on-diagonal heat kernels on self-similar fractals

on-diagonal heat kernels on self-similar fractals

Dedicated to my mother on the occasion of her 65th birthday

Abstract. Letpt(x, y) be the canonical heat kernel associated with a self- similar Dirichlet form on a self-similar fractal and letds denote the spectral dimension of the Dirichlet space, so thatt^ds/2pt(x, x) is uniformly bounded from above and below by positive constants fort∈(0,1]. In this article it is proved that, under certain mild assumptions onpt(x, y), for a “generic” (in particular, almost every) pointxof the fractal,p(·)(x, x)neither varies regu- larly at 0 (and hence the limit limt↓0t^ds/2pt(x, x) doesnotexist)noradmits a periodic functionG:R→Rsuch thatpt(x, x) =t^−ds/2G(−logt) +o(t^−ds/2) ast↓0. This result is applicable to most typical nested fractals (butnot to thed-dimensional standard Sierpi´nski gasket withd≥2 at this moment) and allgeneralized Sierpi´nski carpets, and the assertion of non-regular variation is established also for post-critically finite self-similar fractals (possibly without good symmetry) possessing a certain simple topological property.

Contents

1. Introduction 2

2. Framework and main results 5

3. Proof of Theorems 2.17 and 2.18 10

4. Post-critically finite self-similar fractals 15

4.1. Harmonic structures and resulting self-similar Dirichlet spaces 15 4.2. Cases with good symmetry and affine nested fractals 16

4.3. Cases possibly without good symmetry 18

5. Sierpi´nski carpets 23

References 28

Version of May 1, 2013.

2010Mathematics Subject Classification. Primary 28A80, 60J35; Secondary 31C25, 37B10.

Key words and phrases. Self-similar fractals, Dirichlet form, heat kernel, oscillation, short time asymptotics, post-critically finite self-similar fractals, generalized Sierpi´nski carpets.

The author was supported in part by SFB 701 of the German Research Council (DFG).

1

1. Introduction

Heat kernels on fractals are believed to exhibit highly oscillatory behavior as opposed to the classical case of Riemannian manifolds. For example, as a general- ization of the results of [12, 30, 7] for the standard Sierpi´nski gasket, Lindstrøm [32] constructed canonical Brownian motion on a certain large class of self-similar fractals callednested fractals, and Kumagai [29] proved that its transition density (heat kernel)p=p_t(x, y) satisfies the two-sidedsub-Gaussian estimate

(1.1) c1.1

t^ds/2exp (

−(ρ(x, y)^dw c_1.1t

)_d¹

w−1

)

≤pt(x, y)≤ c1.2

t^ds/2exp (

−(ρ(x, y)^dw c_1.2t

)_d¹

w−1

) . Herec_1.1, c_1.2∈(0,∞) are constants,d_s∈[1,∞) andd_w∈[2,∞) are also constants called the spectral dimension and the walk dimension of the fractal, respectively, andρis a suitably constructed geodesic metric on the fractal which is comparable to some power of the Euclidean metric¹. Later Fitzsimmons, Hambly and Kumagai [10] extended these results to a larger class of self-similar fractals calledaffine nested fractals. In particular, given an affine nested fractalK, for anyx∈Kwe have (1.2) c1.1≤t^ds/2pt(x, x)≤c1.2, t∈(0,1],

and then it is natural to ask how t^ds/2p_t(x, x) behaves as t ↓ 0 and especially whether the limit

(1.3) lim

t↓0t^ds/2p_t(x, x)

exists or not. As Barlow and Perkins conjectured in [7, Problem 10.5] in the case of the Sierpi´nski gasket, this limit was believed not to exist for most self-similar fractals, but this problem had remained open until the author’s recent paper [21].

It was proved in [21] that, under very weak assumptions on the affine nested fractalK, the limit (1.3) does not exist for “generic” (hence almost every)x∈K, and that the same is true for any x ∈ K when K is either the d-dimensional standard (level-2) Sierpi´nski gasket with d≥2 or the N-polygasket withN ≥3, N/46∈N(see Figure 2 below). The proofs of these facts, however, heavily relied on the two important features of affine nested fractals — they arefinitely ramified (i.e.

can be made disconnected by removing finitely many points) andhighly symmetric.

In particular, the results of [21] were not applicable to self-similar fractals without these properties like Hata’s tree-like set, which admits no isometric symmetry as shown in Proposition 4.17 below, and the Sierpi´nski carpet, which is infinitely ramified (see Figure 1).

The purpose of this paper is twofold. First, we replace the assumptions of finite ramification and symmetry of the fractal with certain properties of the heat kernel which are expected to be much robuster in many cases. In particular, our main results imply the non-existence of the limit (1.3) for “generic” pointsxin the cases of Hata’s tree-like set and of the Sierpi´nski carpet. Secondly, we establish not only the non-existence of the limit (1.3) but also more detailed descriptions of the oscillation ofpt(x, x) ast↓0 for “generic” pointsxof the self-similar fractal.

More specifically, let K be the self-similar set determined by a finite family {Fi}i∈S of injective contraction maps on a complete metric space, so thatK is a

1To be precise, the heat kernel estimate in [29] had been presented in terms of the Euclidean metric, and the geodesic metricρwas constructed later in [10].

Figure 1. Examples of self-similar fractals within the reach of the main results of this paper. From theleft, two-dimensional level-3 Sierpi´nski gasket, pentagasket (5-polygasket), Hata’s tree-like set and Sierpi´nski carpet

compact metrizable topological space satisfying K = ∪

i∈SFi(K), and let V0 be the set of boundary points ofK(see Definition 2.3 for the precise definition ofV0).

AssumeK6=V₀, letµbe a Borel measure onK satisfyingµ(F_w1◦ · · · ◦F_wm(K)) = µ_w1· · ·µ_wm for anyw₁. . . w_m∈∪

n∈NSⁿfor some (µ_i)_i∈S∈(0,1)^Swith∑

i∈Sµ_i= 1, and assume that (E,F) is a self-similar symmetric regular Dirichlet form on L²(K, µ) with resistance scaling factor rgiven by r=(

µ^2/d_i ^s−1)

i∈S for someds∈ (0,∞) (see Definition 2.7 for details). Further assuming that (K, µ,E,F) admits a continuous heat kernelp=p_t(x, y) and that the upper inequality of (1.1) holds for t∈(0,1] for somed_w∈(1,∞) and a suitable metricρonKsatisfyingµ(B_s(x, ρ))≤ c1.3s^dsdw/2, (s, x)∈(0,1]×K, Bs(x, ρ) :={y ∈K |ρ(x, y)< s}, we establish the following assertions as the main results of this paper:

(NRV) p_(·)(x, x)does not vary regularly at0 for “generic”x∈K, if

(1.4) lim sup

t↓0

pt(y, y)

pt(z, z) >1 for somey, z ∈K\V0.

(NP) “Generic”x∈K does notadmit a periodic function G:R→Rsuch that p_t(x, x) =t^−ds/2G(−logt) +o(t^−ds/2) ast↓0, if

(1.5)

lim inf

t↓0

pt(y, y)

pt(z, z) >1 for somey, z ∈K\V0. (1.6)

Note that we still have the on-diagonal estimate (1.2) in this situation as shown in Proposition 2.16 below, and recall (see e.g. [9, Section VIII.8]) that a Borel measurable function f : (0,∞)→(0,∞) is said tovary regularly at 0 if and only if the limit limt↓0f(αt)/f(t) exists in (0,∞) for any α∈(0,∞). In particular, if x∈Kandp(·)(x, x) does not vary regularly at 0, then it also follows that the limit (1.3) does not exist. Note also that a log-periodic behavior of the form (1.5) is valid when xis the fixed point of Fw₁◦ · · · ◦Fw_m for some w1. . . wm ∈∪

n∈NSⁿ by Proposition 3.7 below, which is a slight generalization of [16, Theorems 4.6 and 5.3]. Such a log-periodic behavior has been observed in various contexts of analysis on fractals such as Laplacian eigenvalue asymptotics on self-similar sets discussed in [27, 16, 19] and long time asymptotics of the transition probability of the simple random walk on self-similar graphs treated in [13, 28]. Contrary to these existent results, the combination of (NRV) and (NP) asserts thatpt(x, x) oscillates ast↓0 in a non-log-periodic but still non-regularly varying way for “generic” x∈ K as long as the assumption (1.6) is satisfied.

In fact, for (NP) we will actually prove the following stronger result: if (1.6) is satisfied, then for “generic”x∈K and any periodic functionG:R→R,

(1.7) lim sup

t↓0

t^ds/2pt(x, x)−G(−logt)≥My,z

2 , whereM_y,z:= lim inf_t↓0t^ds/2(

p_t(y, y)−p_t(z, z))

∈(0,∞)with y, z as in (1.6).

The proof of (NRV) and (NP) relies only on the self-similarity of the Dirichlet space, the joint continuity of the heat kernel and its sub-Gaussian upper bound, which are all known to hold quite in general, and is free of extra a priori assumptions.

Instead, however, we still need certain topological properties of the fractal K to verify (1.4) or (1.6). Roughly speaking, (1.4) can be verified if the local geometry of KaroundFw₁◦· · ·◦Fw_m(x) is not the same for allx∈V0andw1. . . wm∈∪

n∈NSⁿ withFw₁◦ · · · ◦Fw_m(x)6∈V0, and so can (1.6) if in addition the fractalK (or more precisely, the Dirichlet space (K, µ,E,F)) has good symmetry.

For example, whenKis the two-dimensional level-3 Sierpi´nski gasket in Figure 1, the barycenter is contained inthree of the cells{Fi(K)|i∈S} but each of the other points of(∪

i∈SFi(V0))

\V0 is contained only intwoof them, which and the dihedral symmetry of K together imply (1.6). The pentagasket also satisfies (1.6) for exactly the same reason, whereas only (1.4) can be verified for Hata’s tree-like set due to the lack of symmetry although (1.6) could actually be the case. For the Sierpi´nski carpet, and its generalizations calledgeneralized Sierpi´nski carpets, (1.6) is proved by using their symmetry under the isometries of the unit cube and the fact that some faces of the cells{F_i(K)|i∈S} are contained only inone cell but the others intwo cells.

Unfortunately, actuallythe author does not have any idea whether (1.4) and (1.6)are valid for thed-dimensional standard (level-2) Sierpi´nski gasket withd≥2;

the argument in the previous paragraph does not work in this case since anyx∈ (∪

m∈N

∪

w1...wm∈S^mFw₁◦ · · · ◦Fw_m(V0))

\V0has exactly two neighboring cells (see Figure 2 below). In fact, it will be proved in a forthcoming paper [22] thatp_(·)(x, x) does not vary regularly at0 for any x∈Kfor certain specific post-critically finite self-similar fractals K where very detailed information on the eigenvalues of the Laplacian is known, including thed-dimensional standard Sierpi´nski gasket. This result alone, however, does not exclude the possibility that (1.4) is not valid.

This article is organized as follows. In Section 2, we introduce our framework of self-similar Dirichlet forms on self-similar sets and give the precise statements of our main results (NRV) and (NP) in Theorems 2.17 and 2.18, respectively. Section 3 is devoted to the proof of Theorems 2.17 and 2.18, and then they are applied to post- critically finite self-similar fractals and generalized Sierpi´nski carpets in Sections 4 and 5, respectively. In Section 4, after recalling basics of self-similar Dirichlet forms on post-critically finite self-similar fractals in Subsection 4.1, we verify (1.6) for those with good symmetry such as affine nested fractals in Subsection 4.2, and (1.4) for those possibly without good symmetry such as Hata’s tree-like set in Subsection 4.3. Finally in Section 5, we first collect important facts concerning generalized Sierpi´nski carpets and their canonical self-similar Dirichlet form and then verify (1.6) for them.

Notation. In this paper, we adopt the following notation and conventions.

(1)N={1,2,3, . . .}, i.e. 06∈N.

(2) The cardinality (the number of elements) of a setAis denoted by #A.

(3) We set sup∅:= 0, inf∅:=∞and seta∨b:= max{a, b} anda∧b:= min{a, b} fora, b∈[−∞,∞]. All functions in this paper are assumed to be [−∞,∞]-valued.

(4) Ford∈N,R^d is always equipped with the Euclidean norm| · |.

(5) Let E be a topological space. The Borelσ-field ofE is denoted byB(E). We set C(E) :={u|u:E→R,uis continuous}, supp_E[u] :={x∈E|u(x)6= 0} and kuk∞:= sup_x∈E|u(x)|foru∈C(E). For A⊂E, intEAdenotes its interior in E.

(6) Let E be a set, ρ : E ×E → [0,∞) and x ∈ E. We set distρ(x, A) :=

infy∈Aρ(x, y) forA⊂E andBr(x, ρ) :={y∈E|ρ(x, y)< r} forr∈(0,∞).

2. Framework and main results

In this section, we first introduce our framework of a self-similar set and a self-similar Dirichlet form on it, and then state the main theorems of this paper.

Let us start with standard notions concerning self-similar sets. We refer to [23, Chapter 1], [25, Section 1.2] and [19, Subsection 2.2] for details. Throughout this and the next sections, we fix a compact metrizable topological space K with

#K≥2, a non-empty finite setS and a continuous injective map Fi:K→K for eachi∈S. We setL:= (K, S,{Fi}i∈S).

Definition 2.1. (1) Let W0 :={∅}, where ∅ is an element called the empty word, letWm:=S^m ={w1. . . wm |wi ∈S fori∈ {1, . . . , m}}for m∈Nand let W_∗ :=∪

m∈N∪{0}Wm. For w ∈ W_∗, the unique m ∈N∪ {0} satisfyingw ∈Wm

is denoted by|w|and called thelength of w. Fori∈S andn∈N∪ {0}we write iⁿ:=i . . . i∈W_n.

(2) We set Σ := S^N ={ω₁ω₂ω₃. . .| ω_i ∈S fori∈ N}, which is always equipped with the product topology, and define theshift mapσ: Σ→Σ byσ(ω₁ω₂ω₃. . .) :=

ω₂ω₃ω₄. . .. Fori∈S we defineσ_i: Σ→Σ byσ_i(ω₁ω₂ω₃. . .) :=iω₁ω₂ω₃. . .. For ω=ω₁ω₂ω₃. . .∈Σ andm∈N∪ {0}, we write [ω]_m:=ω₁. . . ω_m∈W_m.

(3) For w =w1. . . wm ∈ W_∗, we setFw := Fw₁ ◦ · · · ◦Fw_m (F_∅ := idK), Kw :=

Fw(K), σw :=σw₁◦ · · · ◦σw_m (σ_∅ := idΣ) and Σw := σw(Σ), and if w 6=∅ then w^∞∈Σ is defined byw^∞:=www . . . in the natural manner.

Definition2.2. Lis called aself-similar structureif and only if there exists a continuous surjective mapπ: Σ→Ksuch thatFi◦π=π◦σi for anyi∈S. Note that suchπ, if exists, is unique and satisfies{π(ω)}=∩

m∈NK_[ω]m for anyω∈Σ.

In what follows we always assume that L is a self-similar structure, so that

#S ≥2 by #K≥2 andπ(Σ) =K. For A⊂K, the closure ofA inK is denoted byA.

Definition 2.3. (1) We define thecritical setC and thepost-critical set P of Lby

(2.1) C:=π⁻¹(∪

i,j∈S, i6=jK_i∩K_j)

and P :=∪

n∈Nσⁿ(C).

Lis called post-critically finite, orp.c.f.for short, if and only ifP is a finite set.

(2) We setV0:=π(P),Vm:=∪

w∈WmFw(V0) form∈NandV_∗:=∪

m∈NVm. (3) We setK^I :=K\V0,K_w^I :=Fw(K^I) forw∈W_∗andV_∗∗:=∪

w∈W_∗Fw(V0).

V0should be considered as the“boundary” of the self-similar setK; recall that K_w∩K_v =F_w(V₀)∩F_v(V₀) for anyw, v∈W_∗with Σ_w∩Σ_v=∅by [23, Proposition 1.3.5-(2)]. Note thatF_w(V₀) =∪

n∈Nπ(σ_w◦σⁿ(C))∈ B(K) for any w∈W_∗ by the compactness of Σ. According to [23, Lemma 1.3.11], V_m−1 ⊂V_m for anym∈N,

and ifV06=∅thenV_∗is dense inK. Furthermore by [19, Lemma 2.11],K_w^I is open inK andK_w^I ⊂K^I for anyw∈W_∗.

Definition2.4. Let (µ_i)_i∈S ∈(0,1)^S satisfy∑

i∈Sµ_i= 1. A Borel probability measure µ on K is called a self-similar measure on L with weight (µ_i)_i∈S if and only if the following equality (of Borel measures onK) holds:

(2.2) µ=∑

i∈S

µ_iµ◦F_i⁻¹. Let (µi)i∈S ∈ (0,1)^S satisfy ∑

i∈Sµi = 1. Then there exists a self-similar measure onLwith weight (µ_i)_i∈S. Indeed, ifν is the Bernoulli measure on Σ with weight (µ_i)_i∈S, thenν ◦π⁻¹ is such a self-similar measure on L; see [23, Section 1.4] for details. Moreover by [25, Theorem 1.2.7 and its proof], ifK6=V0andµis a self-similar measure onLwith weight (µ_i)_i∈S, thenµ(K_w) =µ_wandµ(F_w(V₀)) = 0 for anyw ∈W_∗, where µ_w :=µ_w1· · ·µ_wm for w=w₁. . . w_m∈ W_∗ (µ_∅ := 1). In particular, a self-similar measure onLwith given weight is unique ifK6=V0.

The following lemmas are immediate from the above-mentioned facts.

Lemma2.5. Assume K6=V₀, let µbe a self-similar measure onL with weight (µ_i)_i∈S and letw∈W_∗. Then∫

K|u◦F_w|dµ=µ⁻_w¹∫

Kw|u|dµfor any Borel measur- ableu:K→[−∞,∞]. In particular, if we setF_w^∗u:=u◦Fwforu:K→[−∞,∞], thenF_w^∗ defines a bounded linear operatorF_w^∗ :L²(K, µ)→L²(K, µ).

Lemma 2.6. Let w ∈ W_∗. For u : K → [−∞,∞], define (Fw)_∗u : K → [−∞,∞] by

(2.3) (F_w)_∗u:=

{

u◦F_w⁻¹ onKw, 0 onK\K_w.

If u is Borel measurable then so is (Fw)_∗u, and if K 6= V0 in addition then

∫

K|(F_w)_∗u|dµ = µ_w∫

K|u|dµ. In particular, if K 6= V₀, then (F_w)_∗ defines a bounded linear operator(F_w)_∗:L²(K, µ)→L²(K, µ).

Next we define the notion of a homogeneously scaled self-similar Dirichlet space and state its basic properties. The following definition is a special case of [19, Definition 3.3]. See [11, Section 1.1] for basic notions concerning regular Dirichlet forms.

Definition 2.7 (Homogeneously scaled self-similar Dirichlet space). Assume K6=V0. Letµbe a self-similar measure onL with weight (µi)i∈S, letds∈(0,∞) and setri:=µ^2/d_i ^s−1fori∈S. (E,F) is called ahomogeneously scaled self-similar Dirichlet form onL²(K, µ)with spectral dimensiondsif and only if it is a non-zero symmetric regular Dirichlet form onL²(K, µ) satisfying the following conditions:

(SSDF1) u◦Fi∈ F for anyi∈S and anyu∈ F ∩C(K).

(SSDF2) For anyu∈ F ∩C(K),

(2.4) E(u, u) =∑

i∈S

1

r_iE(u◦Fi, u◦Fi).

(SSDF3) (F_i)_∗u∈ F for any i∈S and anyu∈ F ∩C(K) with supp_K[u]⊂K^I. If (E,F) is a homogeneously scaled self-similar Dirichlet form on L²(K, µ) with spectral dimensiond_s, then (L, µ,E,F) is called ahomogeneously scaled self-similar Dirichlet space with spectral dimension d_s, and we call (µ_i)_i∈S itsweight.

In the rest of this section, we assume that (L, µ,E,F) is a homogeneously scaled self-similar Dirichlet space with weight (µi)i∈S and spectral dimensionds. Then by [19, Lemma 5.5], (SSDF1) and (SSDF2) still hold ifF ∩C(K) is replaced with F. Lemma 2.8. (E,F)is conservative (i.e. 1∈ F and E(1,1) = 0) and strongly local. Moreover,V₀6=∅.

Proof. Since K is compact and (E,F) is regular, F ∩ C(K) is dense in (C(K),k · k∞), so that there existsu∈ F ∩C(K) such thatk1−uk∞≤1/2. Thus 1 = min{2u,1} ∈ F, and then it easily follows from (SSDF2) and ∑

i∈Sr_i⁻¹ =

∑

i∈Sµ¹_i^−2/ds >1 thatE(1,1) = 0. Moreover, (E,F) is local by [19, Lemma 3.4], and it is also easily seen to be strongly local by virtue of its conservativeness.

SupposeV0=∅, so that π: Σ→K is a homeomorphism by [23, Proposition 1.3.5-(3)]. Then sinceKw is compact and open, we easily see from the conserva- tiveness of (E,F) and [11, Theorem 1.4.2-(ii) and Exercise 1.4.1] that1K_w∈ F and E(1K_w,1K_w) = 0 for anyw∈W_∗. This fact together with the denseness of the lin- ear span of{1K_w}w∈W_∗ in L²(K, µ) yieldsF =L²(K, µ) andE= 0, contradicting

the assumption that (E,F) is non-zero.

We need to introduce several geometric notions to formulate the assumption of a sub-Gaussian heat kernel upper bound which is required for our main results. We refer the reader to [25, Sections 1.1 and 1.3] and [19, Section 2] for further details.

Definition 2.9. (1) Letw, v ∈W_∗, w =w1. . . wm, v =v1. . . vn. We define wv ∈W_∗ by wv :=w1. . . wmv1. . . vn (w∅ :=w, ∅v :=v). We write w≤v if and only if w = vτ for some τ ∈ W_∗. Note that Σ_w∩Σ_v = ∅ if and only if neither w≤v norv≤w.

(2) A finite subset Λ ofW_∗is called apartition ofΣ if and only if Σ_w∩Σ_v=∅for anyw, v∈Λ withw6=v and Σ =∪

w∈ΛΣ_w.

(3) Let Λ₁,Λ₂ be partitions of Σ. We say that Λ₁is arefinement ofΛ₂, and write Λ₁≤Λ₂, if and only if for eachw¹∈Λ₁ there exists w²∈Λ₂ such thatw¹≤w².

Definition2.10. (1) Setγ_w:=µ^1/dw ^s forw∈W_∗. We define Λ₁:={∅}, (2.5) Λ_s:={w|w=w₁. . . w_m∈W_∗\ {∅},γ_w1...wm−1 > s≥γ_w}

for eachs∈(0,1), and S:={Λs}s∈(0,1]. We callS thescale on Σ associated with (L, µ,E,F).

(2) For each (s, x)∈(0,1]×K, we define Λ⁰_s,x:={w∈Λs|x∈Kw},Us⁽⁰⁾(x) :=

∪

w∈Λ⁰_s,xK_w, and inductively forn∈N,

(2.6) Λⁿ_s,x:={w∈Λ_s|K_w∩U_s⁽ⁿ⁻¹⁾(x)6=∅} and U_s⁽ⁿ⁾(x) := ∪

w∈Λⁿ_s,x

K_w.

Clearly lims↓0min{|w| |w∈Λs}=∞, and it is easy to see that Λsis a partition of Σ for anys∈(0,1] and that Λs₁≤Λs₂ for anys1, s2∈(0,1] withs1≤s2. These facts together with [23, Proposition 1.3.6] imply that for anyn∈N∪ {0} and any x∈K,{Us⁽ⁿ⁾(x)}s∈(0,1] is non-decreasing ins and forms a fundamental system of neighborhoods ofx in K. Note also that Λⁿ_s,x and Us⁽ⁿ⁾(x) are non-decreasing in n∈N∪ {0}for any (s, x)∈(0,1]×K.

We would like to considerUs⁽ⁿ⁾(x) to be a “ball of radiusscentered atx”. The following definition formulates the situation where Us⁽ⁿ⁾(x) may be thought of as actual balls with respect to a distance function onK.

Definition 2.11. (1) Letρ:K×K→[0,∞). For α∈(0,∞),ρis called an α-qdistance onK if and only if ρ^α :=ρ(·,·)^α is a distance on K. Moreover, ρ is called aqdistance onKif and only if it is anα-qdistance onKfor someα∈(0,∞).

(2) A qdistanceρonKis calledadapted toSif and only if there existβ₁, β₂∈(0,∞) andn∈Nsuch that for any (s, x)∈(0,1]×K,

(2.7) Bβ₁s(x, ρ)⊂U_s⁽ⁿ⁾(x)⊂Bβ₂s(x, ρ).

Ifρis anα-qdistance onKadapted toS, thenρ^αis compatible with the original topology ofK, since{Us⁽ⁿ⁾(x)}s∈(0,1] is a fundamental system of neighborhoods of xin the original topology ofK.

Definition 2.12. We say thatS is locally finite with respect to L, or simply (L,S) islocally finite, if and only if sup{#Λ¹_s,x|(s, x)∈(0,1]×K}<∞.

Note that by [23, Lemma 1.3.6], (L,S) is locally finite if and only if sup{#Λⁿ_s,x| (s, x) ∈ (0,1]×K} <∞ for any n ∈ N. The local finiteness of (L,S) is closely related with local behavior ofµ. In fact, we have the following proposition.

Proposition 2.13. Set γ := mini∈Sγi and let n ∈ N∪ {0}. Then for any (s, x)∈(0,1]×K,

(2.8) γ^dss^ds#Λⁿ_s,x≤µ(

U_s⁽ⁿ⁾(x))

≤s^ds#Λⁿ_s,x.

In particular, for fixed n ∈ N, (L,S) is locally finite if and only if there exist cV,n∈(0,∞)such that µ(

Us⁽ⁿ⁾(x))

≤cV,ns^ds for any (s, x)∈(0,1]×K, Proof. We easily see from the definition (2.5) of Λsthat

(2.9) γ^dss^ds < µw≤s^ds, s∈(0,1], w∈Λs.

Since µ(Kw) = µw and µ(Fw(V0)) = 0 for any w ∈ W_∗ by the assumption that K6=V0, (2.9) implies that for any (s, x)∈(0,1]×K,

γ^dss^ds#Λⁿ_s,x≤ ∑

w∈Λⁿ_s,x

µ_w= ∑

w∈Λⁿ_s,x

µ(K_w) =µ(

U_s⁽ⁿ⁾(x))

≤s^ds#Λⁿ_s,x, proving (2.8). The latter assertion is immediate from (2.8).

Next we prepare fundamental conditions for our main results concerning the heat kernel of (K, µ,E,F).

Definition 2.14 (CHK). We say that (K, µ,E,F) satisfies (CHK), or sim- ply (CHK) holds, if and only if the Markovian semigroup {T_t}t∈(0,∞) onL²(K, µ) associated with (E,F) admits acontinuous integral kernelp, i.e. a continuous func- tionp=p_t(x, y) : (0,∞)×K×K →R such that for any u∈L²(K, µ) and any t∈(0,∞),

(2.10) Ttu=

∫

K

pt(·, y)u(y)dµ(y) µ-a.e.

Such p, if exists, is unique and satisfies pt(x, y) = pt(y, x) ≥0 for any (t, x, y) ∈ (0,∞)×K×Kby a standard monotone class argument. pis called the(continuous) heat kernel of (K, µ,E,F).

Definition2.15 (CUHK). We say that (L, µ,E,F)satisfies (CUHK), or sim- ply (CUHK)holds, if and only if (L,S) is locally finite, (K, µ,E,F) satisfies (CHK) and there existd_w∈(1,∞), a (2/d_w)-qdistanceρonKadapted toSandc_2.1, c_2.2∈ (0,∞) such that for any (t, x, y)∈(0,1]×K×K,

(2.11) p_t(x, y)≤c_2.1t^−ds/2exp (

−c_2.2

(ρ(x, y)² t

)_dw−1¹ ) . Note that (CUHK) remains the same if we replacet^−ds/2with 1/µ(

B^√_t(x, ρ)) in (2.11) and omit the condition that (L,S) is locally finite; indeed, this equivalence easily follows from Definition 2.11-(2), Proposition 2.13 and [19, Proposition 5.8].

Proposition 2.16. Suppose that (CUHK) holds. Then there existc2.3, c2.4∈ (0,∞)such that for anyx∈K,

(2.12) c2.3≤t^ds/2pt(x, x)≤c2.4, t∈(0,1].

Proof. t^ds/2pt(x, x)≤c2.1 for any (t, x)∈(0,1]×K by (2.11). For the lower bound we follow [24, Proof of Theorem 2.13]. Let ρ be the qdistance onK as in Definition 2.15. Since (L,S) is assumed to be locally finite, Definition 2.11-(2) and Proposition 2.13 easily imply thatµ(Br(x, ρ))≤c2.5r^ds for any (r, x)∈(0,∞)×K for some c2.5 ∈(0,∞), and the same calculation as [24, Proof of Lemma 4.6-(1)]

shows that ∫

K\B_δ√t(x,ρ)pt(x, y)dµ(y) ≤ 1/2 for any (t, x) ∈ (0,1]×K for some δ ∈ (0,∞). Now for (t, x) ∈ (0,1]×K, the conservativeness of (E,F) yields

∫

Kpt(x, y)dµ(y) = 1, and hence 1

2 ≤1−

∫

K\B_δ^√_t(x,ρ)

p_t(x, y)dµ(y) =

∫

B_δ^√_t(x,ρ)

p_t(x, y)dµ(y)

≤

√ µ(

B_δ^√_t(x, ρ)) ∫

K

pt(x, y)²dµ(y)≤√

c2.5δ^dst^ds/2p2t(x, x)

by the symmetry and the semigroup property of the heat kernel p, proving the

lower inequality in (2.12).

Now we are in the stage of stating the main theorems of this paper. Note that any Borel measure on Kvanishing on V_∗(∈ B(K)) is of the form ν◦π⁻¹ withν a Borel measure on Σ, sinceπ|Σ\π−1(V_∗): Σ\π⁻¹(V_∗)→K\V_∗is a homeomorphism.

Recall the following notions: a Borel measureν on Σ is calledσ-ergodicif and only ifν◦σ⁻¹=ν andν(A)ν(Σ\A) = 0 for anyA∈ B(Σ) with σ⁻¹(A) =A, and it is said to havefull support if and only if ν(U)>0 for any non-empty open subsetU of Σ. Recall also that we setK^I :=K\V0andV_∗∗:=∪

w∈W_∗Fw(V0).

Theorem 2.17. Suppose that (CUHK) holds and that

(2.13) lim sup

t↓0

pt(y, y)

pt(z, z) >1 for somey, z∈K^I.

Then there existsNRV ∈ B(K)satisfyingV_∗∗⊂NRV andν◦π⁻¹(NRV) = 0for any σ-ergodic finite Borel measureν onΣwith full support, such thatp_(·)(x, x)doesnot vary regularly at0for anyx∈K\NRV. In particular, the limitlimt↓0t^ds/2pt(x, x) does notexist for any x∈K\NRV.

Note that (2.13) does not hold if and only if limt↓0pt(y, y)/pt(z, z) = 1 for any y, z∈K^I.

Theorem 2.18. Suppose that (CUHK) holds and that

(2.14) lim inf

t↓0

pt(y, y)

pt(z, z) >1 for somey, z∈K^I.

Then there exists N_P∈ B(K)satisfyingV_∗∗⊂N_Pandν◦π⁻¹(N_P) = 0for any σ- ergodic finite Borel measureν onΣwith full support, such that for anyx∈K\NP

and any periodic functionG:R→R,

(2.15) lim sup

t↓0

t^ds/2p_t(x, x)−G(−logt)≥M_y,z 2 , whereMy,z:= lim inft↓0t^ds/2(

pt(y, y)−pt(z, z))

∈(0,∞)with y, z as in (2.14).

Note that by (2.12), for each y, z ∈ K, lim inf_t↓0p_t(y, y)/p_t(z, z) > 1 if and only if lim inf_t↓0t^ds/2(

p_t(y, y)−p_t(z, z))

∈(0,∞).

Remark 2.19. Let y, z ∈ K^I be as in (2.13) or (2.14). Then the sets NRV

in Theorem 2.17andNP in Theorem 2.18can be given explicitly in terms of (and hence can be determined solely by) y, z and π; see (3.8), Lemmas 3.10 and 3.12 below.

The proof of Theorems 2.17 and 2.18 is given in the next section. As we will see in Sections 4 and 5, the conditions (2.13) and (2.14) are satisfied for many typical examples such as most nested fractals andall generalized Sierpi´nski carpets.

3. Proof of Theorems 2.17 and 2.18

Throughout this section, we fix a homogeneously scaled self-similar Dirichlet space (L= (K, S,{Fi}i∈S), µ,E,F) with weight (µi)i∈S and spectral dimension ds

and assume that (CUHK) holds withdw andρas in Definition 2.15.

Definition 3.1. LetU be a non-empty open subset of K. We defineµ|U :=

µ|_B(U),

(3.1) FU :={u∈ F ∩C(K)|supp_K[u]⊂U} and E^U :=E|FU×FU, where the closure is taken in the Hilbert space F with inner product E1(u, v) :=

E(u, v) +∫

Kuvdµ. (E^U,FU) is called thepart of the Dirichlet form(E,F)on U. Since u = 0 µ-a.e. on K\U for any u ∈ FU, we can regard FU as a linear subspace of L²(U, µ|U) in the natural manner. Under this identification, we have the following lemma.

Lemma 3.2. Let U be a non-empty open subset of K. Then (E^U,FU) is a strongly local regular Dirichlet form on L²(U, µ|U) whose associated Markovian semigroup{T_t^U}t∈(0,∞) admits a unique continuous integral kernelp^U =p^U_t(x, y) : (0,∞)×U ×U → R, called the Dirichlet heat kernel on U, similarly to (2.10).

Moreover,0≤p^U_t(x, y) =p^U_t(y, x)≤p_t(x, y)for any(t, x, y)∈(0,∞)×U×U. Proof. Recall Lemma 2.8. The regularity of (E,F) yields that of (E^U,FU) by (3.1) and [11, Lemma 1.4.2-(ii)], and the strong locality of (E,F) implies that of (E^U,FU). Since (E,F) is conservative, a continuous integral kernel p^U of{T_t^U}t∈(0,∞)exists by [19, Lemma 7.11-(2)] and (CUHK), and a monotone class argument immediately shows the uniqueness of suchp^U. Finally, the last assertion easily follows from [25, (C.2)] and a monotone class argument again.

Lemma 3.3. Let U be a non-empty open subset of K. Then for any(t, x, y)∈ (0,∞)×U×U,

(3.2) p_t(x, y)−p^U_t(x, y)≤ sup

s∈[t/2, t]

sup

z∈U\U

p_s(x, z) + sup

s∈[t/2, t]

sup

z∈U\U

p_s(z, y).

Proof. This is immediate from [15, Theorem 5.1] (or [14, Theorem 10.4]), the continuity of the heat kernelspt(x, y) andp^U_t(x, y) and the compactness ofU.

Lemma 3.4. Let w∈W_∗. Then for any(t, x, y)∈(0,∞)×K^I×K^I, (3.3) (γ_w²t)^ds/2p^K

I w

γ_w²t

(Fw(x), Fw(y))

=t^ds/2p^K_t ^I(x, y).

Proof. Fw|K^I : K^I → K_w^I is clearly a homeomorphism, and F_w^∗ defines a bijection F_w^∗ : L²(K_w^I, µ|K_w^I)→ L²(K^I, µ|K^I) such that F_w^∗(FK^I_w) = FK^I by [19, Lemma 5.5]. Moreover, γ_w^ds∫

K^I(F_w^∗u)²dµ= ∫

K_w^I u²dµ for any u ∈ L²(K_w^I, µ|K_w^I) by Lemma 2.5 andγ_w^ds−2E(F_w^∗u, F_w^∗u) = E(u, u) for anyu∈ FK_w^I by (SSDF2). It easily follows from these facts and [11, Lemma 1.3.4-(i)] thatF_w^∗T_γ^K₂^wI

wt=T_t^KIF_w^∗ for any t∈(0,∞), which and the uniqueness of the continuous heat kernelsp^KwI and

p^KI imply (3.3).

Lemma 3.5. There exists c3.1 ∈ (0,∞) such that for any x ∈ K and any w∈W_∗,

(3.4) dist_ρ(

F_w(x), F_w(V₀))

≥c_3.1γ_wdist_ρ(x, V₀).

Proof. Let β1, β2 ∈ (0,∞) and n ∈ N be as in Definition 2.11-(2) for the qdistanceρ, letx∈K,w∈W_∗ and setδ:= distρ(x, V0). The assertion is obvious for x ∈ V₀. Assuming x ∈ K\V₀ = K^I, by K = U₁⁽ⁿ⁾(x) ⊂ B_β2(x, ρ) we have ρ(x, y)< β₂ for any y ∈ K and hence δ ∈(0, β₂) (recall that V₀ 6= ∅ by Lemma 2.8), andU_δ/β⁽ⁿ⁾

2(x)∩V₀=∅sinceU_δ/β⁽ⁿ⁾

2(x)⊂B_δ(x, ρ)⊂K^I. Then an induction in keasily shows that Λ^k_γ

wδ/β₂,F_w(x)={

wvv∈Λ^k_δ/β

2,x

}for any k∈ {0, . . . , n}, and hence

Bγ_wδβ₁/β₂(Fw(x), ρ)⊂U_γ⁽ⁿ⁾

wδ/β2(Fw(x)) =Fw

(U_δ/β⁽ⁿ⁾

2(x))

⊂K_w^I =Kw\Fw(V0).

Thus ρ(F_w(x), y) ≥ γ_wδβ₁/β₂ = (β₁/β₂)γ_wdist_ρ(x, V₀) for any y ∈ F_w(V₀) and

(3.4) follows withc_3.1:=β₁/β₂.

Lemma3.6. There existc3.2, c3.3∈(0,∞)such that for any(t, x)∈(0,1]×K^I and any w∈W_∗,

(γ_w²t)^ds/2p_γ2 wt

(Fw(x), Fw(x))

−t^ds/2pt(x, x)

≤c3.2exp

(−c3.3distρ(x, V0)^dw−12 t^−dw−11 )

. (3.5)

Proof. We easily see from (2.11), Lemmas 3.3 and 3.5 that, with c3.2 :=

2^1+ds/2c_2.1 andc_3.3:=c_2.2c

dw−12

3.1 , for anyw∈W_∗ and any (t, x)∈(0, γ_w⁻²]×K^I, 0≤(γ_w²t)^ds/2

( p_γ2

wt

(Fw(x), Fw(x))

−p^K

I w

γ_w²t

(Fw(x), Fw(x)))

≤c3.2exp

(−c3.3distρ(x, V0)^dw−12 t^−dw−11 )

, (3.6)

which together with Lemma 3.4 immediately shows (3.5).