Polya and Weinberger [PPW56] proved the gap estimate of consecutive eigenvalues for a bounded domain in R 2 , generalized to R n by Thompson [Tho69], that for any k ≥ 1, This was improved by Hile and Protter [HP80].A sharp inequality was proved by Yang [Yan91,CY07] that As is well-known, see e.g.[Ash99], Yang's inequality implies the Payne-Polya-Weinberger inequality etc.These are called universal inequalities for eigenvalues since they are independent of the domain Ω.See [AB91, AB92, AB94, AB96, HS97, Ash99, Ash02, CY05, AB07] for more results regarding Euclidean spaces.
Universal inequalities have been generalized to eigenvalues of Laplace-Beltrami operators on Riemannian manifolds.In particular, Yang's inequality has been proved for space forms.For the unit n-sphere, Cheng and Yang [CY05] proved that k i=1 For H n , the n-dimensional hyperbolic space of sectional curvature −1, Cheng and Yang [CY09] proved that Note that (n−1) 2 4 is the bottom of the spectrum of H n .For a general Riemannian manifold, Chen and Cheng [CC08] proved a variant of Yang's inequality using related geometric quantities via isometric embedding into the Euclidean space.For universal inequalities on manifolds, we refer the readers to [Li80, YY80, Leu91, Har93, HM94, CY06, Har07, SCY08, CY09, ESHI09, CZL12, CP13, CZY16].
In this paper, we study universal inequalities for eigenvalues on graphs, in particular Cayley graphs of discrete groups.We recall the setting of general networks.A network is a pair (V, c) where V is a countable set and c : V × V → [0, ∞) is called the conductance.The conductance must satisfy 0 ≤ c(x, y) = c(y, x) < ∞ (symmetric) and and π(x) := y c(x, y) < ∞ for every x.We write x ∼ y to indicate c(x, y) > 0 (in which case we say that x ∼ y is an edge in the network).A network naturally provides a reversible Markov chain, whose transition matrix is given by P (x, y) = c(x,y) π(x) .The (normalized) Laplacian is the operator ∆ = I − P, where I denotes the identity operator, i.e.
∆f (x) = We denote by L 2 (V, π) the Hilbert space of L 2 summable functions on V, equipped with the inner product It is well-known, the Laplacian ∆ is a bounded self-adjoint operator on L 2 (V, π), whose spectrum is contained in [0, 2].We write λ min for the bottom of the spectrum of ∆.
The Laplacian with Dirichlet boundary condition on finite subsets of networks has been investigated in the literature, see e.g.[Dod84, Fri93, CG98, CY00, BHJ14].For finite Ω ⊂ V, the Laplacian with Dirichlet boundary conditions on Ω, denote by ∆ Ω , is defined as the Laplacian ∆ restricted to the subspace The eigenvalues of ∆ Ω , called Dirichlet eigenvalues on Ω, are ordered by where | • | denotes the cardinality of the subset.We are interested in proving universal inequalities on graphs, in particular Yang-type inequalities (1) and (2).Due to the discrete nature of graphs, some modification is required.
Definition 1 We say that the network (V, c) satisfies Yang's inequality (resp.the Yang-type inequality) with constant C Y (resp.C Y T ) if the following holds for any finite subset Ω ⊂ G: (3) (resp. Since λ i ≤ 2, for any i ≥ 1, one easily sees that in case of λ min = 0, the Yang-type inequality implies Yang's inequality with C Y = C Y T + 2. Following the arguments in [Yan91, Ash99, CY07], the first author et al. [HLS17] proved that the integer lattice Z n , a discrete analog of R n , satisfies Yang-type inequality, with constant C Y T = 4 n .Recently, Kobayashi [Kob20] proved the Yang-type inequality for the eigenvalues of the Laplacian (not Dirichlet eigenvalues) of a finite edge-transitive graph.
Note that Z n can be regarded as a Cayley graph of a free Abelian group.In this paper, we prove Yang-type inequalities for more general Cayley graphs of finitely generated infinite groups.

Amenable groups
Our first result regards amenable groups.Let G be a finitely generated amenable group.Consider some probability measure µ on G (which we think of as a non-negative function µ : G → [0, 1] such that x µ(x) = 1).Assume that µ is symmetric, i.e. µ(x) = µ(x −1 ) for all x ∈ G. Then µ induces a corresponding Cayley graph (or network) by setting the conductances c(x, y) = µ(x −1 y).This network corresponds to the µ-random walk on G.This network is denoted by (G, µ).
Theorem 2 Let G be a finitely generated infinite amenable group.Let µ be a symmetric probability measure on G, and consider the Cayley network (G, µ) of G with respect to µ. Set µ * := inf 1 =y∈supp(µ) µ(y).
For finitely generated groups with Abelian quotients, i.e. those groups which admit homomorphisms onto Z n for some n, we prove the Yang-type inequality with C Y T = 4 n for specific µ-random walks, see Theorem 6.This extends the result for Z n from [HLS17].

Free groups
Next, we consider Yang-type inequalities on regular trees, which can be regarded as Cayley graphs of free groups.Let T d , d ≥ 3, be a d-regular tree with the conductances of the edges c(x, y) = 1 {x∼y} 1 d , which is a discrete analog of hyperbolic space H d .The Laplacian corresponds to the generator of the simple random walk on T d .As is wellknown, the bottom of the spectrum of . Following the arguments in [CY09], we prove the following result.In order to generalize the result to Cayley graphs of amenable groups, i.e.Theorem 2, we use harmonic cocycles as test functions.The existence of harmonic cocycles for amenable groups was proved by [Mok95,KS97].
For H n , Cheng and Yang [CY09] used Busemann functions of geodesic rays to prove Yang-type inequality (2).To extend the result to T d , i.e.Theorem 3, we use the discrete analogs of Busemann functions as test functions.
The paper is organized as follows: In next section, we introduce some basic facts on networks.In Section 3, we prove the useful estimate of eigenvalues for general networks, Lemma 4. Section 4 is devoted to the proofs of main results, Theorem 2 and Theorem 3.
In the last section, we derive some applications of the Yang-type inequality, such as the Paley-Polya-Weinberger inequality and the Hile-Protter inequality, etc.
2 Notation and basic operators

Γ calculus
Let (V, c) be a network on the set of vertices V with the conductance c.We allow c(x, x) > 0, which corresponds to a self-edge at x ∈ V .
Recall the inner product on functions defined in the introduction Accordingly we write ||f || 2 = ||f || 2 π := f, f , and the space of L 2 summable functions is given by L The Dirichlet energy is defined to be ), then it is not difficult to prove the "integration by parts" formula, Define the so called carré du champ operator (at x ∈ V ) as follows: and Γ(f ) := Γ(f, f ).Note that Γ is symmetric and bi-linear.
Finally we define the scalar-valued (non-linear) functional:

Identities
In this section we summarize a few identities which we will require in the analysis below.
All are straightforward and easy to prove, and hold for all f, g ∈ L 2 (V, π). x,y We also may compute,

Universal inequality
The following is an analogue of [CY06, Proposition 1].It is the main estimate which will imply our results.
Let (V, c) be a network.Let Ω ⊂ V be a finite subset of size n = |Ω|.Let u 1 , . . ., u n be an orthonormal basis of eigenvectors for ∆ Ω defined on the subspace L 2 (Ω) of L 2 (V, π); that is, Since the Laplacian is self-adjoint, such an orthonormal basis exists, λ i ∈ R and u i are real valued.
We call such a collection (λ i , u i ) n i=1 the Dirichlet system for Ω.
Lemma 4 Let (V, c) be a network.Let Ω ⊂ V be a finite subset of size n = |Ω|.Let (λ i , u i ) n i=1 be the Dirichlet system for Ω.

⊓ ⊔
Let H be a Hilbert space and α : V → H.We extend the definitions of the inner product and of Γ, Λ by defining Here u : V → R is any (finitely supported) real valued function.With this notation, we have the following theorem generalizing Lemma 4.
Theorem 5 Let (V, c) be a network.Let Ω ⊂ V be a finite subset of size n = |Ω|.Let (λ i , u i ) n i=1 be the Dirichlet system for Ω.Let H be a Hilbert space and let α : V → H.
Then for any k < n, Note that when H = R this is exactly Lemma 4.
Proof.Let h ∈ H be any non-zero vector.Define the function α ′ : V → R by α ′ (x) = α(x), h H . Plugging this into Lemma 4 we see that we only need to compute Γ(α ′ ), Λ(α ′ , u i ), Γ(α ′ , u i ), ∆α ′ .It is simple to verify that Summing this over h in an orthonormal basis for H, we have the theorem.⊓ ⊔ 4 The proof of main results

Amenable groups
One application of Theorem 5 is for the case of amenable groups.Given a finitely generated group, there is a natural network one may define.Actually, the initial data is a finitely generated group G and a probability measure µ on G, which is assumed to be symmetric, i.e. µ(x) = µ(x −1 ).This measure is used to construct the random walk on G, which is just the Markov chain with transition matrix P (x, y) = µ(x −1 y).This Markov chain is precisely the reversible Markov chain associated to the network on G given by conductances c(x, y) = µ(x −1 y).We denote this network by (G, µ), and call it the Cayley network of G with respect to µ. (Since µ is a probability measure, in this case π(x) = 1 for all x.) For a probability measure µ on G, define µ(y).
Note that µ has finite support if and only if µ * > 0.
Recall that Kesten's amenability criterion [Kes59] states that the bottom of the spectrum of ∆ is 0 if and only if G is an amenable group.
We are now ready to prove Theorem 2. Since the G-action is unitary, we may compute that Now, if u is an eigenfunction of unit length, with ∆u = λu, then since for any 1 = y ∈ supp(µ), Since G acts unitarily on H, we have by Jensen's inequality, Plugging all the above into Theorem 5 we arrive at where we have used that λ k+1 − λ i ≤ 2. This completes the proof.⊓ ⊔

Groups with Abelian quotients
For general groups with Abelian quotients, we can prove the Yang-type inequality, analogous to the result in [HLS17].
Theorem 6 Let G be a finitely generated group.Let α : G → Z n be a surjective homomorphism.Let S = {s 1 , . . ., s n , k 1 , . . ., k m } be a generating set for G so that (α(s j )) n j=1 is the standard basis of Z n , and such that α(k j ) = 0 for all j = 1, . . ., m.Let µ be a symmetric measure supported on S∪S −1 .Let ε = 1− n j=1 (µ(s j )+µ(s −1 j )).(e.g. one may take µ(k Then, the network (G, µ) satisfies the following: For any finite Ω ⊂ G and k < |Ω|, , we get the Yang-type inequality up to an ε-defect, with constant at most 4 n .Remark 8 The case G ∼ = Z n was already treated in [HLS17], where the same result was shown, using similar methods.This is the case ε = 0 and µ(s j ) = µ(s −1 j ) = 1 2n in the above theorem.
Proof.The main advantage of α being a homomorphism is that µ(y)α(y) = ±µ(s j )e j y = (s j ) ±1 , 0, otherwise., where {e j } n j=1 is the standard basis of Z n .Thus, for the Euclidean Hilbert space for any x ∈ G. Also, ∆α ≡ 0. Now, if u is an eigenfunction of unit length, with ∆u = λu, then Γ(α), As in the proof of Theorem 2, Plugging all of this into Theorem 5, we arrive at

Trees
In this section, we prove the Yang-type inequality for d-regular tree T d , d ≥ 3, with the conductances of the edges c(x, y) = 1 {x∼y} 1 d .
Proof of Theorem 3. Fix a ray to infinity, and an origin o.Let b be the Buseman function corresponding to the ray with b(o) = 0.That is: x n+1 ∼ • • • be an infinite simple path, so x i = x j for all i = j.Becase T d is a tree, this path is necessarily a geodesic: the distance between x j , x i in the graph is always |j − i|.This path is the ray mentioned above.Now, for any j ≥ 0 set b(x j ) := −j.It is also simple to check that the function (which corresponds to choosing ξ = 1, maximizing the above expression) then ∆f = λf .Coincidentally, this is the bottom of the L 2 spectrum of ∆, i.e.
For any x let x be the unique vertex with b( x) = b(x) − 1.For a function f let f (x) := f ( x).Note that as x ranges over the whole graph, the pair (x, x) ranges over all edges in the graph, each edge counted exactly once in the direction of decreasing the Buseman function b.Thus, Also, the map x → x is a (d − 1)-to-1 map.So, Let u be an eigenfunction ∆u = λu.Note that Thus, , so using (15) and ( 16), assuming that ||u|| = 1, Combining this with (17), (18), and plugging into Lemma 4, we have that: where we used

Applications of Yang-type inequalities
In this section, we derive some applications of the Yang-type inequality on graphs.
Let (V, c) be the network with the bottom of the spectrum λ min .For any finite subset Ω, let {λ i } |Ω| i=1 be the Dirichlet eigenvalues of the Laplace on Ω. Set By the trace of the Laplacian, Corollary 9 Suppose that the network (V, c) satisfies the Yang-type inequality (3).Then for any finite subset Ω, Proof.This follows from the Yang-type inequality (3) for k = 1.

⊓ ⊔
The Yang-type inequality implies the following result, which is a discrete analog of Yang's second inequality.
Corollary 10 Suppose that the network (V, c) satisfies the Yang-type inequality (3).Then for any finite subset Ω, if .
Proof.Let C = C Y T .Without loss of generality, we may assume that λ k+1 > λ 1 , otherwise the result is trivial.By the Yang-type inequality (3), Note that the function which implies that b i is non-increasing.Using Chebyshev's inequality, i.e. Thus, , which proves the theorem.

⊓ ⊔
By the above result, we derive the following inequality, a discrete analog of the Hile-Protter inequality.
Corollary 11 Suppose that the network (V, c) satisfies the Yang-type inequality (3).Then for any finite subset Ω, if Proof.Without loss of generality, we may assume that λ where we used Jensen's inequality for g(x).By Corollary 10, where we used Chebyshev's inequality in the last line.
By plugging it into (21), we prove the result.

⊓ ⊔
This result yields a discrete analog of the Paley-Polya-Weinberger inequality.
Proof.Without loss of generality, we assume that λ k < λ k+1 .By Corollary 11, Then we have Now we prove an upper bound estimate for λ k .

Theorem 3
The network given by the simple random walk on the d-regular tree T d (where d > 2) satisfies theYang-type inequality with constant C Y T = 8 √ d−1 d .We sketch the proof strategies of Theorem 2 and Theorem 3: By the variational principle, for an upper bound estimate of eigenvalues, it suffices to construct appropriate test functions.Following the arguments in [Yan91, CY06], for any network and any test function α : V → R, we prove the Dirichlet eigenvalues satisfy some crucial estimate involving α, see Lemma 4, a discrete analog of [CY06, Proposition 1].This enables us to derive the Yang-type inequality with choice of α with nice properties for ∆α and the gradient of α.For R n or Z n , as in [Yan91, CY07, HLS17], linear functions are good candidates for test functions.
Furthermore, for any vertex z, let z * be the closest vertex to z from the above path.Set b(z) = b(z * ) + dist(z, z * ).The important properties of b are thus: b : T d → Z is a function such that b(o) = 0 and such that every vertex x has d − 1 neighbors y ∼ x with b(y) = b(x) + 1, and exactly one neighbor x ∼ x with b( x) = b(x) − 1.One easily sees that 2Γ(b)(x) = 1 ∀x ∈ T d .