Contents Shi-LeiKongandKa-SingLau CRITICALEXPONENTSOFINDUCEDDIRICHLETFORMSONSELF-SIMILARSETS

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CRITICAL EXPONENTS OF INDUCED DIRICHLET FORMS ON SELF-SIMILAR SETS

Shi-Lei Kong and Ka-Sing Lau

Abstract

In [KLW], we studied certain random walks on the hyperbolic graphs X associated with the self-similar setsK, and showed that the discrete energy E_X onX has an induced energy formE_K on K. The domain ofE_K is a Besov space Λ^α,β/2_2,2 where α is the Hausdorff dimension ofK andβ is a parameter determined by the “return ratio” of the random walk. In this paper, we consider the functional relationship of E_X and E_K. In particular, we investigate the critical exponents of the β in the domain Λ^α,β/2_2,2 in order for E_K to be a Dirichlet form. We provide some criteria to determine the critical exponents through the effective resistance of the random walk on X, and make use of certain electrical network techniques to calculate the exponents for some concrete examples.

1 Introduction

The theory of Dirichlet forms on a metric measure space was originated in the seminal work of Beurling and Deny [BeD] as a generalization of the Laplacian on R^d. Motivated by the development of fractals, in the past three decades, there has been extensive study and development of this form from analytic and probabilistic points of view. In [Jo], Jonsson studied the local regular Dirichlet form (E,D) on the Sierpinski gasketK. He showed that the domain D_K is the Besov space Λ^α,β/2_2,∞

where α = log 3/log 2 and β = log 5/log 2. This consideration was extended by Pietruska-Paluba to nested fractals and d-sets [P1, P2]. It has also been studied amply in the general setting of metric measure space together with the heat kernel estimates (e.g., [GH1, GH2, GHL1, GHL2, GHL4]). In another direction, St´os [St]

investigated the Dirichlet forms on a d-set K as a stable process subordinate to the Brownian motion onK. He showed that E_K^(β),0< β <2, is non-local, and the domain is another type of Besov space Λ^α,β/2_2,2 . For the associated stable processes, the heat kernel was studied in detail by Chen and Kumagai [CK] on a d-set with 0 < β < 2. Recently, there is a great interest to study non-local regular Dirichlet forms and the associated heat kernels of the jump processes on metric measure spaces (e.g., [CKW1, CKW2, GHL3, GHH1, GHH2, HK]).

Let K be a compact subset in R^d and ν be an associated measure satisfying ν(B(x, r)) r^α for any ball B(x, r) with center at x ∈ K and radius r > 0 (by f g, we mean f and g are positive functions, and C⁻¹g ≤ f ≤ Cg for some C > 0). We define the Besov space Λ^α,β/2_2,2 to be the Banach space contained in L²(K, ν) with norm defined by

kukΛ^α,β/2_2,2 =kuk_L2+ Z Z

K×K

|u(ξ)−u(η)|²

|ξ−η|^α+β dν(ξ)dν(η)1/2

. (1.1)

(Note that ν×ν vanishes on the diagonal.) It is easy to see that Λ^α,β/2_2,2 , β >0 is decreasing with increasingβ, and it can be trivial for sufficiently largeβ. We define thecritical exponent β^∗ ofK by

β^∗ = sup{β >0 : Λ^α,β/2_2,2 contains nonconstant functions}.

The critical exponent plays an important role in the study of the Laplacian and Brownian motion on fractals. It is where a local regular Dirichlet form (i.e., the Laplacian) is expected to occur, and the domain is Λ^α,β_2,∞^∗^/2. The β^∗ is referred to as the “walk dimension” of the underlying set K as it is the scaling exponent for the space-time relation of the diffusive process: Ex(|X_t−x|²) ≈ t^2/β^∗. For the special cases, we know that for R^d, β^∗ = 2, and for the d-dimensional Sierpinski gasket,β^∗ = log(d+ 3)/log 2 [Jo] (for planar SG, β^∗= log 5/log 2≈2.322), and for

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the Sierpinski carpet, it was estimated that β^∗ ≈2.031 [B, BB]. In [GHL1], it was proved that 2≤β^∗ ≤α+ 1 under the assumption that a sub-Gaussian heat kernel exists together with a chain condition. Despite the many developments, it is still an open question whether a Laplacian will exist on some more general fractal sets.

In [KLW], we studied non-local Dirichlet forms with another approach. For a self-similar set K with the open set condition (OSC), there is a hyperbolic graph (X,E) (augmented tree) on the symbolic space ofK, and the hyperbolic boundary is H¨older equivalent toK[Ka,LW1,LW2]. We introduced a class of transient reversible random walks with return ratio λ∈(0,1), called λ-natural random walk (λ-NRW) on (X,E) with conductance c(x,y) depending on λ (see Section 2). We showed that the Martin boundary, the hyperbolic boundary andK are homeomorphic, and the hitting distribution ν is the normalized α-Hausdorff measure where α is the dimension ofK. Moreover, by using a theory of Silverstein [Si], it was proved that the graph energy

E_X^(λ)[f] = 1 2

X

x,y∈Xc(x,y)|f(x)−f(y)|² (1.2) defined by the random walk induces a positive definite bilinear form on K,

E_K^(β)(u, v) Z Z

K×K

(u(ξ)−u(η))(v(ξ)−v(η))

|ξ−η|^α+β dν(ξ)dν(η),

with β = logλ/logr, where r is the minimum of the contraction ratios of the IFS maps. Clearly the domainD_K^(β) ={u∈L²(K, µ) :E_K^(β)[u]<∞}is the Besov space Λ^α,β/2_2,2 .

In this paper we continue the investigation of the above induced bilinear func- tionalE_K^(β). Our investigation has two main focuses, namely, to establish the functional relationship of the discrete energy E_X^(λ) and the induced E_K^(β), then use it to study the critical exponents ofD^(β)_K (i.e., Λ^α,β/2_2,2 ). LetD_X^(λ)be the domain ofE_X^(λ), and letDH^(λ)_X denote the class of harmonic functions inD^(β)_K . Foru∈ D^(β)_K , we useHuto denote the Poisson integral ofuonX, and forf ∈ D^(λ)_X , let Trf(ξ) = limxn→ξf(xn).

By imposing a norm onD^(λ)_X , we prove (Theorem 3.5, Corollary 3.6)

Theorem 1.1. Suppose K is a self-similar set and assume that the OSC holds.

Then for a λ-NRW with λ∈(0, r^α), we have D^(β)_K = Tr(DH^(λ)_X ). Moreover, Tris a Banach space isomorphism on DH^(λ)_X , andTr⁻¹ =H.

The condition λ∈(0, r^α) in Theorem 1.1 will be used throughout the paper. It implies thatβ > α, and functions in D^(β)_K are H¨older continuous (Proposition 2.5);

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moreover, the convergence rate (λ/r^α)ⁿ is essential when we consider functions in D_X that tend to the boundary K.

To consider the critical exponent of D^(β)_K , we introduce some finer classification of the domains. We let

β^∗₁ := sup{β >0 :D_K^(β)∩C(K) is dense inC(K)}, β^∗₂ := sup{β >0 : dimD_K^(β) =∞}

β^∗₃ := sup{β >0 :D_K^(β) contains nonconstant functions},

Clearly we have 2 ≤ β₁^∗ ≤ β₂^∗ ≤ β₃^∗ ≤ ∞, and β₃^∗ = β^∗ for the β^∗ defined above.

In the standard cases, these three exponents are equal, but there are also examples that they are different [GuL]. It is one of the main purposes of this paper to discuss these exponents and to provide some criteria to determine them. Our approach relies on the effective resistance. We useR^(λ)(ξ, η) to denote the effective resistance for ξ, η ∈ K (see Section 4), and note that the infinite word i^∞ of {S_i}^N_i=1 will represent an element in K.

Theorem 1.2. With the assumptions as in Theorem 1.1, thenD^(λ)_K consists of only constant functions if and only ifR^(λ)(i^∞, j^∞) = 0 for all i, j= 1,· · · , N.

Consequently, if we let λ^∗₃ = sup{λ > 0 : R^(λ)(i^∞, j^∞) = 0 ∀ i, j = 1,· · ·, N}, thenβ^∗₃ = logλ^∗₃/logr (Thereom 5.4). Moreover ifβ₃^∗ > αand K is connected, we have β^∗₂ = β₃^∗ (Theorem 5.6). For β₁^∗, we give a result on the post critically-finite (p.c.f.) sets [Ki1]. We letV0 denote the “boundary” of K.

Theorem 1.3. If in addition, K is a p.c.f. set and satisfies another mild geometric condition (see Theorem 5.9). Then if

R^(λ−)(ξ, η)>0, ∀ ξ6=η ∈V0,

for some 0< < λ, then D^(β)_K is dense in C(K) with supremum norm.

Consequently, if λ^∗₁ := inf{λ >0 :R^(λ)(ξ, η) >0, ∀ξ 6=η ∈V₀} ∈ (0, r^α), then β₁^∗ = logλ^∗₁/logr.

A challenging task is to determine the effective resistance R^(λ)(i^∞, j^∞) (or R^(λ−)(ξ, η) for ξ, η ∈ V₀) to be = 0 or > 0 in the above theorems. For this we make use of the basic tools in the electrical network theory (series and parallel laws, ∆-Y transform, as well as cutting and shorting) to provide a device for such estimation, which is applied to some special cases as examples.

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For the organization of the paper, in Section 2, we summarize the needed results from [KLW]. In Section 3, we prove some basic results on the extension of functions onD^(β)_K be the Poisson integral, and the limit of function inD^(λ)_X , and prove Theorem 1.1. We define and justify the limiting effective resistance in Section 4, and prove Theorems 1.2 and 1.3 in Section 5. In Section 6, we make use of the electrical techniques to give some implementations of the theorems by some examples. Some remarks and open problems are provided in Section 7.

2 Preliminaries

We will give a brief summary of the background results in [KLW] for the convenience of the reader, and all the unexplained notations can be found there. Let {S_i}^N_i=1, N ≥2, be an iterated function system (IFS) of contractive similitudes on R^d with contraction ratios{r_i}^N_i=1, and letK be theself-similar set. Let Σ^∗ be the symbolic space ofK. Letr = min{r_i :i= 1,· · ·, N}. Forn≥1, define

J_n={x=i₁· · ·i_k∈Σ^∗ :r_x≤rⁿ< r_i₁···ik−1}, (2.1) and J₀ ={ϑ} by convention. Consider the modified symbolic spaceX =S∞

n=0J_n, which has a tree structure with an edge setE_v. The tree structure can be strength- ened to a hyperbolic graph by adding more horizontal edges on the tree according to the neighboring cells on each level n [Ka, LW1, LW2]. According to [LW2], we define

E_h={(x,y)∈ J_n× J_n: dist(Sx(K), Sy(K))≤κrⁿ, n≥0},

where κ > 0 is arbitrary but fixed. Let E = E_v ∪E_h, and call it an augmented tree. It was shown that (X,E) is a hyperbolic graph in the sense of Gromov [Wo1].

In this case, the horizontal geodesic is uniformly bounded, and for any x,y ∈ X, the canonical geodesic consists of three segments [x,u,v,y], where [x,u],[v,y] are vertical paths in E_v, [u,v] is a horizontal geodesic in J_` with the smallest`. Using this geodesic, the Gromov product(x|y) =`−h/2, where h is the length of [u,v].

We define a visual metric ρ_a(x,y) =e^−a(x|y) onX for somea >0. Let Xb_H be the completion, and define the hyperbolic boundary ∂HX = XbH \X. Then ∂HX is a compact metric space.

A geodesic ray {x_n}_n is a sequence of words with xn = i1i2· · ·in ∈ J_n. If ξ ∈ ∂_HX has a canonical representation i₁i₂· · · ∈ Σ^∞, then {x_n}_n converges to ξ, and ξ ∈ Sxn(K) for all n. It follows that for any other geodesic ray {y_n}_n converging toξ, we have x_n∼_hy_n. In the sequel, we will make use of the geodesic rays frequently to relate functions on X and K. We call the sequence {κ_n}^∞_n=1 a

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κ-sequence if each κ_n is a selection map from K to J_n, such that for each ξ ∈ K, (κn(ξ))^∞_n=1 is a geodesic ray converging to ξ. It follows from the above that

Lemma 2.1. For any IFS {S_i}^N_i=1, let (X,E) be the hyperbolic graph as defined above. Let E be a closed subset K. Then for any two κ-sequences {κ_n}^∞_n=1 and {κ⁰_n}^∞_n=1, we have

κ⁰_n(E)⊂ {x∈ J_n:d(x, κ_n(E))≤1} for each n, where d(·,·) is the graph metric on (X,E).

Theorem 2.2. [LW2] For any IFS {S_i}^N_i=1, let (X,E) be the hyperbolic graph as defined above. Then the hyperbolic boundary is H¨older equivalent to the self-similar setK, i.e., for the canonical map ι:∂_HX −→K,

ρa(ξ, η)(=e^−a(x|y)) |ι(ξ)−ι(η)|^−a/^log^r.

Throughout, we will always assume that the IFS {S_i}^N_i=1 satisfies the open set condition (OSC) [Fa]. In this case, the self-similar set K has Hausdorff dimension α which is uniquely determined by PN

i=1r^α_i = 1.

In [KLW], we introduced a class of reversible random walks on the augmented tree (X,E): forλ∈(0,1), we set theconductance c:E→(0,∞) such that

c(x,x⁻) =r_x^αλ^−|x|, and c(x,y)r^α_xλ^−|x|, x∼_hy∈X\ {ϑ}, (2.2) wherex⁻is the parent ofx,r_x:=r_i₁· · ·r_i_m forx=i₁· · ·i_m. We define thenatural random walkwith return ratioλ∈(0,1) (λ-NRW) to be the Markov chain{Z_n}^∞_n=0 onX with transition probabilityP(x,y) =c(x,y)/m(x) ifx∼y, and 0 otherwise, wherem(x) =P

y:x∼yc(x,y) is thetotal conductanceatx∈X. Note that the chain moves forward and backward with a ratio λ ∈ (0,1); hence {Z_n}^∞_n=0 is transient.

LetMdenote the Martin boundary, and letZ∞ be the pointwise limit of{Z_n}^∞_n=0, starting from ϑ.

Theorem 2.3. [KLW] Let{S_i}^N_i=1 be an IFS satisfying the open set condition, and let {Z_n}^∞_n=0 be a λ-NRW. Then

(i) the distribution ν of Z∞ onM is the α-Hausdorff measure;

(ii) the Martin kernel K(x, ξ)λ^|x|−(x|ξ)r^−α(x|ξ);

(iii) the Martin boundary M, the hyperbolic boundary ∂HX, and the self-similar set are all homeomorphic.

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It follows from part (iii) that we can carry Doob’s discrete potential theory onto the self-similar set K. We denote the space ofharmonic functions (w.r.t. P) on X by H(X) ={f ∈`(X) :P f =f}, where`(X) is the collection of real functions on X, and P f(x) =P

y∈XP(x,y)f(y). The Poisson integral for u∈L¹(K, ν) is Hu(·) =

Z

K

K(·, ξ)u(ξ)dν(ξ)∈ H(X). (2.3) The graph energyof f ∈`(X) is given by

E_X[f] = 1 2

X

x,y∈X:x∼y

c(x,y)(f(x)−f(y))², (2.4) and the domain of E_X is D_X = {f ∈ `(X) : E_X[f] < ∞}. Using Silverstein’s theorem [Si], together with Theorem 2.3 (iii), we obtain an induced quadratic form on K.

Theorem 2.4. [KLW] Under the assumptions in Theorem 2.3, the graph energy in (2.4) induces an energy formE_K[u] given by

E_K[u] :=E_X[Hu] = m(ϑ) 2

Z Z

K×K

By definition, the domain of E_K is D_K = {u ∈ L²(K, ν) : Hu ∈ D_X}. It follows from the above that D_K is the Besov space Λ^α,β/2_2,2 . If we define kuk²_E

K = E_K[u] +kuk²_L₂_(K,ν), then (D_K,k · k_E_K) is a Banach space, and is equivalent to Λ^α,β/2_2,2 . Forγ >0, we let

C^γ(K) ={u∈C(K) :kuk_Cγ :=kuk_∞+ esssup_ξ,η∈K|u(ξ)−u(η)|

|ξ−η|^γ <∞} (2.6) denote the H¨older space. We will use the following result frequently. It was proved in [GHL1] (the assumption of heat kernel stated there is not needed in the proof) that

Proposition 2.5. If β > α, then for all u∈L²(K, ν), kuk_C^γ ≤Ckuk

Λ^α,β/2_2,2 (2.7)

withγ = (β−α)/2. Consequently, Λ^α,β/2_2,2 ,→C^γ is an imbedding.

It follows that for α < β < β₁^∗, D_K ∩C(K) = D_K is trivially dense in D_K under the normk · k_E_K, and inC(K) under the supremum norm. This implies that (E_K,D_K) is a non-local regular Dirichlet form.

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3 Harmonic functions and trace functions

In this section, we will set up a natural relation between harmonic functions on X with finite graph energy and continuous functions onK with finite induced energy (Theorem 3.5). First we use Theorem 2.3(ii) to provide a “uniform tail estimate” of the Martin kernel. As in [KLW, Section 5], we introduce a projectionι:X→K by selectingι(x)∈S_x(O∩K) arbitrarily, where Ois an open set in the OSC satisfying O∩K 6=∅.

Proposition 3.1. Let {S_i}^N_i=1 be an IFS satisfying the OSC, and let {Z_n}^∞_n=0 be a λ-NRW on the augmented tree (X,E). Then for any , δ > 0, there exists a positive integern0 such that for any x∈X and|x| ≥n0, K(x, ξ)≤ε holds for any ξ∈K\B(ι(x), δ).

Proof. It follows from Theorem 2.3(ii) that

K(x, ξ)≤C₁λ^|x|(λr^α)^−(x|ξ), x∈X, ξ∈K.

Note that (x|ξ)≤(ι(x)|ξ) by [KLW, Lemma 3.9(ii)]. Hence forξ ∈K\B(ι(x), δ), r^−(x|ξ)≤r^{−(ι(x)|ξ)}≤C2|ι(x)−ξ|⁻¹ ≤C2δ⁻¹,

(the second inequality follows from Theorem 2.2 (see also [LW2, Theorem 1.2]).

Hence forε >0, we can pick positive integern0 such that the last inequality in the following holds:

K(x, ξ)≤C1λⁿ⁰r^−(α+log^λ/^log^r)(x|ξ)≤C1λⁿ⁰(C2δ⁻¹)^α+log^λ/log^r≤ε.

Let νx,x∈X, denote the hitting distribution ofZ∞on K, starting fromx. As K(x,·) =dν_x/dν, the above result shows that the mass of the distribution ν_x will concentrate around ι(x) (equivalently, Sx(K)) as |x| → ∞. We have a Fatou-type theorem as a corollary.

Corollary 3.2. Suppose {S_i}^N_i=1 satisfies OSC, and let {Z_n}^∞_n=0 be a λ-NRW on the augmented tree (X,E). Then for u ∈ C(K) and ε > 0, there exists a positive integer n₀ such that

|Hu(x)−u(ξ)| ≤ε (3.1)

whenever |x| ≥n0 andξ∈Sx(K). In particular, limn→∞Hu(xn) =u(ξ) uniformly for ξ∈K, where {x_n}_n is a geodesic ray converging to ξ.

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0 hold for any x ∈ X with

|x| ≥n0 and ξ∈K\B(ι(x), δ). Then for |x| ≥n0, by using the usual technique of splitting the following integral onK intoK∩B(ι(x), δ) andK\B(ι(x), δ), we can show that

|Hu(x)−u(ι(x))| ≤ Z

K

|K(x, η)(u(η)−u(ι(x)))|dν(η)≤ε

Hence for ξ∈S_x(K),

|Hu(x)−u(ξ)| ≤ |Hu(x)−u(ι(x))|+|u(ι(x))−u(ξ)| ≤ 2ε 3 +ε

3 =ε, and (3.1) holds. For the last statement, let {x_n}_n be a geodesic ray converging to ξ, then x_n = i₁· · ·i_n, and this i₁i₂· · · ∈ Σ^∞ is a representation of some ξ⁰ with ξ⁰ ∈ Sxn(K), and ξ⁰ = ξ in ∂HX. Hence by (3.1), we have limn→∞Hu(xn) = u(ξ⁰) =u(ξ), and the convergence is uniform onξ.

In the rest of this section, we assume that the λ-NRW has a return ratio λ ∈ (0, r^α). Then β = logλ/logr > α, and Proposition 2.5 applies.

Lemma 3.3. Suppose {S_i}^N_i=1 satisfies OSC, and let {Z_n}^∞_n=0 be aλ-NRW on the augmented tree (X,E) with λ∈(0, r^α). Then for f ∈ D_X,

(i) there exists C >0 (depend onf) such that for any geodesic ray {x_n}_n,

|f(x_n+1)−f(xn)| ≤C(λ/r^α)^n/2, and hence lim

n→∞f(x_n) exists;

(ii) for two equivalent geodesic rays (x_n)_n and (y_n)_n, lim

n→∞f(x_n) = lim

n→∞f(y_n).

Proof. (i) Letτ =λ/r^α <1. For a geodesic ray (xn)n, since

|f(xn+1)−f(xn)| ≤ s

E_X[f]

c(xn+1,xn) ≤C(λ/r^α)^n/2 =Cτ^n/2, hence the sequence (f(xn))n converges in an exponential rate.

(ii) For two equivalent geodesic rays (x_n)_nand (y_n)_n that converge to the same ξ, if they are distinct, thenxn∼_hyn for all n(or by Lemma 2.1). Then

|f(x_n)−f(y_n)| ≤ s

E_X[f]

c(x_n,y_n) ≤C⁰τ^n/2, (3.2) which tends to 0 asn→ ∞. Hence the two limits are equal.

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With the assumption as in Lemma 3.3, we can define a linear map Tr : D_X →

`(K) (called atrace map) by

(Trf)(ξ) = lim

n→∞f(x_n), ξ ∈K, (3.3)

where (xn)nis a geodesic ray that converges to ξ. We call Trf thetrace functionof f. By Lemma 3.3(ii), the limit in (3.3) is “uniform” in the sense as follows.

Proposition 3.4. Suppose {S_i}^N_i=1 satisfies OSC, and let {Z_n}^∞_n=0 be a λ-NRW with ratio λ ∈ (0, r^α) on the augmented tree (X,E). Then the limit in (3.3) is uniform, in the sense that for f ∈ D_X and ε >0, there exists a positive integern0

such that

|f(x)−Trf(ξ)| ≤ε (3.4)

whenever |x| ≥n0 and ξ∈Sx(K). Moreover,Trf is continuous onK.

Proof. The first part follows from Lemma 3.3. To prove the second part, forε >0, by (3.4), we can pick a positive integern0 such that|f(x)−Trf(ξ)|< ε/3 whenever

|x| ≥ n0 and ξ ∈ Sx(K). Since the length of the horizontal geodesics in (X,E) is uniformly bounded [LW1, LW2], say by M. Also let C be a constant such that c(x,y) ≥C⁻¹(r^α/λ)^|x| for all x ∼_h y. By assumption τ :=λ/r^α <1. We choose n1≥n0 such thatMp

CE_X[f]τⁿ¹ < ε/3.

As |ξ −η| r^(ξ|η) (Theorem 2.2), we can pick δ > 0 such that (ξ|η) ≥ n1

whenever |ξ −η| < δ. Now for ξ, η ∈ K with |ξ −η| < δ, consider a canonical geodesic [ξ,u,v, η] with horizontal geodesicπ(u,v) = [u =u₀,u₁, . . . ,u_k=v] (see Section 2). Then|u| ≥(ξ|η)≥n1, and hence

|Trf(ξ)−Trf(η)| ≤ |Trf(ξ)−f(u)|+|f(u)−f(v)|+|f(v)−Trf(η)|

< ε 3 +

k−1

X

i=0

|f(u_i)−f(u_i+1)|+ ε 3

< 2ε

3 +Mp

CE_X[f]τⁿ¹ < ε. (by (3.2)) This concludes that Trf ∈C(K).

Theorem 3.5. Suppose {S_i}^N_i=1 satisfies the OSC, and let {Z_n}^∞_n=0 be a λ-NRW with ratio λ∈ (0, r^α) on the augmented tree (X,E). Then Tr(DH_X) = D_K where DH_X is the class of harmonic functions in D_X. More precisely, TrHu = u for u∈ D_K, and HTrf =f for f ∈ DH_X.

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Proof. Foru∈ D_K, by definition we haveHu∈ DH_X. Note thatD_K∩C(K) =D_K, as D_K = Λ^α,β/2_2,2 can be imbedded into the H¨older space C^(β−α)/2(K) if β > α (Proposition 2.5). By Corollary 3.2, we have TrHu=u.

For f ∈ DH_X, let u = Trf ∈ C(K) (by Proposition 3.4). For any ε > 0, by Corollary 3.2 and Proposition 3.4, there exists a positive integer n₀ such that for

|x| ≥n0 and ξ∈Sx(K),

|f(x)−u(ξ)|< ε

2 and |Hu(x)−u(ξ)|< ε

2. (3.5)

Suppose that f 6=Hu. Assume without loss of generality thatf(x0)> Hu(x0) for somex₀ ∈ J_m. Leta_n= maxx∈Jn(f(x)−Hu(x)), n≥1. Note that f−Huis harmonic. By the Maximum Principle of harmonic functions, we regardJ_n+1 as the boundary of Xn+1 =Sn+1

k=0J_k. Then an+1 ≥maxx∈Xn(f(x)−Hu(x)) = an, thus the sequence{a_n} is non-decreasing. Hence infn≥ma_n =a_m >0. This contradicts that limn→∞a_n= 0 by (3.5). We conclude that f =Hu=HTrf.

In the above theorem, we can actually give a norm on D_X so that H :D_K → DH_X is a Banach space isomorphism. Indeed, by Proposition 3.4 and the continuity of functions in D_K, we know that functions in D_X are bounded. Fix w ∈ (0, r^α), let kfk²_`₂_(X,w)=P

x∈X|f(x)|²w^|x|, and definek · k_E_X on D_X by kfk²_E

X =E_X[f] +kfk²_`2(X,w), (3.6) Then it is direct to check thatkfk²_E

X defines a complete norm on D_X.

Corollary 3.6. With the same assumption as in Theorem 3.5 and let w∈(0, r^α).

Then for all u∈L²(K, ν),

kHuk_`2(X,w)≤Ckuk_L2(K,ν). (3.7)

Consequently, H : (D_K, k · k_E_K)→(DH_X, k · k_E_X) is an isomorphism.

Proof. LetF(x,y) denote the probability that the random walk ever visitsy from x. Forn≥1 and|y|> n, by [KLW, Theorem 4.6],

F(ϑ,y) = X

x∈Jn

Fn(ϑ,x)F(x,y) = X

x∈Jn

r^α_xF(x,y)≥r^α(n+1) X

x∈Jn

F(x,y).

HenceP

x∈JnK(x, ξ) =P

x∈Jn

F(x,y)

F(ϑ,y) ≤r^−α(n+1). It follows that foru∈L²(K, ν), kHuk²_`2(X,w) = X

x∈X Ex(u(Z∞))2

w^|x| ≤ X

x∈X Ex(u(Z∞)²) w^|x|

= X^∞

n=0wⁿ X

x∈Jn

Z

K

K(x, ξ)|u(ξ)|²dν(ξ) ≤ Ckuk²_L2(K,ν).

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where C=r^−αP∞

n=0(w/r^α)ⁿ. Asw/r^α <1, this yields (3.7). In view of Theorem 3.5, the norm isomorphism of the map H : D_K → DH_X follows from this and E_K(u, v) =E_X(Hu, Hv), and the open mapping theorem.

4 Effective resistances of E

_X

In this section, we will set up the limiting effective resistance for the λ-NRW on the augmented tree (X,E) in order to prepare for the investigation of the critical exponents ofD_K in the next section.

We will start with a general situation. Let V be a finite graph with a reversible Markov chain with conductance c(x, y), x, y∈V. Let `(V) denote the class of real valued functions on V, and let E_V(f) be the graph energy of f. For any V₁ ⊂ V, it is well-known that each f ∈ `(V1) has an harmonic extension to V, which has a minimal energy among all g ∈`(V) with g|_V₁ = f. In the following, we give an expression of the minimal energy in terms of the conductance c(x, y) of the chain on V.

Proposition 4.1. LetV be a finite set, andV =V1∪V₂with#V1≥2. Assume that there is a reversible Markov chain on V with conductancec(·,·). Then for f ∈V₁,

min

E_V[g] :g∈`(V), g|_V₁ =f = 1 2

X

x,y∈V1,x6=yc∗(x, y)(f(x)−f(y))², (4.1) where c∗(x, y) = c(x, y) +P

z,w∈V2c(x, z)GV2(z, w)P(w, y), x, y ∈ V1, x 6= y (here G_V₂(·,·) is the Green function of the random walk restricted toV2), and it defines a conductance function on V₁.

Proof. LetF^V¹(x, y) =Px(ZtV1 =y) wheretV1 is the first hitting time ofV1. Then F^V¹(z, y) =X

w∈V2

GV2(z, w)P(w, y), ∀x∈V2, y∈V1.

We can check directly from the definition thatc∗(x, y) = c∗(y, x), x, y ∈V₁,using the reversibility of the chain (i.e.,m(x)P(x, z) =m(z)P(z, x) andm(z)G_V₂(z, w) = m(w)GV2(w, z)). Hencec∗(x, y) defines a conductance on V1.

To prove (4.1), we let h(·) = P

y∈V1F^V¹(·, y)f(y) ∈ `(V). Then it is easy to check thathis the unique function such thatP h=honV2andh=f onV1. Hence E_V[h] = min{E_V[g] :g∈`(V), g|_V₁ =f}. Observe that

E_V[h] = 1 2

X

x,y∈V

c(x, y)(h(x)−h(y))² = X

x,y∈V

c(x, y)(h(x)−h(y))h(x)

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Hence

E_V[h] = X

x∈V1

h(x)X

y∈V

c(x, y)(h(x)−h(y)) (byP h=h on V₂)

= X

x∈V₁

f(x) X

y∈V₁

c(x, y)(f(x)−f(y)) + X

y∈V₂

c(x, y)X

z∈V₁

F^V¹(y, z)(f(x)−f(z))

= X

x,y∈V1

f(x)(f(x)−f(y))

c(x, y) + X

z∈V2

c(x, z)F^V¹(z, y)

(switch y andz)

= X

x,y∈V1

c∗(x, y)f(x)(f(x)−f(y))

= 1 2

X

x,y∈V₁

c∗(x, y)(f(x)−f(y))². (use c∗(x, y) =c∗(y, x))

This yields (4.1).

For a finite connected graph (X,E) with conductances, the effective resistance between two disjoint nonempty subsetsA,B⊆X is given by

RX(E, F) = (min{E_X[f] :f ∈`(X) withf = 1 onE, and f = 0 on F})⁻¹. (4.2) Also we setRX(E, F) = 0 ifE∩F 6=∅by convention. ClearlyRX(·,·) is symmetric, and the energy minimizer in (4.2) is unique, bounded in between 0 and 1, and is harmonic onX\(E∪F).

For theλ-NRW on (X,E), for convenience and the simplicity in the estimations, we will assume slightly more that the conductance on the horizontal edges satisfies c(x,y) =r^α|x|λ^−|x| forx∼_h y∈X\ {ϑ}. (4.3) (we use in (2.2)), and there is no change of the results. Let {κ_n}^∞_n=1 be a κ- sequencedefined in Section 2. For any two closed subsets Φ, Ψ⊆K, we define the level-nresistance between them (depend on κ_n) by

R^(λ)_n (Φ,Ψ) :=R_X_n(κ_n(Φ), κ_n(Ψ)), (4.4) whereX_n:=∪ⁿ_k=0J_k and has same conductance restricted fromX.

Theorem 4.2. Suppose {S_i}^N_i=1 satisfies the OSC, and let {Z_n}^∞_n=0 be a λ-NRW on the augmented tree (X,E) with λ∈(0, r^α). Then for any two closed subsets Φ, Ψ⊆K, the limitlimn→∞R^(λ)n (Φ,Ψ) exists, and is independent of the choice of the κ-sequence.

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We will prove a technical lemma first. For E, F ⊂ J_n such that in the graph distance, dist(E, F)>2, we define

∂E={x∈ J_n: dist(x, E) = 1}, ∂F ={x∈ J_n: dist(x, F) = 1}. (4.5) LetE_n:=E_n(E, F) = min{E_X_n[f] :f ∈`(X_n), f = 1 on E, f = 0 onF}, and let f be the energy minimizing function.

Lemma 4.3. Consider theλ-NRW on(X,E)withλ∈(0, r^α). Let{E_n}_n≥1,{F_n}_n≥1 be two sequences such that E_n, F_n ⊂ J_n, and lim infn→∞dist(E_n, F_n) > 2. If sup_n≥1E_n(En, Fn)<∞, then for anyε >0, there exists n0, such that for n≥n0,

X

x∈∂En

X

y∈Xn\En

c(x,y)(1−f(y))²< ε, X

x∈∂Fn

X

y∈Xn\Fn

c(x,y)f(y)² < ε,

where f is the energy minimizer ofE_n:=E_n(E_n, F_n).

Proof. We observe that E_n= 1

2 X

x,y∈Xn

c(x,y)(f(x)−f(y))²

= X

x,y∈Xn

c(x,y)f(x)(f(x)−f(y))

= X

x∈En

X

y∈Xn

c(x,y)(1−f(y)) ≥ (r^α/λ)ⁿ X

y∈∂En

(1−f(y)) (4.6)

(the third equality holds because f is harmonic on Xn\(En∪Fn)). Thus f(x) ≥ 1−(λ/r^α)ⁿE_n forx∈ ∂E_n. Using a similar argument, and that f is harmonic on Xn\(En∪Fn∪∂En), for large n, we have

E_n≥ 1 2

X

x,y∈X_n

c(x,y)(f(x)−f(y))²− X

x∈∂En

X

y∈E_n

c(x,y)(f(x)−f(y))²

= X

x∈∂En

f(x) X

y∈Xn\En

c(x,y)(f(x)−f(y)) + X

y∈E_n

c(y,y⁻)(1−f(y⁻))

≥ 1−(λ/r^α)ⁿE_n X

x∈∂En

X

y∈Xn\En

c(x,y)(f(x)−f(y)) (4.7)

(The inequality is valid because forx∈∂E_n, by harmonicity,P

y∈X_n\E_nc(x,y)(f(x)−

f(y)) =−P

y∈Enc(x,y)(f(x)−f(y)) =−P

y∈Enc(x,y)(f(x)−1)≥0).

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Now we use (4.6) and (4.7) to make the final estimate:

X

x∈∂En

X

y∈Xn\En

c(x,y)(1−f(y))²

≤ λⁿ r^α(n+1)

X

x∈∂E_n

X

y∈Xn\En

c(x,y)(1−f(y)) 2

= λⁿ r^α(n+1)

X

x∈∂En

X

y∈Xn\En

c(x,y) (1−f(x)) + (f(x)−f(y))2

≤ λⁿ r^α(n+1)

kE_n+ E_n 1−(λ/r^α)ⁿE_n

2

=:ε(n) by (4.6), (4.7)

(4.8) where k = sup_x∈X#{y :x ∼_h y} (as the graph (X,E) has bounded degree, and c(x,y) > 0 only when x ∼_h y or y = x⁻). Hence we can choose n0 such that ε(n)< εforn > n0. Analogously, using 1−f instead off, we obtain the estimate forF as well.

Proof of Theorem 4.2. We fix a λ ∈ (0, r^α) and omit the superscript (λ) in this proof. First we fix a κ-sequence {κ_n}^∞_n=0, and prove that limn→∞Rn(Φ,Ψ) exists.

For brevity, we write Φ_n := κ_n(Φ) and Ψ_n := κ_n(Ψ). If Φ∩Ψ 6= ∅, then by the property of geodesic rays in (X,E), for anyn, either Φn∩Ψ_n6=∅or min{d(x,y) :x∈ Φn,y∈Ψn}= 1 (by Lemma 2.1). In both situations, we have limn→∞Rn(Φ,Ψ) = 0 (for the second case, by (4.2),R_n(Φ,Ψ)≤(r^αnλ⁻ⁿ)⁻¹ = (λ/r^α)ⁿ).

Hence we assume that Φ∩Ψ = ∅. Then there exists ` >0 such that for n≥`, dist(Φn,Ψn)>3. By (4.2) and (4.4), forn≥`,

R_n(Φ,Ψ) = (min{E_X_n[f] :f = 1 on Φ_n, andf = 0 on Ψ_n})⁻¹. (4.9) Let E_n denote the minimal energy, and let f_n ∈`(X_n) be the energy minimizer in (4.9). Let{n_k}_k≥1 withn_k≥`be the subsequence such that limk→∞R_n_k(Φ,Ψ) = lim sup_n→∞Rn(Φ,Ψ)>0.(otherwise limn→∞Rn(Φ,Ψ) = 0). Then sup_kE_n_k <∞.

Forn < n_k and ξ∈K, by Lemma 3.3(i), we have

|f_n_k(κn(ξ))−fn_k(κn_k(ξ))| ≤

nk−n

X

m=1

|f_n_k(κn+m−1(ξ))−fn_k(κn+m(ξ))|

≤ Cλ r^α

n/2

:=ε(n). (4.10)

As λ∈(0, r^α), limn→∞ε(n) = 0. LetV₁ = Φ_n∪Ψ_n. Then for sufficiently large n

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and n_k> n, we have ε(n)< ¹₂, and

E_n_k ≥ E_X_n[f_n_k]≥min{E_X_n[f] :f ∈`(X_n), f =f_n_k onV₁}

= 1 2

X

x,y∈V1

c∗(x,y)(fnk(x)−fnk(y))² (by Proposition 4.1)

≥ X

x∈Φn

X

y∈Ψn

c∗(x,y)(fnk(x)−fnk(y))²

≥ X

x∈Φn

X

y∈Ψn

c∗(x,y)((1−2ε(n))² (by (4.10))

= min{E_X_n[f] :f ∈`(X_n), f|_Φ_n = 1−2ε(n), f|_Ψ_n = 0}

(by Proposition 4.1)

= (1−2ε(n))²E_n.

Therefore, R_n(Φ,Ψ)≥(1−2ε(n))²R_n_k(Φ,Ψ) for any large n and n_k > n. Taking limit, we have

lim inf

n→∞ Rn(Φ,Ψ)≥ lim

k→∞Rnk(Φ,Ψ) = lim sup

n→∞ Rn(Φ,Ψ).

Hence limn→∞R_n(Φ,Ψ) exists.

Next we show that the above limit is independent of the choice of theκ-sequence.

For this, we define

∂Φ_n={x∈ J_n:d(x,Φ_n) = 1}, ∂Ψ_n={x∈ J_n:d(x,Ψ_n) = 1}

as in (4.5). For any other κ-sequences {κ⁰_n}_n, it follows from Lemma 2.1 that κ⁰_n(Φ)⊂Φn∪∂Φn and κ⁰_n(Ψ)⊂Ψn∪∂Φn. Hence it suffices to show that

n→∞lim R_X_n(Φ_n∪∂Φ_n,Ψ_n∪∂Φ_n) = lim

n→∞R_n(Φ,Ψ). (4.11) Without loss of generality, we assume limn→∞R_n(Φ,Ψ) > 0. Then sup_nE_n <

∞. Let h_n ∈ `(X_n) with h_n = 1 on ∂Φ_n, h_n = 0 on ∂Ψ_n, and h_n = f_n on Xn\(∂Φn∪∂Ψn).

0≤RXn(Φn∪∂Φn,Ψn∪∂Φn)⁻¹−Rn(Φ,Ψ)⁻¹ ≤ E_X_n[hn]− E_X_n[fn].

Then by Lemma 4.3, for given ε, and for large n, E_X_n[h_n]− E_X_n[f_n] ≤ 2ε. This implies (4.11) and proves the theorem.

Theorem 4.2 implies the following definition is well defined.

Definition 4.4. With the same assumption as in Theorem 4.2, we define the (limiting) effective resistance between two closed subsets Φ andΨ in K by

R^(λ)(Φ,Ψ) := lim

n→∞R_n^(λ)(Φ,Ψ). (4.12)

(We omit the superscript (λ) if there is no confusion.)

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5 The critical exponents of D

_X

We use the trace function to establish a basic result on the existence of nonconstant functions in D_K.

Theorem 5.1. With the same assumption as in Theorem 4.2, supposeΦ,Ψare two closed subsets ofK satisfying R(Φ,Ψ)>0. Then there exists u:=u_Φ,Ψ∈ D_K such that u= 1 onΦ, and u = 0 onΨ. Moreover, uΦ,Ψ is the unique energy minimizer in D_K in the following sense

R(Φ,Ψ)⁻¹ =E_K[uΦ,Ψ] = inf{E_K[u⁰] :u⁰ ∈ D_K with u⁰ = 1 on Φ, u⁰ = 0 on Ψ}.

(5.1) Proof. First we show that the set on the right in (5.1) is non-empty. Clearly Φ∩Ψ =

∅ (otherwise R(Φ,Ψ) = 0). Fix a κ-sequence {κ_n}_n. As in the proof of Theorem 4.2, there exists a positive integer` such that κn(Φ)∩κn(Ψ) = ∅ for alln≥`, let f_n ∈ `(X_n) be the energy minimizer for κ_n(Φ) and κ_n(Ψ) as in (4.9). We extend f_n toX by setting f_n(x) = 0 forx∈X\X_n, then f_n is harmonic on Xn−1. Note that 0 ≤ fn ≤ 1 for all n ≥ `. Hence for each x ∈ X, there exists a convergent subsequence of {f_n(x)}_n≥`. By the diagonal argument, we can find a subsequence {f_n_k}_k≥1 withn₁≥`such thatf_n_k converges to a functionf ∈`(X) pointwise. We claim that

(a) f ∈ DH_X and 0≤f ≤1 on X;

(b) For any ξ∈Φ, limn→∞f(κ_n(ξ)) = 1;

(c) For anyη∈Ψ, limn→∞f(κ_n(η)) = 0.

In fact, as f_n_k is harmonic on X_n_k−1, the pointwise limitf is harmonic onX. For k≥1, let g_k be the function on the edge setEdefined by: for (x,y)∈E,

g_k(x,y) =

( c(x,y)(f_n_k(x)−f_n_k(y))², ifx,y∈X_n_k,

0, otherwise.

ThenE_n_k :=E_X_nk[f_n_k] = ¹₂P

(x,y)∈Eg_k(x,y), and limk→∞g_k(x,y) =c(x,y)(f(x)− f(y))². By Fatou’s Lemma, we have

E_X[f] = 1 2

X

(x,y)∈E

c(x,y)(f(x)−f(y))²= 1 2

X

(x,y)∈E

k→∞lim g_k(x,y)

≤ 1 2lim inf

k→∞

X

(x,y)∈E

g_k(x,y) = lim

k→∞E_n_k =R(Φ,Ψ)⁻¹<∞. (5.2) Hence (a) follows.

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To prove (b), observe that R(Φ,Ψ)>0 implies that sup_k≥1E_n_k <∞. Hence for any k≥1,n < n_k and ξ ∈Φ, by Lemma 3.3(i)

|f_n_k(κn(ξ))−1| ≤

nk−1

X

m=n

|f_n_k(κm(ξ))−fnk(κm+1(ξ))| ≤C1(λ/r^α)^n/2.

Lettingk→ ∞, we have |f(κ_n(ξ))−1| ≤C₂(λ/r^α)^n/2, hence (b) follows by letting n→ ∞. With a similar argument, we can also conclude (c).

By the claim and Theorem 3.5, let u = Trf ∈ D_K. Then 0 ≤ u ≤ 1 on K, u(ξ) = limn→∞f(κ_n(ξ)) = 1 for all ξ ∈Φ, and u(η) = limn→∞f(κ_n(η)) = 0 for all η∈Ψ.

Now we complete the proof of the theorem. By (5.2), E_K[uΦ,Ψ] = E_X[fΦ,Ψ] ≤ R(Φ,Ψ)⁻¹. For the reverse inequality, it suffices to show that R(Φ,Ψ)⁻¹ ≤ E_K[u]

for all u∈ D_K with u= 1 on Φ andu= 0 on Ψ. Fix a κ-sequence {κ_n}_n. For any ε∈ (0,¹₂), by Proposition 3.2, there exists a positive integer n0 = n0(ε) such that

|Hu(κ_n(ξ))−u(ξ)| ≤ε whenever n≥ n0 and ξ ∈ K. TakingV1 =κn(Φ)∪κn(Ψ) and g=Hu|_V₁ as in Proposition 4.1, then we have, forn≥n₀,

min{E_X_n[f] :f ∈`(X_n), f =Hu on V₁}= 1 2

X

x,y∈V1

c∗(x,y)(Hu(x)−Hu(y))². (5.3) Hence

E_X_n[Hu]≥min{E_X_n[f] :f ∈`(X_n), f =Hu onV₁}

≥ X

x∈κ_n(Φ)

X

y∈κ_n(Ψ)

c∗(x,y)(Hu(x)−Hu(y))² (by (5.3))

≥ X

x∈κn(Φ)

X

y∈κn(Ψ)

c∗(x,y) (1−ε)−ε2

= (1−2ε)²Rn(Φ,Ψ)⁻¹.

As ε can be arbitrarily small, we have R(Φ,Ψ)⁻¹ ≤ limn→∞E_X_n[Hu] = E_K[u].

Hence (5.1) follows.

The uniqueness of u_Φ,Ψ as an energy minimizer follows from the fact thatE_K is strictly convex in D_K.

The function f ∈ DH_X thus constructed is called aharmonic function induced by Φ and Ψ. The function u= Trf ∈ D_K is referred as theenergy minimizer of Φ and Ψ. We denote them byfΦ,Ψ and uΦ,Ψ respectively.

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Corollary 5.2. With the same assumption as in Theorem 4.2, the following con- ditions are equivalent: for two distinct points ξ, η∈K,

(i) there exists u∈ D_K with range [0,1] such that u(ξ) = 1 and u(η) = 0;

(ii) there exists u∈ D_K such that u(ξ)6=u(η);

(iii) R(ξ, η)>0.

In this case, R(ξ, η) = sup|u(ξ)−u(η)|²

E_K(u, u) : u∈ D_K, E_K(u, u)>0 .

Proof. Note that (i) ⇒ (ii) is trivial, and (iii) ⇒ (i) follows from Theorem 5.1.

We need only prove (ii) ⇒ (iii). We observe that the given u ∈ D_K is continuous (Proposition 2.5). Fix any κ-sequence, by Corollary 3.2, there exists n₀ >0 such that for n ≥ n₀, |Hu(κ_n(ξ))−u(ξ)| ≤ ¹₃|u(ξ)−u(η)|, and the same for η. Hence

|Hu(κ_n(ξ))−Hu(κn(η))| ≥ ¹₃|u(ξ)−u(η)|. Then by (4.2), forn > n0

|u(ξ)−u(η)|²

9Rn(ξ, η) ≤ |Hu(κ_n(ξ))−Hu(κ_n(η))|² Rn(ξ, η)

≤c(κ_n(ξ), κ_n(η))|Hu(κ_n(ξ))−Hu(κ_n(η))|²≤ E_X_n[Hu].

Taking the limit onn, we have |u(ξ)−u(η)|²

9R(ξ,η) ≤ E_K[u]<∞. HenceR(ξ, η)>0.

Corollary 5.3. With the same assumption as in Theorem 4.2, if R^(λ)(ξ, η) = 0, thenβ^∗₁ ≤logλ/logr where β₁^∗:= sup{β >0 :D_K^(β)∩C(K) is dense inC(K)}.

Proof. IfR^(λ)(ξ, η) = 0, then every u∈ D_K must satisfy u(ξ) =u(η), so D_K is not dense in C(K), which implies β₁^∗≤logλ/logr.

Remark. For the implication of (ii)⇒(iii) in Corollary 5.2, we can omitλ∈(0, r^α) (i.e., β > α), but consider u ∈ D_K ∩C(K), and replace R(ξ, η) by R(ξ, η) :=

lim infn→∞Rn(ξ, η), then the implication still holds. Consequently, Corollary 5.3 is still valid.

In the following, we will apply Corollary 5.2 to give some criteria to determine the critical exponents for β₂^∗ := sup{β >0 : dimD^(β)_K =∞} and β₃^∗ := sup{β >0 : D^(β)_K contains nonconstant functions}.

Leti_n=ii· · ·i∈ J_ndenote the unique word in levelnconsisting of symboli∈Σ, and leti^∞=ii· · · ∈Σ^∞(identified with the pointξ= limnS_i_n(K) inK). Then for two distinct symbols i, j ∈ Σ, we use R(i^∞, j^∞) to denote the effective resistance for the corresponding two points inK, andR(i^∞, j^∞) = limn→∞Rn(i^∞, j^∞).

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Theorem 5.4. With the same assumption as in Theorem 4.2, then D_K consists of only constant functions if and only if

R^(λ)(i^∞, j^∞) = 0, ∀i, j∈Σ. (5.4) Consequently, β₃^∗ = logλ^∗₃/logr if

λ^∗₃ := sup{λ >0 :R^(λ)(i^∞, j^∞) = 0, ∀i, j∈Σ} ∈(0, r^α), (5.5) and β₃^∗=∞ if the above set of λis empty.

Proof. If for some i, j∈Σ, R(i^∞, j^∞)>0, then there existsu∈ D_K with u(i^∞)6=

u(j^∞) by Proposition 5.1 (or by Corollary 5.2 (iii)⇒ (ii)). Thus it suffices to show that (5.4) impliesD_K={constant functions}.

First we claim that for u ∈ C(K), if u(xi^∞) = u(xj^∞) for any x ∈ Σ^∗ and i, j∈Σ, thenu is a constant function. Indeed, let c=u(1^∞), then choosing x=ϑ we have u(i^∞) = c for any i ∈ Σ. Choosing x = i we have u(ij^∞) = u(i^∞) = c hence u(xj^∞) = c for any x ∈ Σ¹ and j ∈ Σ. Following the similar arguments, inductively we haveu(xj^∞) =c for anyx∈Σ^∗ and j∈Σ. By continuity, u≡cis constant.

For nonconstant u∈C(K), by the claim we can pick x∈Σ^∗ and i, j∈Σ such that u(xi^∞)6=u(xj^∞). We telescope u on the cellS_x(K) to get eu=u◦S_x. Then eu(i^∞)6=eu(j^∞). By Proposition 5.3 (or by Corollary 5.2 (ii)⇒(iii)) and assumption (5.4), we must haveeu /∈ D_K. Note that

E_K[u]≥c1

Z

Sx(K)

Z

Sx(K)

|u(ξ)−u(η)|²

|ξ−η|^α+β dν(ξ)dν(η)

≥c₂ Z

K

Z

K

|eu(ξ)−eu(η)|²

|ξ−η|^α+β dν(ξ)dν(η)≥c₃E_K[eu], (5.6) henceu /∈ D_K. Finally asD_K∩C(K) =D_K by Theorem 2.2,D_K contains constant functions only.

Next we will show that β₂^∗ =β₃^∗ under the connectedness of the self-similar set.

The following lemma is a key step to include more non-trivial functions inD_K. Lemma 5.5. With the same assumption as in Theorem 4.2, supposeξ ∈K and Ψ is a closed subset in K satisfying R(ξ,Ψ) >0. Let u = uξ,Ψ ∈ D_K be the limiting harmonic function. Then for η ∈ K such that 0< u(η)<1, we have R(η,Ψ)>0 and R(ξ,Ψ∪ {η})>0.