Analysis and geometry of the measurable Riemannian
structure on the Sierpiński gasket
Naotaka Kajino
Abstract. This expository article is devoted to a survey of existent results
concerning the measurable Riemannian structure on the Sierpiński gasket andto a brief account of the author’s recent result on Weyl’s eigenvalue asymptotics
of its associated Laplacian. In particular, properties of the Hausdorﬀ measurewith respect to the canonical geodesic metric are described in some detail asa key step to the proof of Weyl’s asymptotics. A complete characterization ofminimal geodesics is newly proved and applied to invalidity of Ricci curvature
lower bound conditions such as the curvature-dimension condition and themeasure contraction property. Possibility of and diﬃculties in extending theresults to other self-similar fractals are also discussed.

Contents
1. Introduction
2. Sierpiński gasket and its standard Dirichlet form
3. Measurable Riemannian structure on the Sierpiński gasket
4. Geometry under the measurable Riemannian structure
5. Short time asymptotics of the heat kernels
5.1. Intricsic metrics and oﬀ-diagonal Gaussian behavior
5.2. One-dimensional asymptotics at vertices
5.3. On-diagonal asymptotics at almost every point
6. Ahlfors regularity and singularity of Hausdorﬀ measure
7. Weyl’s Laplacian eigenvalue asymptotics
8. Connections to general theories on metric measure spaces
8.1. Identification of Dirichlet form as Cheeger energy
8.2. Invalidity of Ricci curvature lower bound9. Possible generalizations to other self-similar fractals
9.1. Sierpiński gaskets

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4
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19
20
22
26
27
28
33
33

Version of March 11, 2013.
2010 Mathematics Subject Classification. Primary 28A80, 35P20, 53C23; Secondary 31C25,
37B10, 60J35.
Key words and phrases. Sierpiński gasket, Dirichlet form, Kusuoka measure, measurable
Riemannian structure, geodesic metric, heat kernel, Weyl’s Laplacian eigenvalue asymptotics,
Ricci curvature lower bound.
The author was supported in part by SFB 701 of the German Research Council (DFG).

1

2

NAOTAKA KAJINO

9.2. Other nested fractals and Sierpiński carpets
Appendix A. Case of the standard Laplacian on the Sierpiński gasket
References

35
36
38

1. Introduction
The purpose of this expository article is to review known results concerning themeasurable Riemannian structure on the Sierpiński gasket (Figure 1) and describe
its connections to general theories of analysis and geometry on metric measurespaces. We also state the author’s recent result on Weyl’s eigenvalue asymptotics
of its associated Laplacian and briefly explain the idea of its proof. In particular, we
present various properties of the Hausdorﬀ measure with respect to the canonicalgeodesic metric as the key facts for the proof of Weyl’s asymptotics.
The notion of the measurable Riemannian structure on the Sierpiński gasket
was first introduced by Kigami [56] on the basis of Kusuoka’s construction in [67]of “weak gradients” for Dirichlet forms on fractals. In [56], Kigami proved that the
Sierpiński gasket can be embedded in R2 by a certain harmonic map, whose image
is now called the harmonic Sierpiński gasket (Figure 2), and that Kusuoka’s “weakgradients” can be identified as the gradients with respect to the (measurable) “Riemannian structure” inherited from R2 through this embedding. (Related results
are also found in Hino [38, 40].) These results are reviewed in Section 3 after abrief account of the Sierpiński gasket and its standard Dirichlet form in Section 2.
Kigami further proved in [58] that the heat kernel associated with this “Riemannian structure” satisfies the two-sided Gaussian bound in terms of the natural
geodesic metric, unlike typical fractal diﬀusions treated e.g. in [11, 64, 26, 7, 8] for
whose transition densities (heat kernels) the two-sided sub-Gaussian bounds hold.
Later in [48] the author proved some more detailed asymptotics of that heat kernelsuch as Varadhan’s asymptotic relation, together with an analytic characterizationof the geodesic metric and slight generalizations and improvements of the resultsin [58]. These results are reviewed in Section 5 following a summary of basic geometric properties of the measurable Riemannian structure in Section 4, where wealso newly prove a complete characterization of minimal geodesics (Theorem 4.19).
Very recently, the author has also proved Weyl’s Laplacian eigenvalue asymptotics for this case, which is to be treated in a forthcoming paper [51]. The proofof Weyl’s asymptotics require some detailed properties of the Hausdorﬀ measurewith respect to the geodesic metric and this is reviewed in Section 6, along withthe singularity of the Hausdorﬀ measure to the energy measures. Then in Section
7, we give the statement of Weyl’s asymptotics and sketch the idea of its proof.
Since the situation of the measurable Riemannian structure on the Sierpiński
gasket looks similar to that of Riemannian manifolds, it is natural to expect close
connections to general theories of analysis and geometry on metric measure spaces
which are not applicable to the case of typical fractal diﬀusions. In fact, Koskela
and Zhou [62, Section 4] recently proved that the theory of diﬀerential calculus
on metric measure spaces, established by Cheeger [19] and developed further by
e.g. Shanmugalingam [86] and Keith [52, 53, 54], is applicable to the measurable
Riemannian structure on the Sierpiński gasket. To be more precise, they prove thatin this case the (1, 2)-Sobolev space equipped with a natural (1, 2)-seminorm, due

ANALYSIS AND GEOMETRY OF MEASURABLE RIEMANNIAN STRUCTURE ON S.G.

Figure 1. Sierpiński gasket

3

Figure 2. Harmonic
Sierpiński gasket

to Cheeger [19, Section 2] and Shanmugalingam [86, Definition 2.5], coincides withthe standard Dirichlet form on the Sierpiński gasket. This result is briefly reviewed
in Subsection 8.1. On the other hand, the notions of Ricci curvature lower bound forgeneral metric measure spaces due to Lott and Villani [75, 74], Sturm [91, 92] andOhta [80] are not applicable to the case of the measurable Riemannian structure.
More precisely, the (harmonic) Sierpiński gasket equipped with the natural geodesic
metric and the “Riemannian volume measure” does not satisfy either the curvature
dimension condition CD(k, N ) of Lott and Villani [75, 74] and Sturm [91, 92]
or the measure contraction property MCP(k, N ) of Ohta [80] and Sturm [92] forany (k, N ) ∈ R × [1, ∞]. We prove this fact in Subsection 8.2 (Theorem 8.25) as
an application of the characterization of minimal geodesics (Theorem 4.19) after areview of the precise definitions of CD(k, N ) and MCP(k, N ) and related results.
Finally, we conclude this paper with a short discussion on possibility of (anddiﬃculties in) extending the above-mentioned results to other self-similar fractals.
In the appendix, we provide a brief review of important results for the Brownian
motion and the standard Laplacian on the Sierpiński gasket, whose associated heat
kernel is known to satisfy the two-sided sub-Gaussian estimate and exhibit various
oscillatory behavior. Those who are not familiar with these results are strongly
recommended to read the appendix directly after Section 2.
Notation. In this article, we adopt the following notation and conventions.
(1) N = {1, 2, 3, . . . }, i.e. 0 6∈ N.
(2) The cardinality (the number of all the elements) of a set A is denoted by #A.
(3) We set sup ∅ := 0 and inf ∅ := ∞. We write a∨b := max{a, b}, a∧b := min{a, b},a+ := a ∨ 0 and a− := −(a ∧ 0) for a, b ∈ [−∞, ∞]. We use the same notations also
for functions. All functions treated in this paper are assumed to be [−∞, ∞]-valued.
(4) Let k ∈ N. The Euclidean inner product and norm on Rk are denoted by h·, ·i
and | · | respectively. For a continuous map γ : [a, b] → Rk , where a, b ∈ R, a ≤ b, let`Rk (γ) be its length with respect to | · |. Let Rk×k be the set of real k × k matrices,which are also regarded as linear maps from Rk to itself through the standard basisof Rk , and set Rk×k:= Rk×k \ {0Rk×k }. For T ∈ Rk×k , let det T be its determinant,0∗
T its transpose, and kT k its Hilbert-Schmidt norm with respect to h·, ·i. The real
orthogonal group of degree k is denoted by O(k).

4NAOTAKA KAJINO(5) Let E be a topological space. The Borel σ-field of E is denoted by B(E). We set
C(E) := {f | f : E → R, f is continuous} and kf k∞ := supx∈E |f (x)|, f ∈ C(E).
For A ⊂ E, its interior in E is denoted by intE A and its boundary in E by ∂E A.
(6) Let (E, ρ) be a metric space. For r ∈ (0, ∞), x ∈ K and A ⊂ E, we set
Br (x, ρ) := {y ∈ E | ρ(x, y) < r}, diamρ A := supy,z∈A ρ(y, z) and distρ (x, A) :=inf y∈A ρ(x, y). For f : E → R we set Lipρ f := supx,y∈E, x6=y |f (x) − f (y)|/ρ(x, y).
A metric ρ0 on E is called comparable to ρ if and only if c1 ρ ≤ ρ0 ≤ c2 ρ for somec1 , c2 ∈ (0, ∞).
2. Sierpiński gasket and its standard Dirichlet form
In this section, we briefly recall basic facts concerning the Sierpiński gasket andits standard Dirichlet form (resistance form). We mainly follow [48, Section 2] forthe presentation of this section and refer the reader to [27, 57, 60, 87] for further
details of each fact.
Definition 2.1 (Sierpiński gasket). Let V0 = {q1 , q2 , q3 } ⊂ R2 be the set of
the three vertices of an equilateral triangle, set S := {1, 2, 3}, and for i ∈ S definefi : R2 → R2 by fi (x) := (x + qi )/2. The Sierpiński gasket (Figure 1) is definedas the self-similar set associated ∪with {fi }i∈S , i.e. the unique non-empty compact
subset K of R2 that satisfies K = i∈S fi (K).
: K → K.
∪ For i ∈ S we set Fi := fi |K ∪Define Vm for m ∈ N inductively by Vm := i∈S Fi (Vm−1 ) and set V∗ := m∈N Vm .
Note that Vm−1 ⊂ Vm for any m ∈ N. K is always regarded as equipped withthe relative topology inherited from R2 , so that Fi : K → K is continuous for eachi ∈ S and V∗ is dense in K.
Definition 2.2. (1) Let W0 := {∅}, where ∅ is an element called the emptymword, let∪ Wm := S = {w1 . . . wm | wi ∈ S for i ∈ {1, . . . , m}} for m ∈ N andW∗ := m∈N∪{0} Wm . For w ∈ W∗ , the unique m ∈ N ∪ {0} with w ∈ Wm isdenoted by |w| and called the length of w. Also for i ∈ S and n ∈ N ∪ {0} we write
in := i . . . i ∈ Wn .
(2) We set Σ := S N = {ω1 ω2 ω3 . . . | ωi ∈ S for i ∈ N}, and define the shift map
σ : Σ → Σ by σ(ω1 ω2 ω3 . . . ) := ω2 ω3 ω4 . . . . Also for i ∈ S we define σi : Σ → Σ byσi (ω1 ω2 ω3 . . . ) := iω1 ω2 ω3 . . . and set i∞ := iii . . . ∈ Σ. For ω = ω1 ω2 ω3 . . . ∈ Σand m ∈ N ∪ {0}, we write [ω]m := ω1 . . . ωm ∈ Wm .
(3) For w = w1 . . . wm ∈ W∗ , we set Fw := Fw1 ◦ · · · ◦ Fwm (F∅ := idK ), Kw :=Fw (K), σw := σw1 ◦ · · · ◦ σwm (σ∅ := idΣ ) and Σw := σw (Σ).
Associated with the triple (K, S, {Fi }i∈S ) is a natural projection π : Σ →K given by the following proposition, which is used to describe the topologicalstructure of K.
Proposition 2.3. There exists a unique continuous surjective ∩map π : Σ → Ksuch that Fi ◦ π = π ◦ σi for any i ∈ S, and it satisfies {π(ω)} = m∈N K[ω]m forany ω ∈ Σ. Moreover, #π −1 (x) = 1 for x ∈ K \ V∗ , π −1 (qi ) = {i∞ } for i ∈ S, andfor m ∈ N and each x ∈ Vm \ Vm−1 there exist w ∈ Wm−1 and i, j ∈ S with i 6= jsuch that π −1 (x) = {wij ∞ , wji∞ }.
Recall the following basic fact ([57, Proposition 1.3.5-(2)]) meaning that V0
should be considered as the “boundary” of K, which we will use below without
further notice: if w, v ∈ W∗ and Σw ∩ Σv = ∅ then Kw ∩ Kv = Fw (V0 ) ∩ Fv (V0 ).

ANALYSIS AND GEOMETRY OF MEASURABLE RIEMANNIAN STRUCTURE ON S.G.

5

As studied in [5, 57, 87], a standard Dirichlet form (to be precise, a resistanceform) (E, F) is defined on the Sierpiński gasket K, as follows. See [57, Chapter2] and [60, Part 1] for general theory of resistance forms. A concise introduction
to the theory of resistance forms is found in [87, Chapter 1], where the theory is
illustrated by treating the particular case of the Sierpiński gasket in detail.
Definition 2.4. Let m ∈ N∪{0}. We define a non-negative definite symmetricbilinear form Em : RVm × RVm → R on Vm by∑
1 1 ( 5 )m(2.1)
Em (u, v) := ·(u(x) − u(y))(v(x) − v(y)),
2 2 3mx,y∈Vm , x∼ymwhere, for x, y ∈ Vm , we write x ∼ y if and only if x, y ∈ Fw (V0 ) for some w ∈ Wm
and x 6= y.
The usual definition of Em does not contain the factor 1/2 so that each edge inmthe graph (Vm , ∼) has resistance (3/5)m . Here it has been added for simplicity ofthe subsequent arguments; see Definition 3.1-(0) below. The factor 3/5, called theresistance scaling factor of the Sierpiński gasket, is specifically chosen for the sakeof the validity of the following proposition.
Proposition 2.5. Let m, n ∈ N ∪ {0}, m ≤ n. Then for each u ∈ RVm ,(2.2)

Em (u, u) = min{En (v, v) | v ∈ RVn , v|Vm = u}and there exists a unique function hm,n (u) ∈ RVn with hm,n (u)|Vm = u such that
Em (u, u) = En (hm,n (u), hm,n (u)). Moreover, hm,n : RVm → RVn is linear.
Let u : V∗ → R. (2.2) implies that {Em (u|Vm , u|Vm )}m∈N∪{0} is non-decreasing
and hence has the limit in [0, ∞]. Moreover, if limm→∞ Em (u|Vm , u|Vm ) < ∞, thenit is not diﬃcult to verify that u is uniformly continuous with respect to any metricon K compatible with the original (Euclidean) topology of K, so that u is uniquely
extended to a continuous function on K. Based on these observations, we can prove
the following theorem; see [57, Chapter 2 and Section 3.3] or [87, Chapter 1] fordetails. Let 1 := 1K denote the constant function on K with value 1.
Theorem 2.6. Define F ⊂ C(K) and E : F × F → R by
(2.3)

F := {u ∈ C(K) | limm→∞ E (m) (u|Vm , u|Vm ) < ∞},E(u, v) := limm→∞ E (m) (u|Vm , v|Vm ) ∈ R,u, v ∈ F.

Then F is a dense subalgebra of C(K), E is a non-negative definite symmetricbilinear form on F, and (E, F) possesses the following properties:(1) {u ∈ F | E(u, u) = 0} = {c1 | c ∈ R} =: R1, and (F/R1, E) is a Hilbert space.
(2) RE (x, y) := supu∈F \R1 |u(x) − u(y)|2 /E(u, u) < ∞ for any x, y ∈ K and RE :K × K → [0, ∞) is a metric on K compatible with the original topology of K.
(3) u+ ∧ 1 ∈ F and E(u+ ∧ 1, u+ ∧ 1) ≤ E(u, u) for any u ∈ F.
(4) F = {u ∈ C(K) | u ◦ Fi ∈ F for any i ∈ S}, and for any u, v ∈ F ,5∑(2.4)
E(u, v) =E(u ◦ Fi , v ◦ Fi ).
3
i∈S

6

NAOTAKA KAJINO(E, F) is called the standard resistance form on the Sierpiński gasket, which isindeed a resistance form on K with resistance metric RE by Theorem 2.6-(1),(2),(3)and F being a dense subalgebra of C(K). Consequently we also have the following
theorem by virtue of [60, Corollary 6.4, Theorems 9.4 and 10.4], where the strong
locality of (E, F) follows from (2.4) and E(1, 1) = 0. See [27, Section 1.1] for thenotions of regular Dirichlet forms and their strong locality.
Theorem 2.7. Let ν be a finite Borel measure on K with full support, i.e. suchthat ν(U ) > 0 for any non-empty open subset U of K. Then (E, F) is a strongly
local regular Dirichlet form on L2 (K, ν), and its associated Markovian semigroup{Ttν }t∈(0,∞) on L2 (K, ν) admits a continuous integral kernel pν , i.e. a continuousfunction pν = pν (t, x, y) : (0, ∞) × K × K → R such that for any f ∈ L2 (K, ν) andany t ∈ (0, ∞),∫(2.5)
Ttν f =pν (t, ·, y)f (y)dν(y)ν-a.e.
K

In the situation of Theorem 2.7, a standard monotone class argument easilyshows that such pν is unique and satisfies pν (t, x, y) = pν (t, y, x) ≥ 0 for any(t, x, y) ∈ (0, ∞) × K × K. Moreover, pν is in fact (0, ∞)-valued by [59, Theorem
A.4]. ν is called the reference measure of the Dirichlet space (K, ν, E, F), and pν iscalled the (continuous) heat kernel associated with (K, ν, E, F). See [60, Theorem
10.4] for other basic properties of pν .
Since we have a regular Dirichlet form (E, F) with compact state space K,
by [27, (3.2.13) and (3.2.14)] we can define E-energy measures as in the following
definition.
Definition 2.8. The E-energy measure of u ∈ F is defined as the unique Borel
measure µhui on K such that
∫
1
(2.6)
f dµhui = E(uf, u) − E(u2 , f )for any f ∈ F.
2
K
We also define λhui to be the unique Borel measure on Σ that satisfies λhui (Σw ) =(5/3)|w| E(u ◦ Fw , u ◦ Fw ) for any w ∈ W∗ , which exists by (2.4) and the Kolmogorov
extension theorem. For u, v ∈ F we set µhu,vi := (µhu+vi − µhu−vi )/4 and λhu,vi :=(λhu+vi − λhu−vi )/4, so that they are finite Borel signed measures on K and on Σ
respectively and are symmetric and bilinear in (u, v) ∈ F × F .
Let u ∈ F. According to [20, Theorem 4.3.8] (see also [16, Theorem I.7.1.1]),
the strong locality of (E, F) implies that the image measure µhui ◦ u−1 on (R, B(R))
is absolutely continuous with respect to the Lebesgue measure on R. In particular,µhui ({x}) = 0 for any x ∈ K. We also easily see the following proposition by using(2.4) and (2.6). Note that π(A) ∈ B(K) for any A ∈ B(Σ) by Proposition 2.3.
Proposition 2.9. λhu,vi = µhu,vi ◦π and λhu,vi ◦π −1 = µhu,vi for any u, v ∈ F .
The definition of the measurable Riemannian structure on the Sierpiński gasket
involves certain harmonic functions. In the present setting, harmonic functions areformulated as follows.
Definition 2.10. (1) We define FB := {u ∈ F | u|K\B = 0} for each B ⊂ K.
(2) Let F be a closed subset of K. Then h ∈ F is called F -harmonic if and only if(2.7) E(h, h) =infu∈F , u|F =h|FE(u, u)or equivalently,

E(h, u) = 0,∀u ∈ FK\F .

ANALYSIS AND GEOMETRY OF MEASURABLE RIEMANNIAN STRUCTURE ON S.G.

7

We set HF := {h ∈ F | h is F -harmonic} and Hm := HVm for each m ∈ N ∪ {0}.
Note that HF is a linear subspace of F for any closed subset F of K and thatHm−1 ⊂ Hm for any m ∈ N. Moreover, we easily have the following proposition by[60, Lemma 8.2 and Theorem 8.4].
Proposition 2.11. Let F be a non-empty closed subset of K.
(1) Let u ∈ F. Then there exists a unique hF (u) ∈ HF such that hF (u)|F = u|F .
Moreover, hF : F → HF is linear.
(2) Let h ∈ HF . Then minF h ≤ h(x) ≤ maxF h for any x ∈ K.
Proposition 2.5 and (2.4) imply the following useful characterizations of Hm .
Proposition 2.12. It holds that for any m ∈ N ∪ {0},Hm = {u ∈ F | E(u, u) = Em (u|Vm , u|Vm )}(2.8)= {u ∈ F | u ◦ Fw ∈ H0 for any w ∈ Wm }.

(2.9)

For each h ∈ H0 , by virtue of h ◦ Fw ∈ H0 , w ∈ W∗ , h|V∗ can be, in principle,
explicitly calculated from h|V0 through simple matrix multiplications, as follows.
Proposition 2.13 ([57, (3.2.3) and Example 3.2.6]). Define





5 0 0
2 2 1
2
1
1
1
2 2 1 , A2 := 0 5 0 , A3 := 1(2.10) A1 :=
5
5
5
2 1 2
1 2 2
0

1
2
0


2
2 ,
5

which we regard as linear maps from RV0 to itself through the standard basis of RV0 .
Then for any u ∈ H0 and any w = w1 . . . wm ∈ W∗ ,(2.11)u ◦ Fw |V0 = Awm · · · Aw1 (u|V0 ).

3. Measurable Riemannian structure on the Sierpiński gasket
This section is devoted to a brief introduction to the notion of the measurable
Riemannian structure on the Sierpiński gasket and its basic properties. We continue
to follow mainly [48, Section 2] and refer to [67, 56, 38] for further details.
We first define a “harmonic embedding” Φ of K into R2 , through which we will
regard K as a kind of “Riemannian submanifold in R2 ” to obtain its measurable
Riemannian structure. We also introduce a measure µ which is regarded as theE-energy measure of the “embedding” Φ and will play the role of the “Riemannianvolume measure”. See [95] for an attempt to generalize the framework of harmonicembeddings and their energy measures to other finitely ramified fractals.
Recall that V0 = {q1 , q2 , q3 }.
Definition 3.1. (0) We define h1 , h2 ∈ F to be the V0 -harmonic functions√
satisfying h1 (q1 ) = h2 (q1 ) = 0, h1 (q2 ) = h1 (q3 ) = 1 and −h2 (q2 ) = h2 (q3 ) = 1/ 3,
so that E(h1 , h1 ) = E(h2 , h2 ) = 1 (recall the factor 1/2 in (2.1)) and E(h1 , h2 ) = 0
by (2.8), and h1 ◦ F1 = (3/5)h1 and h2 ◦ F1 = (1/5)h2 by (2.11).
(1) We define a continuous map Φ : K → R2 and a compact subset KH of R2 by(3.1)

Φ(x) := (h1 (x), h2 (x)), x ∈ K

andKH := Φ(K).

KH is called the harmonic Sierpiński gasket (Figure 2). We also set q̂i := Φ(qi ) fori ∈ S, so that {q̂1 , q̂2 , q̂3 } = Φ(V0 ) is the set of vertices of an equilateral triangle.

8

NAOTAKA KAJINO(2) We define finite Borel measures µ on K and λ on Σ by
(3.2)

µ := µhh1 i + µhh2 iandλ := λhh1 i + λhh2 i ,respectively, so that λ = µ ◦ π and λ ◦ π −1 = µ by Proposition 2.9. µ is called theKusuoka measure on the Sierpiński gasket.
Notation. In what follows h1 , h2 always denote the V0 -harmonic functions
given in Definition 3.1-(0). We often regard {h1 , h2 } as an orthonormal basis of(H0 /R1, E). Moreover, we set√
and
SH0 := {h ∈ H0 | khkE = 1}.
(3.3)
kukE := E(u, u), u ∈ F
The following proposition, which is in fact an easy consequence of Proposition
2.13, provides an alternative geometric definition of KH , and essentially as its
corollary we also see the injectivity of Φ (Theorem 3.3), Proposition 3.4 below andthat µhhi has full support for any h ∈ H0 \ R1.
Proposition 3.2 ([56, §3]). Define
(
)
(
3/5 0
3/10
√
(3.4) T1 :=, T2 :=
0 1/5− 3/10√√)()− 3/10
3/10
3/10, T3 := √
1/2
3/10
1/2
(1 0)
and set Tw := Tw1 · · · Twm for w = w1 . . . wm ∈ W∗ (T∅ := 0 1 ). Also for i ∈ S
define Hi : R2 → R2 by Hi (x) := q̂i + Ti (x − q̂i ). Then the following hold:( θ − sin θ )(1) T2 = R 23 π T1 R− 23 π and T3 = R− 23 π T1 R 23 π , where Rθ := cosfor θ ∈ R.
sin θ cos θ∗∗(2) For any w ∈ W∗ , Tw := (Tw ) is equal to the matrix representation of the linear
map Fw∗ : H0 /R1 → H0 /R1, Fw∗ h := h ◦ Fw by the basis {h1 , h2 } of H0 /R1.
(3) Hi ◦ Φ =∪ Φ ◦ Fi and hence Hi ◦ (Φ ◦ π) = (Φ ◦ π) ◦ σi for any i ∈ S. In particular,
KH = i∈S Hi (KH ), i.e. KH is the self-similar set associated with {Hi }i∈S .
Theorem 3.3 ([56, Theorem 3.6]). Φ : K → KH is a homeomorphism.
Proposition 3.4. µ(Kw ) = λ(Σw ) = (5/3)|w| kTw k2 for any w ∈ W∗ .
Moreover, we have the following theorem due to Kusuoka [67] (see [48, Theorem
6.8] for an alternative simple proof based on (2.4) and the strong locality of (E, F)).
Recall that σ : Σ → Σ is the shift map defined by σ(ω1 ω2 ω3 . . . ) := ω2 ω3 ω4 . . . .
Theorem 3.5 ([67, §6, Example 1]). λ is σ-ergodic, that is, λ ◦ σ −1 = λ andλ(A)λ(Σ \ A) = 0 for any A ∈ B(Σ) with σ −1 (A) = A.
We also remark the following fact due to Hino [38].
Theorem 3.6 ([38, Theorem 5.6]). Let h ∈ H0 \ R1. Then µ and µhhi aremutually absolutely continuous.
Now we can introduce the measurable Riemannian structure on K, which isformulated as a Borel measurable map Z : K → R2×2 , as follows. Recall thatπ|Σ\π−1 (V∗ \V0 ) is injective by Proposition 2.3.
Proposition 3.7 ([67, §1], [56, Proposition B.2]). Define ΣZ ∈ B(Σ) andKZ ∈ B(K) by{}∗
T[ω]m T[ω]m
2×2(3.5) ΣZ := ω ∈ Σ ZΣ (ω) := lim
existsin
R, KZ := π(ΣZ ).
m→∞ kT[ω] k2m

ANALYSIS AND GEOMETRY OF MEASURABLE RIEMANNIAN STRUCTURE ON S.G.

9

Then λ(Σ \ ΣZ ) = µ(K \ KZ ) = 0, ZΣ (ω) is an orthogonal projection of rank 1 forany ω ∈ ΣZ , π −1 (V∗ ) ⊂ ΣZ and ZΣ (ω) = ZΣ (τ ) for ω, τ ∈ π(−1 (x),) x ∈ V∗ \ V0 .
Hence setting Zx := ZΣ (ω), ω ∈ π −1 (x) for x ∈ KZ and Zx := 10 00 for x ∈ K \KZ
gives a well-defined Borel measurable map Z : K → R2×2 , x 7→ Zx .
Theorem 3.8 ([56, §4]). Set CΦ1 (K) := {v ◦ Φ | v ∈ C 1 (R2 )}. Then for eachu ∈ CΦ1 (K), ∇u := (∇v) ◦ Φ is independent of a particular choice of v ∈ C 1 (R2 )satisfying u = v ◦ Φ. Moreover, CΦ1 (K) ⊂ F, CΦ1 (K)/R1 is dense in (F/R1, E),
and for any u, v ∈ CΦ1 (K),∫(3.6)
dµhu,vi = hZ∇u, Z∇vidµandE(u, v) =hZ∇u, Z∇vidµ.
K

In view of Theorem 3.8, especially (3.6), we may regard Z as defining a “onedimensional tangent space Im Zx of K at x with the metric inherited from R2 ” forµ-a.e. x ∈ K in a measurable way, with µ considered as the associated “Riemannianvolume measure” and Z∇u as the “gradient vector field” of u ∈ CΦ1 (K). Then the
Dirichlet space associated with this “Riemannian structure” is (K, µ, E, F).
Remark 3.9. (1) By [56, Theorem B.5-(1)], Σ \ ΣZ is dense in Σ and hence
K \ KZ is dense in K. In other words, there exists a dense set of points x of K
where the notion of the tangent space Im Zx at x does not make sense.
(2) Z|KZ : KZ → R2×2 is discontinuous. Indeed, let n ∈ N ∪ {0} and set xn :=F1n 3 (q2(), so) that limn→∞( xn)= q1 . Then it easily follows from( )(3.4) and (3.5) that
Zq1 = 10 00 and Zxn = 00 01 , which does not converge to 10 00 = Zq1 as n → ∞.
e
As a matter of fact, any u ∈ F admits a natural “gradient vector field” ∇u,thereby (3.6) extended to functions in F, as in the following theorem whose essential
part is due to Hino [38, Theorem 5.4]; see [48, Theorem 2.17] for details.
Theorem 3.10. Let h ∈ H0 \ R1. Then for any u ∈ F the following hold:e(1) For µ-a.e. x ∈ K, there exists ∇u(x)∈ Im Zx such that for any ω ∈ π −1 (x),
(3.7)

sup
y∈K[ω]meu(y) − u(x) − h∇u(x),Φ(y) − Φ(x)i = o(kT[ω]m k)

as m → ∞.

ee 2 dµ.
Such ∇u(x)∈ Im Zx as in (3.7) is unique for each x ∈ KZ , and dµhui = |∇u|du−1(2) For µhhi -a.e. x ∈ K, there exists dh (x) ∈ R such that for any ω ∈ π (x),
(3.8)

sup
y∈K[ω]mu(y) − u(x) −du(x)(h(y) − h(x)) = o(kh ◦ F[ω]m kE )dhSuch dudh (x) ∈ R as in (3.8) is unique for each x ∈ K, and dµhui =as m → ∞.
( du )2dhdµhhi .

In fact, Theorem 3.10 has been recently improved by Koskela and Zhou [62]e and du are given in terms of
where the reminder estimates for the derivatives ∇udh
the associated geodesic metrics; see Theorem 8.3 below.
Remark 3.11. (1) As mentioned in [48, Remark 2.20], the “gradient vector
e in Theorem 3.10-(1) coincides with the “weak gradient” Y (· ; u) defined
field” ∇uby Kusuoka [67, Lemma 5.1] (see also [58, Definition 4.11]).
(2) The rank of the matrix Z, which is 1 µ-a.e. in the present case, is closely related
to the martingale dimension of the associated diﬀusion process. The martingale
dimension of a symmetric diﬀusion process is formally defined as the maximal

10

NAOTAKA KAJINO

number of martingale additive functionals which are independent in the sense ofstochastic integral representation, and intuitively it corresponds to the “maximal
dimension of the tangent space” over the state space. For the purpose of analytic
characterization of martingale dimension, Kusuoka [67, 68] introduced the notionof index for certain strongly local symmetric regular Dirichlet forms on a certainclass of self-similar fractals and identified it as the martingale dimension of theassociated diﬀusion. Hino [38, Definitions 2.9, 3.3 and Theorem 3.4] has recently
extended these results to general strongly local symmetric regular Dirichlet forms,
where the index is defined through certain matrix-valued measurable maps similar
to Z as above whose entries are the Radon-Nikodym derivatives of energy measures.
The index of a non-degenerate elliptic symmetric diﬀusion on a smooth manifold
is easily seen to be equal to the dimension of the manifold, whereas it is diﬃcult todetermine the exact value of the index for diﬀusions on fractals. In our case of thestandard resistance form (E, F) on the Sierpiński gasket, it follows from rank Z = 1,µ-a.e., that the index is 1, and the same is true also for the k-dimensional Sierpiński
gasket with k ≥ 3, as shown in [67, §6, Example 1]. Hino [37, 39] has recently
proved that the index of a point-recurrent self-similar diﬀusion (to be precise, theindex of the resistance form associated with a regular harmonic structure — see[57, Chapter 3]) on a post-critically finite self-similar set is always 1. This result inparticular applies to Brownian motion on aﬃne nested fractals, whose construction
is essentailly due to Lindstrøm [73]; see [57, Section 3.8], [64, 26] and references
therein for details concerning aﬃne nested fractals and Brownian motion on them.
In the case of the canonical Dirichlet form on a generalized Sierpiński carpet,
which was constructed in [6, 8, 69] and is known to be unique by [9], Hino has also
proved in [39, Theorem 4.16] that the index is less than or equal to the spectral
dimension ds of the generalized Sierpiński carpet. Note that this result gives only
an upper bound for the index, so that the exact value of the index for generalized
Sierpiński carpets is still unknown, except when ds < 2, which implies that the index
is 1. (A brief summary of important facts concerning the canonical Dirichlet formon generalized Sierpiński carpets, as well as pictures of some typical generalized
Sierpiński carpets, is available in [50, Section 5].)(3) [38, Theorem 5.4], from which Theorem 3.10 follows, was stated and proved
only for (regular harmonic structures on) post-critically finite self-similar sets. In
fact, Hino [40, Theorem 3.4] has recently generalized it to general strongly local
symmetric regular Dirichlet forms with finite index. See [40] for details.
4. Geometry under the measurable Riemannian structureThis section is a brief summary of the results in [48, Section 3], which areslight improvements of those in [58, Sections 3 and 5] and concern basic geometricproperties of K under the measurable Riemannian structure.
We start with the definition of the canonical geodesic metrics associated with
the Dirichlet spaces (K, µ, E, F) and (K, µhhi , E, F), h ∈ H0 \ R1.
Definition 4.1. Let h ∈ H0 \ R1. We define the harmonic geodesic metric ρH
on K and the h-geodesic metric ρh on K by respectively(4.1) ρH (x, y) := inf{`H (γ) | γ : [0, 1] → K, γ is continuous, γ(0) = x, γ(1) = y},(4.2) ρh (x, y) := inf{`h (γ) | γ : [0, 1] → K, γ is continuous, γ(0) = x, γ(1) = y}

ANALYSIS AND GEOMETRY OF MEASURABLE RIEMANNIAN STRUCTURE ON S.G. 11

for x, y ∈ K, where we set `H (γ) := `R2 (Φ ◦ γ) and `h (γ) := `R (h ◦ γ) for eachcontinuous map γ : [a, b] → K, a, b ∈ R, a ≤ b.
ρH was first introduced by Kigami in [58, Section 5], and the author adopted his
idea to define ρh in [48]. As observed in [48, Section 3] and reviewed below, ρh plays
the role of the canonical geodesic metric for the Dirichlet space (K, µhhi , E, F), as ρH
does for (K, µ, E, F), and (K, ρh , µhhi ) possesses most of the fundamental geometricproperties in common with (K, ρH , µ). The generalization to (K, ρh , µhhi ), where infact the constants involved are all independent of h ∈ SH0 , played essential roles in
the proofs of the main results of [48], and it does also in the proofs of the author’s
recent results in [51], which are reviewed in Sections 6 and 7 below.
Remark 4.2. Note that ρH is diﬀerent from the “harmonic metric” ρΦ on K
introduced in [56, Definition 3.8], which is defined by(4.3)

ρΦ (x, y) := |Φ(x) − Φ(y)|,x, y ∈ K.

ρΦ is a metric on K compatible with the original topology of K by Theorem 3.3
and satisfies ρΦ ≤( ρH , but ρH is notto ρΦ . Indeed,
as noted in [58, p.
) comparable()
800, Remark], ρΦ F1n (q2 ), F1n (q3 ) /ρH F1n (q2 ), F1n (q3 ) = O(3−n ) as n → ∞.
In practice, we need to relate the metrics ρH and ρh suitably to the cell structureof K to obtain various fundamental inequalities such as volume doubling property ofmeasures and weak Poincaré inequality. In [59], Kigami proposed a systematic wayof describing the geometry of a self-similar set using the cell-structure and applied
it to establish reasonable suﬃcient conditions for sub-Gaussian bounds of the heat
kernel associated with a self-similar Dirichlet form. We follow his framework to
describe the relation between the cell structure of K and the metrics ρH and ρh .
Definitions 4.3, 4.4, 4.6, 4.8 and Proposition 4.5 below are adopted from [59].
Definition 4.3. (1) Let w, v ∈ W∗ , w = w1 . . . wm , v = v1 . . . vn . We define
wv ∈ W∗ by wv := w1 . . . wm v1 . . . vn (w∅ := w, ∅v := v). We write w ≤ v if andonly if w = vτ for some τ ∈ W∗ . Note that Σw ∩ Σv = ∅ if and only if neitherw ≤ v nor v ≤ w.
(2) A finite subset Λ of W∗ is called∪ a partition of Σ if and only if Σw ∩ Σv = ∅ forany w, v ∈ Λ with w 6= v and Σ = w∈Λ Σw .
(3) Let Λ1 , Λ2 be partitions of Σ. We say that Λ1 is a refinement of Λ2 , and writeΛ1 ≤ Λ2 , if and only if for each w1 ∈ Λ1 there exists w2 ∈ Λ2 such that w1 ≤ w2 .
If Λ1 ≤ Λ2 , then we have a natural surjection Λ1 → Λ2 by which w1 ∈ Λ1 ismapped to the unique w2 ∈ Λ2 such that w1 ≤ w2 , and in particular, #Λ1 ≥ #Λ2 .
Definition 4.4. (1) A family S = {Λs }s∈(0,1] of partitions of Σ is called a scaleon Σ if and only if S satisfies the following three properties:(S1) Λ1 = W0 (= {∅}). Λs1 ≤ Λs2 for any s1 , s2 ∈ (0, 1] with s1 ≤ s2 .
(S2) min{|w| | w ∈ Λs } → ∞ as s ↓ 0.
(Sr) Each s ∈ (0, 1) admits ε ∈ (0, 1 − s] such that Λs0 = Λs for any s0 ∈ (s, s + ε).
(2) A function l : W∗ → (0, 1] is called a gauge function on W∗ if and only if
l(wi) ≤ l(w) for any (w, i) ∈ W∗ × S and limm→∞ max{l(w) | w ∈ Wm } = 0.
There is a natural one-to-one correspondence between scales on Σ and gauge
functions on W∗ , as in the following proposition. See [59, Section 1.1] for a proof.

12

NAOTAKA KAJINO

Proposition 4.5. (1) Let l be a gauge function on W∗ . For s ∈ (0, 1], define(4.4)

Λs (l) := {w | w = w1 . . . wm ∈ W∗ , l(w1 . . . wm−1 ) > s ≥ l(w)}where l(w1 . . . wm−1 ) := 2 when w = ∅. Then the collection S(l) := {Λs (l)}s∈(0,1] isa scale on Σ. We call S(l) the scale induced by the gauge function l.
(2) Let S = {Λs }s∈(0,1] be a scale on Σ. Then there exists a unique gauge function
lS on W∗ such that S = S(lS ). We call lS the gauge function of the scale S.
Definition 4.6. Let S = {Λs }s∈(0,1] be a scale on Σ. For s ∈ (0, 1] and x ∈ K,
we define∪∪(4.5)
Ks (x, S) :=Kw ,Us (x, S) :=Kw .
w∈Λs , x∈Kww∈Λs , Kw ∩Ks (x,S)6=∅

Ks (x, S) and Us (x, S) are clearly non-decreasing in s ∈ (0, 1], and it immediately
follows from [57, Proposition 1.3.6] that {Ks (x, S)}s∈(0,1] and {Us (x, S)}s∈(0,1] arefundamental systems of neighborhoods of x in K.
Proposition 2.3 easily yields the following lemma.
Lemma 4.7. Let S = {Λs }s∈(0,1] be a scale on Σ, let s ∈ (0, 1], x ∈ K and w ∈Λs . Then #{v ∈ Λs | Kv ∩ Ks (x, S) 6= ∅} ≤ 6 and #{v ∈ Λs | Kw ∩ Kv 6= ∅} ≤ 4.
Definition 4.8. Let S = {Λs }s∈(0,1] be a scale on Σ. A metric ρ on K is called
adapted to S if and only if there exist β1 , β2 ∈ (0, ∞) such that
(4.6)

Bβ1 s (x, ρ) ⊂ Us (x, S) ⊂ Bβ2 s (x, ρ),(s, x) ∈ (0, 1] × K.

Lemma 4.9. Let S = {Λs }s∈(0,1] be a scale on Σ with gauge function l and letρ be a metric on K adapted to S. Then ρ is compatible with the original topologyof K, and diamρ Kw ≤ β2 l(w) for any w ∈ W∗ , where β2 ∈ (0, ∞) is as in (4.6).
Proof. See [48, Lemma 3.7].



Next we define scales on Σ to which the metrics ρH and ρh , h ∈ SH0 , areadapted (recall (3.3) for SH0 ).
Definition 4.10. (1) We define SH = {ΛHscale on Σ induceds }s∈(0,1] to be the√by the gauge function lH : W∗ → (0, 1], lH (w) := kTw k ∧ 1 = (3/5)|w| µ(Kw ) ∧ 1.
(2) Let h ∈ SH0 . We define Sh = {Λhs }s∈(0,1] to be the scale on Σ induced by the√gauge function lh : W∗ → (0, 1], lh (w) := kh ◦ Fw kE = (3/5)|w| µhhi (Kw ).
As we will state in Theorem 4.15 below, ρH and ρh introduced in Definition 4.1,
where h ∈ SH0 , are indeed metrics on K adapted to SH and Sh respectively andthe infimums in (4.1) and (4.2) are achieved by a specific class of paths in K. The
key to these results is the next theorem, which requires the following definition.
Definition 4.11. (1) For x, y ∈ R2 , we set xy := {x + t(y − x) | t ∈ [0, 1]},
which is also regarded as the map [0, 1] 3 t 7→ (1 − t)x + ty ∈ R2 .
mm
(2) Let m ∈ N∪{0} and x, y ∈ Vm , x ∼ y, where ∼ is as in Definition 2.4. We define
w(x, y) to be the unique w ∈ Wm such that x, y ∈ Fw (V0 ). Note that xy ⊂ Kw(x,y) .
√√Theorem 4.12 ([94], [58, Theorem 5.4]). Set I := [−1/ 3, 1/ 3]. ThenΦ(q2 q3 ) = {(ϕ(t), t) | t ∈ I} for some ϕ : I → R, and the following hold:√√(1) ϕ is C 1 but not C 2 , ϕ0 is strictly increasing and ϕ0 (±1/ 3) = ±1/ 3.
(2) q2 q3 ⊂ KZ and (ϕ0 (t), 1) ∈ Im ZΦ−1 (ϕ(t),t) for any t ∈ I.

ANALYSIS AND GEOMETRY OF MEASURABLE RIEMANNIAN STRUCTURE ON S.G. 13(3) KH ⊂ {(s, t) ∈ R × I | s ≤ ϕ(t)}, or equivalently, h1 ≤ ϕ ◦ h2 .
Definition 4.13. (1) Let m ∈ N ∪ {0}. A sequence Γ = {xk }N
k=0 ⊂ Vm ,mwhere N ∈ N, is called an m-walk if and only if xk−1 ∼ xk for k ∈ {1, . . . , N } andw(xk−1 , xk ) 6= w(xk , xk+1 ) for k ∈ {1, . . . , N − 1}. For such Γ we define continuousb : [0, `H (Γ)] → K bymaps Γ : [0, N ] → K and Γ(4.7)

Γ(t) := (k − t)xk−1 + (t − k + 1)xk ,t ∈ [k − 1, k], k ∈ {1, . . . , N },

b :=
and Γ
where ϕΓ is the homeomorphism ϕΓ : [0, N ] → [0, `H (Γ)],ϕΓ (t) := `H (Γ|[0,t] ); note that `H (Γ) < ∞ by Theorem 4.12 and Proposition 3.2.
(2) Let γ : [a, b] → K be continuous, a, b ∈ R, a < b. For m ∈ N ∪ {0}, γ is()
b `H (Γ) t−a , t ∈ [a, b] for somecalled a harmonic m-geodesic if and only if γ(t) = Γb−am-walk Γ. γ is called a harmonic geodesic if and only if there exist n ∈ N ∪ {0} and{am }m≥n , {bm }m≥n ⊂ [a, b] with limm→∞ am = a and limm→∞ bm = b such thatam+1 ≤ am < bm ≤ bm+1 and γ|[am ,bm ] is a harmonic m-geodesic for each m ≥ n.
Γ ◦ ϕ−1Γ ,Then Theorem 4.12 together with Propositions 3.2 and 3.7 immediately yields
the following proposition.
Proposition 4.14. If γ : [0, 1] → K is a harmonic geodesic, then Φ ◦ γ|(0,1) is
C and for any t ∈ (0, 1), γ(t) ∈ KZ , (Φ◦γ)0 (t) ∈ Im Zγ(t) and |(Φ◦γ)0 (t)| = `H (γ).
1Theorem 4.15 ([58, Theorems 5.1 and 5.11], [48, Propositions 3.15 and 3.16]).
Let h ∈ SH0 and let h denote any one of H and h.
(1) ρh is a metric on K satisfying(4.8)

B√2s/50 (x, ρh ) ⊂ Us (x, Sh ) ⊂ B10s (x, ρh ),(s, x) ∈ (0, 1] × K.

h
: [0, 1] → K(2) For each x, y ∈ K with x 6= y, there exists a harmonic geodesic γxyhhhsuch that γxy (0) = x, γxy (1) = y and ρh (x, y) = `h (γxy ). Moreover, if m ∈ N ∪ {0}hand x, y ∈ Vm then we can take a harmonic m-geodesic as γxy.

Remark 4.16. Let γ : [0, 1] → K be a harmonic geodesic and let h ∈ H0 \ R1.
Then we easily see from Theorem 4.12 that the set {t ∈ (0, 1) | (h ◦ γ)0 (t) = 0}
is discrete (see [48, (3.15)]), so that ϕh : [0, 1] → [0, `h (γ)], ϕh (t) := `h (γ|[0,t] ),is strictly increasing. Therefore γ admits a parametrization by `h -length given by−1γ ◦ ϕ−1h : `h (γ ◦ ϕh |[0,t] ) = t for any t ∈ [0, `h (γ)].
The proof of Theorem 4.15 is based on the following lemma, which in turn isan easy consequence of Theorem 4.12.
Lemma 4.17 ([58, Lemma 5.6], [48, Lemma 3.18]). Let h ∈ SH0 and let hdenote any one of H and h. Let w ∈ W∗ and x, y ∈ Fw (V0 ), x 6= y. Then
(4.9) `h (xy) = inf{`h (γ) | γ : [0, 1] → Kw , γ is continuous, γ(0) = x, γ(1) = y},
√
√
2
4 6
(4.10)
lh (w) ≤ `h (xy) ≤lh (w).
10
3
The proof of (4.9) for `H utilizes the following theorem from convex geometry as
well, which is quoted in [58, Theorem 5.2] without reference. Recall that a subset Γof R2 is called a Jordan curve if and only if it is homeomorphic to {x ∈ R2 | |x| = 1}.
For such Γ, its length `R2 (Γ) with respect to the Euclidean norm | · | is naturally
defined, and according to the Jordan curve theorem (see e.g. [77, Section 4]), R2 \ Γconsists of two connected components whose boundaries in R2 are both equal to Γ.

14

NAOTAKA KAJINO

Theorem 4.18. Let C be a compact convex subset of R2 with intR2 C 6= ∅.
(1) ∂R2 C is a Jordan curve and `R2 (∂R2 C) < ∞.
(2) Let D be a compact subset of R2 such that ∂R2 D is a Jordan curve and C ⊂ D.
If C 6= D, or equivalently if ∂R2 C 6= ∂R2 D, then `R2 (∂R2 C) < `R2 (∂R2 D).
Proof. (1) An elementary argument shows that ∂R2 C is a Jordan curve (seee.g. [15, Theorems 6.7 and 11.9]), and it follows from (2) that `R2 (∂R2 C) < ∞.
(2) Note that for q ∈ intR2 D, ∂R2 D regarded as a closed curve in R2 \ {q} is
not homotopic in R2 \ {q} to a constant map, by virtue of the Schönflies theorem[77, Section 10, Theorem 4] saying that every injective continuous map from a
Jordan curve to R2 is the restriction of a homeomorphism from R2 to R2 . On
the basis of this fact, the assertion can be verified by using [15, Theorem 7.9]
to approximate ∂R2 C by convex polygons whose vertices belong to ∂R2 C and by
applying the arguments in [15, Proof of Theorem 7.11 and Exercise 7.4].

In fact, we can also prove the following characterization of shortest paths withrespect to `H by using Theorems 4.12 and 4.18, as follows.
Theorem 4.19. Let γ : [0, 1] → K be continuous and satisfy ρH (γ(0), γ(1)) =`H (γ) > 0. Then there exist a unique harmonic geodesic γ
b : [0, 1] → K and aunique non-decreasing continuous surjection ϕγ : [0, 1] → [0, 1] such that γ = γb ◦ϕγ .
Moreover, if m ∈ N ∪ {0} and γ(0), γ(1) ∈ Vm , then γb is a harmonic m-geodesic.
Theorem 4.19 has important applications to the invalidity of various geometricconditions on (K, ρH , µ) related to Ricci curvature lower bound; see Subsection 8.2
below.
We need the following lemma for the proof of Theorem 4.19.
Lemma 4.20. Let w ∈ W∗ and x, y ∈ Fw (V0 ), x 6= y. Let a, b ∈ R, a < b andlet γ : [a, b] → Kw be a continuous injective map with γ(a) = x and γ(b) = y. If`H (γ) ≤ `H (xy), then γ([a, b]) = xy.
Proof. Let C, D be the compact subsets of R2 with intR2 C 6= ∅ 6= intR2 D
whose boundaries in R2 are the Jordan curves Φ(x)Φ(y) ∪ Φ(xy) and Φ(x)Φ(y) ∪Φ ◦ γ([a, b]), respectively. Then C is convex and C ⊂ D by virtue of the rotational
symmetry of KH , Proposition 3.2-(3), Theorem 4.12 and the Jordan curve theorem.
Therefore if `H (γ) ≤ `H (xy), then `R2 (∂R2 D) ≤ `R2 (∂R2 C), hence ∂R2 D = ∂R2 C by
Theorem 4.18-(2), and thus γ([a, b]) = xy.
Proof of Theorem 4.19. For uniqueness, let γb and ϕγ be as in the assertion,
so that ϕγ (0) = 0, ϕγ (1) = 1 and `H (bγ ) = `H (γ). Let t ∈ [0, 1] and choose s ∈ [0, 1]
so that t = `H (γ|[0,s] )/`H (γ). Then `H (γ|[0,t] ) = `H (bγ |[0,ϕγ (t)] ) = `H (γ)ϕγ (t) by
Proposition 4.14, hence t = ϕγ (s) and γb(t) = γ(s), proving the uniqueness assertion.
For existence, define a non-decreasing continuous surjection ϕγ : [0, 1] → [0, 1]
by ϕγ (t) := `H (γ|[0,t] )/`H (γ). Since γ(s) = γ(t) for any s, t ∈ [a, b] with ϕγ (s) =ϕγ (t), there exists a unique map γ
b : [0, 1] → K such that γ = γb ◦ ϕγ , and then γb iscontinuous and `H (bγ |[0,t] ) = t`H (γ) for any t ∈ [0, 1]. In particular, `H (bγ |[s,t] ) > 0
for any s, t ∈ [0, 1] with s < t, which together with `H (bγ ) = `H (γ) = ρH (γ(0), γ(1)),γb(0) = γ(0) and γb(1) = γ(1) shows that γ
b is injective.
To see that γ
b is a harmonic geodesic, set n := inf{m ∈ N∪{0} | #bγ −1 (Vm ) ≥ 2}
−1
(n < ∞ by γb(0) 6= γb(1)), and for m ≥ n set am := min γb (Vm ) and bm :=ANALYSIS AND GEOMETRY OF MEASURABLE RIEMANNIAN STRUCTURE ON S.G. 15

max γ
b−1 (Vm ), so that am+1 ≤ am < bm ≤ bm+1 for any m ≥ n. The injectivity ofγb easily yields limm→∞ am = 0, limm→∞ bm = 1 and #b
γ −1 (Vm ) < ∞.
−1
N
Let m ≥ n, N := #bγ (Vm ) − 1 and let {tk }k=0 be the strictly increasing
enumeration of the elements of γb−1 (Vm ), so that t0 = am and tN = bm . Also
set xk := γb(tk ) for k ∈ {0, . . . , N } and Γ := {xk }N
k=0 ⊂ Vm . We claim that()
b `H (Γ) t−am for any t ∈ [am , bm ]. Indeed, letΓ is an m-walk and that γb(t) = Γbm −ammk ∈ {1, . . . , N }. It easily follows from γb((tk−1 , tk ))∩Vm = ∅ that xk−1 ∼ xk and thatγb([tk−1 , tk ]) ⊂ Kw(xk−1 ,xk ) , and then we also have `H (bγ |[tk−1 ,tk ] ) = ρH (xk−1 , xk ) by`H (bγ ) = ρH (bγ (0), γb(1)). If k < N and w(xk−1 , xk ) = w(xk , xk+1 ), then Lemma 4.20
γ |[tk−1 ,tk+1 ] ), contradicting `H (bγ ) = ρH (bγ (0), γb(1)).
implies `H (xk−1 xk+1 ) < `H (bThus Γ is an m-walk. Moreover, since `H (bγ |[tk−1 ,tk ] ) = ρH (xk−1 , xk ) ≤ `H (xk−1 xk ),Lemma 4.20 yields γb([tk−1 , tk ]) = xk−1 xk , which together with `H (bγ |[0,t] ) = t`H (bγ ),t ∈ [0, 1], easily implies the above claim. Thus γb|[am ,bm ] is a harmonic m-geodesic forany m ≥ n and hence γ
b is a harmonic geodesic. If m ∈ N ∪ {0} and γ(0), γ(1) ∈ Vm ,then m ≥ n, am = 0 and bm = 1, so that γ
b is a harmonic m-geodesic.
Remark 4.21. If h ∈ H0 \R1, then the assertions of Theorem 4.19 and Lemma
4.20 are not valid for `h and ρh .

Proof. Noting that minV0 h < maxV0 h by Proposition 2.11-(2), let i, j ∈ S
be such that h(qi ) = minV0 h and h(qj ) = maxV0 h and let {k} = S \ {i, j}. We
first assume h(qi ) + h(qj ) 6= 2h(qk ). By considering −h and qj instead of h and qi
if necessary, we may assume that h(qi ) + h(qj ) < 2h(qk ). Let U be the connectedcomponent of h−1 ((−∞, h(qk )) with qi ∈ U . Then Theorem 4.12 and the rotational
symmetry of KH easily imply that h is strictly decreasing on qk qi and on qk Fkn (qj )for some n ∈ N. Therefore Kkn \ {qk } ⊂ U by the strong maximum principle [57,
Theorem 3.2.14] for h ◦ Fkn , and hence ρh (qi , x) = h(x) − h(qi ) for any x ∈ Kkn
by [48, (4.13)] (see (5.8) below). A similar argument for −h ◦ Fkn together with
[48, (4.13)] also implies that ρh (qk , x) = h(qk ) − h(x) for any x ∈ Kkm for somem ≥ n. Thus by Theorem 4.15 and Remark 4.16, for any x ∈ Kkm \ {qk } there
exists a continuous injective map γx : [0, 1] → K with γx (0) = qi , γx (1/2) = x
and γx (1) ∪= qk such that `h (γx ) = h(qk ) − h(qi ) = `h (qi qk ) = ρh (qi , qk ). Now if
x ∈ Kkm \ w∈W∗ Fw (q1 q2 ∪ q2 q3 ∪ q3 q1 ), then the∪conclusions of Theorem 4.19 andLemma 4.20 are not valid for γx since γb((0, 1)) ⊂ w∈W∗ Fw (q1 q2 ∪ q2 q3 ∪ q3 q1 ) andhence x 6∈ γb((0, 1)) for any harmonic geodesic γ
b : [0, 1] → K.
If h(qi ) + h(qj ) = 2h(qk ), then since h ∈ H{qi ,qj } by the axial symmetry of Kand (E, F), it follows similarly to [48, Proposition 4.9] (see Proposition 5.8 below)that ρh (qi , x) = h(x) − h(qi ) and ρh (qj , x) = h(qj ) − h(x) for any x ∈ K. The restof the proof goes in exactly the same way as in the previous paragraph.
At the last of this section, we state the volume doubling property and the weak
Poincaré inequality of (K, µ, E, F) and (K, µhhi , E, F) under the metrics ρH and ρh ,respectively. The following lemma is essential for the proofs of those properties.
Lemma 4.22 ([48, Lemma 3.9], cf. [58, Section 3]). Let h ∈ SH0 .
(1) For any (w, i) ∈ W∗ × S,
3
1
3
1
(4.11)µ(Kw ) ≤ µ(Kwi ) ≤ µ(Kw ),kTw k ≤ kTwi k ≤ kTw k,
15
5
5
5
1
3
1
3
(4.12)µhhi (Kw ) ≤ µhhi (Kwi ) ≤ µhhi (Kw ),lh (w) ≤ lh (wi) ≤ lh (w).
15
5
5
5

16

NAOTAKA KAJINO(2) If w, v ∈ W∗ satisfies |w| = |v| and Kw ∩ Kv 6= ∅, then
(4.13)µhhi (Kw ) ≤ 9µhhi (Kv ),lh (w) ≤ 3lh (v)andlH (w) ≤ 3lH (v).

Then we can verify the following proposition on the basis of Lemma 4.22 inexactly the same ways as [59, Proofs of Theorems 1.3.5 and 1.4.3].
Proposition 4.23 ([48, Proposition 3.10], cf. [58, Theorem 6.2]). (1) Thereexists cG ∈ (0, ∞) such that for any g, h ∈ SH0 ,µhgi (Kw ) ≤ cG µhgi (Kv )(4.14)hwhenever either w, v ∈ ΛHs or w, v ∈ Λs for some s ∈ (0, 1] and Kw ∩ Kv 6= ∅.
(2) Let κ := log5 15 and κ̂ := log5/3 15. Then there exists cv ∈ (0, ∞) such that for
any g, h ∈ SH0 , any x ∈ K and any s, t ∈ (0, 1] with s ≤ t,( t )κ̂( t )κν(Us (x, S)), µhgi (Ut (x, S)) ≤ cv
µhgi (Us (x, S)),(4.15) ν(Ut (x, S)) ≤ cvss

where (ν, S) denotes any one of (µ, SH ) and (µhhi , Sh ).
Remark 4.24. The powers κ and κ̂ in (4.15) are best possible. See [48, Remark
3.11] for details.
Now we conclude the volume doubling property of (K, ρH , µ) and (K, ρh , µhhi )as an immediate consequence of Proposition 4.23-(2) and Theorem 4.15-(1).
Theorem 4.25 ([58, Theorem 6.2], [48, Theorem 3.19]). As in Proposition
4.23-(2) let κ := log5 15 and κ̂ := log5/3 15. Then there exists cV ∈ (0, ∞) such thatfor any g, h ∈ H0 \ R1, any x ∈ K and any r, s ∈ (0, ∞) with r ≤ s,( s )κ( s )κ̂(4.16) ν(Bs (x, ρ)) ≤ cV
ν(Br (x, ρ)), µhgi (Bs (x, ρ)) ≤ cV
µhgi (Br (x, ρ)),rr
where (ν, ρ) denotes any one of (µ, ρH ) and (µhhi , ρh ).
Finally we state the weak Poincaré inequality of (K, µ, E, F) and (K, µhhi , E, F).
Proposition 4.26 ([48, Proposition 3.20]). Let cP := 34 106 c4G with cG as in
Proposition 4.23-(1). Let h ∈ H0 \ R1 and let (ν, ρ) denote any one of∫(µ, ρH ) and−1udν,(µhhi , ρh ). Then for any (r, x) ∈ (0, ∞)×K, with uν,ρr,x := ν(Br (x, ρ))Br (x,ρ)
∫
(
)
2
2
√
(4.17)u − uν,ρu ∈ F.
r,x dν ≤ cP r µhui B250 2r (x, ρ) ,Br (x,ρ)Proposition 4.26 is easily proved by using Lemma 4.7, Theorem 4.15-(1), Lemma
4.22-(1), Proposition 4.23-(1) and the following fact implied by the definition of theresistance metric RE : for any w ∈ W∗ and any x, y ∈ Kw ,( 3 )|w|()(4.18) |u(x) − u(y)|2 ≤ RE Fw−1 (x), Fw−1 (y) E(u ◦ Fw , u ◦ Fw ) ≤ 3µhui (Kw );5
note that we easily have diamRE K ≤ 3 by using [57, Lemma 3.3.5].
Notation. In what follows we will use the constants κ = log5 15, κ̂ = log5/3 15,
cG and cV appearing in Proposition 4.23 and Theorem 4.25 without further notice.

ANALYSIS AND GEOMETRY OF MEASURABLE RIEMANNIAN STRUCTURE ON S.G. 17

5. Short time asymptotics of the heat kernels
In this section, we review known results on short time asymptotic behavior of
the heat kernels pµ and pµhhi , h ∈ H0 \R1, mainly following [48, Sections 4–6]. The
results concern three diﬀerent aspects of the asymptotics: oﬀ-diagonal Gaussian
behavior, one-dimensional behavior at vertices and non-integer-dimensional µ-a.e.
on-diagonal behavior, which are reviewed separately in each of the following threesubsections.
5.1. Intricsic metrics and oﬀ-diagonal Gaussian behavior. Let us start
this subsection with the following standard definition.
Definition 5.1. Let ν be a finite Borel measure on K with full support. Define(5.1)ρν (x, y) = sup{u(x) − u(y) | u ∈ F , µhui ≤ ν},x, y ∈ K.

Clearly, ρν (x, y) = ρν (y, x) ∈ [0, ∞), ρν (x, x) = 0 and ρν (x, y) ≤ ρν (x, z) + ρν (z, y)
for any x, y, z ∈ K; in fact, ρν (x, y)2 ≤ ν(K)RE (x, y). ρν is called the intrinsic
metric of the Dirichlet space (K, ν, E, F) or simply the ν-intrinsic metric on K.
As suggested by the results of [89, 90, 83, 44], oﬀ-diagonal Gaussian behaviorof the Markovian semigroup of a strong local Dirichlet space is described best by
the associated intrinsic metric. On the other hand, it is highly non-trivial to give areasonable geometric characterization of the intrinsic metric for concrete examples.
For the canonical Dirichlet space associated with a smooth Riemannian manifold,
it is not diﬃcult to see that the intrinsic metric is equal to its Riemannian distance;see [79] and references therein for related results on Riemannian manifolds. The
same is in fact true also for (K, µ, E, F) and (K, µhhi , E, F), h ∈ H0 \ R1, as follows.
Theorem 5.2 ([48, Theorem 4.2]). Let h ∈ H0 \ R1 and let (ν, ρ) denote any
one of (µ, ρH ) and (µhhi , ρh ). Then ρ = ρν . Moreover, ρ(x, ·) ∈ F and µhρ(x,·)i = ν
for any x ∈ K.
Then in view of Theorem 4.25 and Proposition 4.26, the general results ofSturm [89, 90] and Ramı́rez [83] together with ρ = ρν imply the following Gaussian
bounds and Varadhan’s asymptotic relation.
Corollary 5.3 ([58, Theorem 6.3], [48, Corollary 4.3]). Let h ∈ H0 \ R1 andlet (ν, ρ) denote any one of (µ, ρH ) and (µhhi , ρh ). Let n ∈ N. Then there exist
cL , cU ∈ (0, ∞) determined solely by κ, cG , cV and cU (n) ∈ (0, ∞) determined solely
by n, κ, cG , cV such that for any (t, x, y) ∈ (0, ∞) × K × K,(
2)((
2 )κ/22)
exp − ρ(x,y)
1 + ρ(x,y)
exp − ρ(x,y)
cL t
t
4t
(
)
cL
(5.2)
≤ pν (t, x, y) ≤ cU √ () () ,ν B√t (x, ρ)ν B√t (x, ρ) ν B√t (y, ρ)((2 )κ/2+n2)
1 + ρ(x,y)
exp − ρ(x,y)
n
t
4t
√ (
(5.3)
∂t pν (t, x, y) ≤ cU (n)) () .
tn ν B√t (x, ρ) ν B√t (y, ρ)Corollary 5.4 ([48, Corollary 4.4]). Let h ∈ H0 \ R1 and let (ν, ρ) denoteany one of (µ, ρH ) and (µhhi , ρh ). Then
(5.4)

lim 4t log pν (t, x, y) = −ρ(x, y)2 ,t↓0x, y ∈ K.

18

NAOTAKA KAJINO

Moreover, according to a recent result [62, Theorem 7.1] of Koskela and Zhou,
we have the following asymptotic behavior of the “logarithmic derivatives of theheat kernels” by virtue of ρ(x, ·) ∈ F and µhρ(x,·)i = ν, x ∈ K; see [76, 88] forthe corresponding pointwise results for the heat kernels on Riemannian manifolds.
Note that distρ (·, A) ∈ F for any A ⊂ K with A 6= ∅ by Proposition 5.6 below.
Corollary 5.5. Let h ∈ H0 \ R1 and let (ν, ρ) denote any one of (µ, ρH ) and(µhhi , ρh ). Then for any A ∈ B(K) with ν(A) > 0, µh4t log Ttν 1A i converges weakly
to µhdistρ (·,A)2 i as t ↓ 0, that is, for any f ∈ C(K),
∫
∫
(5.5)
lim
f dµh4t log Ttν 1A i =f dµhdistρ (·,A)2 i .
t↓0

K

K

The proof of ρ ≤ ρν and that of ρ(x, ·) ∈ F and µhρ(x,·)i = µ for x ∈ K
are based on Theorem 3.10 and Theorem 4.15-(2), whereas the converse inequalityρν ≤ ρ is an immediate consequence of the following proposition.
Proposition 5.6 ([48, Proposition 4.10]). Let h ∈ H0 \R1 and let (ν, ρ) be any
one of (µ, ρH ) and (µhhi , ρh ). Then {u ∈ F | µhui ≤ ν} = {u ∈ C(K) | Lipρ u ≤ 1}.
Proposition 5.6 is proved by using Theorem 4.15-(2) and Proposition 5.8 below
to reduce the proof to the case of the heat kernels on one-dimensional intervals. We
need the following lemma for the statement of Proposition 5.8.
Lemma 5.7 (cf. [60, Theorem 10.4]). Let ν be a finite Borel measure on K withfull support, let U be a non-empty open subset of K and set ν|U := ν|B(U ) and E U :=E|FU ×FU . Then (E U , FU ) is a strong local regular Dirichlet form on L2 (U, ν|U )whose associated Markovian semigroup {Ttν,U }t∈(0,∞) admits a unique continuous
UU
integral kernel pU
ν = pν (t, x, y) : (0, ∞) × U × U → [0, ∞), and pν is extended to aUcontinuous function on (0, ∞)×K×K by setting pν := 0 on (0, ∞)×(K×K\U ×U ).
U
pU
ν is called the heat kernel associated with (U, ν|U , E , FU ).
Proposition 5.8 ([48, Proposition 4.9]). Let h ∈ H0 \R1, i ∈ S, b ∈ (h(qi ), ∞)and set a := h(qi ). Suppose that the connected component U of h−1 ((−∞, b)) withqi ∈ U satisfies U ∩ V0 = {qi }. Let p[a,b) = p[a,b) (t, x, y) : (0, ∞) × [a, b] × [a, b] →[0, ∞) be the heat kernel for d2 /dx2 on [a, b] with Neumann (reflecting) boundary
condition at a and Dirichlet (absorbing) boundary condition at b. Then
(5.6)µhhi ◦ (h|U )−1 = E(h, hi1 )1[a,b] dx(dx is the Lebesgue measure on R),

(5.7)

i −1
p[a,b) (t, a, h(x)),
pUµhhi (t, qi , x) = E(h, h1 )

(5.8)

ρh (qi , x) = ρµhhi (qi , x) = h(x) − a,(t, x) ∈ (0, ∞) × U ,x ∈ U,

where U denotes the closure of U in K.
5.2. One-dimensional asymptotics at vertices. As observed from the picture of the harmonic Sierpiński gasket KH (Figure 2), for x ∈ V∗ , suﬃciently smallneighborhoods of Φ(x) in KH are geometrically very close to the “tangent line of
KH at Φ(x)”. As reflections of this geometric intuition, the Kusuoka measure µ andthe associated heat kernel pµ exhibit sharp one-dimensional behavior, as follows.
Theorem 5.9 ([48, Theorem 5.3]). The limit limr↓0 µ(Br (x, ρH ))/r =: 2ξx ∈(0, ∞) exists for any x ∈ V∗ .

ANALYSIS AND GEOMETRY OF MEASURABLE RIEMANNIAN STRUCTURE ON S.G. 19

Theorem 5.10 ([48, Theorem 5.8]). Let x ∈ V∗ . Choose ζx = (ζx1 , ζx2 ) ∈ Im Zx
so that |ζx | = 1 and set hx := ζx1 (h1 − h1 (x)1) + ζx2 (h2 − h2 (x)1). Then there exist
tx , rx , cx ∈ (0, ∞) such that for any δ ∈ (0, 1] and any (t, y) ∈ (0, tx ] × Brx (x, ρH ),(2 )((y)2 )) exp − hx (y)( κ̂−12(κ̂−1)
exp − hx4t4(1+δ)tκ√√(5.9) pµ (t, x, y) −,≤ cx,δ t 2 + δ |hx (y)| κ+1
ξx 4πtξx 4πt√15κ̂
where cx,δ := cx /δ 4 κ+ 2 +2 . In particular, limt↓0 4πtpµ (t, x, x) = 1/ξx .
Theorem 5.11 ([48, Theorem 5.16]). Let x ∈ V∗ and α ∈ (−1, ∞). Then∫∫−y 2 /41ααedy.
(5.10)
lim α/2
ρH (x, y) pµ (t, x, y)dµ(y) =
|y| √
t↓0 t
4π
K
R
The key to the proofs of Theorems 5.9, 5.10 and 5.11 is again reduction to the“direction of the tangent line Im Zx = Rζx of KH at Φ(x)”, that is, to the caseof µhhx i , pµhhx i and ρhx , based on a suitable modification [48, Proposition 5.4] of
Proposition 5.8 for hx on a suﬃciently small neighborhood of x.
5.3. On-diagonal asymptotics at almost every point. As we saw in Subsections 5.1 and 5.2, the heat kernel pµ of (K, µ, E, F) satisfies the Gaussian bounds
and Varadhan’s asymptotic relation of exactly the same forms as those for the heat
kernels on Riemannian√manifolds, and pµ (t, x, x) is asymptotically equivalent to a
constant multiple of 1/ 4πt as t ↓ 0 for each x ∈ V∗ . On the other hand, we cannot
expect such a smooth behavior of pµ (t, x, x) for generic x ∈ K. Indeed, we have thefollowing result. Recall that λ = µ ◦ π. Note also that 2 log25/3 5 = 1.5181 . . . < 2.
Theorem 5.12 ([48, Theorem 6.1 and Proposition 6.6]). Let h ∈ H0 \ R1 andlet (ν, ρ) denote any one of (µ, ρH ) and (µhhi , ρh ). Define
1 ∑5(5.11)
η := infλ(Σw ) log kTw kanddloc := 2 + η −1 log .
m∈N 2m
3
w∈Wm

√
[
)
Then η = lim 2m w∈Wm λ(Σw ) log kTw k ∈ log 53 , log 35 , dloc ∈ (1, 2 log25/3 5]m→∞andlog ν(Br (x, ρ))
2 log pν (t, x, x)(5.12)lim= lim= dlocν-a.e. x ∈ K.
r↓0
t↓0
log r
− log t∑1

The key step for the proof of Theorem 5.12 is the following proposition, which
can be verified by using Lemma 4.7, Proposition 4.23-(1), (4.8), (4.16) and (4.11).
Proposition 5.13 ([48, Proposition 6.4]). Let x ∈ K and ω ∈ π −1 (x). Then1limr↓0 log µ(Br (x, ρH ))/ log r exists if and only if limm→∞ m
log kT[ω]m k exists, andif either of these two limits exists then
(5.13)

log 53limr↓0 log µ(Br (x, ρH ))=2+∈ [1, 2 log25/3 5].
1
log r
limm→∞ mlog kT[ω]m k

1
log kT[ω]m k
It is immediate from (5.11) that η < log 35 . Since η = limm→∞ m
for λ-a.e. ω ∈ Σ by the σ-ergodicity of λ (Theorem 3.5) and Kingman’s subadditive
ergodic theorem [23, Theorem 10.7.1], Theorem 5.12 is now easily proved by using
Proposition 5.13 (as well as its counterpart for (µhhi , ρh )) and (5.2).

20

NAOTAKA KAJINO

6. Ahlfors regularity and singularity of Hausdorﬀ measure
In this and the next sections, we review the author’s recent unpublished results
to be treated in a forthcoming paper [51] which mainly concerns Weyl’s Laplacian
eigenvalue asymptotics for the Dirichlet space (K, µ, E, F). A crucial fact for Weyl’s
asymptotics is the Ahlfors regularity of (K, ρH ) and of (K, ρh ) uniform in h ∈ SH0 ,which we explain in some detail in this section as a preparation for the next section.
We also see that the Hausdorﬀ measures on (K, ρH ) and (K, ρh ) (of the appropriate
dimension) are singular with respect to the Kusuoka measure µ.
Let us first recall the following standard notations and definitions. See e.g. [24,Chapters 2 and 3] and references therein for details of Hausdorﬀ measure, Hausdorﬀ
dimension and box-counting dimension; note that the definitions there apply to any
metric space although they are stated only for the Euclidean spaces.
Notation. Let (E, ρ) be a metric space and let A ⊂ E be non-empty.
(1) For α ∈ (0, ∞), the α-dimensional Hausdorﬀ measure on E with respect to ρ isdenoted by Hρα and the Hausdorﬀ dimension of A with respect to ρ by dimH (A, ρ).
(2) The lower and upper box-counting dimensions of A with respect to ρ are denotedby dimB (A, ρ) and dimB (A, ρ), respectively. If they are equal, their common value,called the box-counting dimension of A with respect to ρ, is denoted by dimB (A, ρ).
Note that 0 ≤ dimH (A, ρ) ≤ dimB (A, ρ) ≤ dimB (A, ρ) ≤ ∞ by [24, (3.17)].
The following theorem was obtained in [48]. Recall (5.11) for the constant dloc .
Theorem 6.1 ([48, Theorem 7.2 and Proposition 7.6]). Set d := dimH (K, ρH ).
Then dloc ≤ d ≤ 2 log25/3 5 and HρdH (K) ∈ (0, ∞). Moreover, for any h ∈ SH0 ,
(6.1)
(6.2)

d = dimB (K, ρH ) = dimB (K, ρh ),−10 −d3s

19 −d≤ #Λhs ≤ #ΛH,s ≤3 ss ∈ (0, 1].

The proof of (6.2) heavily relies on the rotational symmetry of K and (E, F),
whereas (6.1) follows from (6.2) by virtue of Lemma 4.7, (4.8) and [47, Proposition
2.24]. In fact, we can further prove the following theorem which asserts that (K, ρH )and (K, ρh ), h ∈ SH0 , are Ahlfors regular with Hausdorﬀ dimension d.
Theorem 6.2. There exist c6.1 , c6.2 ∈ (0, ∞) such that for any h ∈ SH0 andany (r, x) ∈ (0, 1] × K,
(6.3)

c6.1 rd ≤ HρdH (Br (x, ρH )) ≤ c6.2 rd ,c6.1 rd ≤ Hρdh (Br (x, ρh )) ≤ c6.2 rd .

In particular, d = dimH (K, ρh ) for any h ∈ H0 \ R1.
The following propositions are the key steps for the proof of Theorem 6.2.
Proposition 6.3. Let h ∈ SH0 and let Λ be a partition of Σ. Then∑
(6.4)
3−29 ≤kh ◦ Fw kdE ≤ 329 .
w∈ΛProposition 6.4. There exist c6.3 , c6.4 ∈ (0, ∞) such that for any h ∈ H0 \ R1
and any w ∈ W∗ ,(6.5)

c6.3 kTw kd ≤ HρdH (Kw ) ≤ c6.4 kTw kd ,(6.6)

c6.3 kh ◦ Fw kdE ≤ Hρdh (Kw ) ≤ c6.4 kh ◦ Fw kdE .

ANALYSIS AND GEOMETRY OF MEASURABLE RIEMANNIAN STRUCTURE ON S.G. 21

Theorem 6.2 is now an easy consequence of Proposition 6.4 together with (4.8),
Lemma 4.7 and Lemma 4.22-(1). Proposition 6.3 follows by applying (6.2) toh◦Fw /kh◦Fw kEh◦Fw /kh◦Fw kE
Λs/kh◦F
with s := minv∈Λ kh ◦ Fv kE for w ∈ Λ and using #Λs/kh◦F
=
w kE
w kE
h
#{v ∈ Λs | v ≤ w} to sum up the resulting inequalities, which is possible since theconstants in (6.2) is independent of h. Then we can also verify Proposition 6.4 on
the basis of Proposition 6.3, Lemma 4.9 and the following lemma, by consideringh ◦ Fw /kh ◦ Fw kE instead of h to localize the argument to Kw (or alternatively, byusing Lemma 7.6 below).
Lemma 6.5 (cf. [48, Lemma 7.8]). Let S = {Λs }s∈(0,1] be a scale on Σ withgauge function l and let ρ be a metric on K adapted to S with β1 , β2 ∈ (0, ∞)α
as in (4.6). Let α, δ, ε ∈ (0, ∞) and let Hρ,δbe the α-dimensional pre-Hausdorﬀ
measure on K with respect to ρ as defined in [24, (2.1)] and [57, Definition 1.5.1].
αIf δ ∈ (0, β1 ) and Hρ,δ(K) < ε, then there exists a partition Λ of Σ such that
∑
−α
α
l(w)
<
4β
ε
and
maxw∈Λ l(w) ≤ β1−1 δ.
1w∈ΛFor h, h⊥ ∈ SH0 with E(h, h⊥ ) = 0, a monotone class argument using Propoddd1−d/2 −1
c6.3 c6.4 HρdH .
sition 6.4 easily shows that 2−d/2 c6.3 c−1
6.4 HρH ≤ Hρh + Hρh⊥ ≤ 2
Furthermore we can also prove the following absolute continuity similar to Theorem
3.6 by using Proposition 6.4 to follow closely [38, Proof of Theorem 5.6].
Theorem 6.6. Let h ∈ H0 \ R1. Then HρdH and Hρdh are mutually absolutely
continuous.
Recall that dloc ≤ d by Theorem 6.1. In fact, here we have the strict inequality,
which also implies the singularity of HρdH and Hρdh , h ∈ H0 \ R1, with respect to µ.
Theorem 6.7. dloc < d.
Corollary 6.8. HρdH is singular to µ, and so is Hρdh for any h ∈ H0 \ R1.
Proof. Let h ∈ H0 \R1 and let (ν, ρ) denote any one of (µ, ρH ) and (µhhi , ρh ).
Define Kdν,ρ
loc ∈ B(K) by{}
log ν(Br (x, ρ))
ν,ρ
loc
(6.7)
Kdloc := x ∈ K lim
=d
.
r↓0
log r
)
)
(
(
= dloc= 0 by Theorems 5.12 and 3.6, whereas dimH Kdν,ρ
Then µ K \ Kdν,ρ
loc , ρlocby [24, Proposition 4.9],
used to verify [24, Covering lemma 4.8] for) (4.16) is( where= 0 by dloc < d. Thus Hρd is singular to µ.

(K, ρ), and hence Hρd Kdν,ρ
loc
In the rest of this section, we briefly explain the idea of the proof of Theorem
6.7. A similar idea was also used in [36, Theorem 4.1 and Proof of Theorem 2.1]
to establish singularity of energy measures on self-similar sets. For m ∈ N, we set∑ ( 5 )m1
kTw k2(6.8)
ηm :=kTw k2 log d,2(2 − d)3HρH (Kw )w∈Wm

1
so that limm→∞ m
ηm = η by Theorem 5.12 and Proposition 6.4. Then for m, n ∈ N,

η(m+1)n =(6.9)( 53 )(m+1)n2(2 − d)( 5 )(m+1)n≥ 3
2(2 − d)

∑
w∈Wmn

∑
w∈Wmn

HρdH (Kw )∑ Hρd (Kwv )kTwv k2
kTwv k2log
HρdH (Kw ) HρdH (Kwv )HρdH (Kwv )Hv∈Wn( 3 )n kTw k2
n log 35
( 3 )n kTw k2
log 5 d= ηmn +,
HρdH (Kw ) 5 d
HρH (Kw )HρH (Kw )2−d

22

NAOTAKA KAJINO

where the inequality is due to the convexity of the function (0, ∞) 3 t 7→ t log t and∑2n2v∈Wn kTwv k = (3/5) kTw k . (6.9) in particular yields()
log 35log 35log 35ηm
1(6.10)
=
η
=
lim
≥
lim
η
+
(m
−
1)
=
1
m→∞ m
m→∞ m
2 − dloc2−d2−dand hence dloc ≤ d, which was proved in [48, Proof of Theorem 7.2] by a diﬀerent
method.
2−d2−dNote that c−1≤ kTw k2 /HρdH (Kw ) ≤ c−1
for any w ∈ W∗
6.4 kTw k
6.3 kTw k
by Proposition 6.4 and that the image Z(KZ ) of the map Z|KZ : KZ → R2×2 isequal to the set of all orthogonal projections on R2 of rank 1 by Theorem 4.12
and the rotational symmetry of KH . By using these facts and the definition of
3.7,(Z in Proposition) we can verify that if we fix a suﬃciently large n ∈ N, thenkTwv k2 /HρdH (Kwv ) v∈W ∈ RWn appearing in the first line of (6.9) is some uniformndistance away from constant vectors for m large enough and for suﬃciently manyw ∈ Wmn . Then the second line of (6.9) has to be smaller than the first by a uniform
constant for m large enough, which together with the same limiting procedure asin (6.10) yields dloc < d. See [51] for the complete proof and further details.
7. Weyl’s Laplacian eigenvalue asymptotics
As already mentioned in the last section, our main concern in this section is
Weyl’s Laplacian eigenvalue asymptotics for the Dirichlet space (K, µ, E, F), whichis the main result of the author’s forthcoming paper [51].
Let us start with the following basic definition. See Lemma 5.7 above for thedefinitions of ν|U , (E U , FU ), {Ttν,U }t∈(0,∞) and pU
ν.
Definition 7.1. Let ν be a finite Borel measure on K with full support and let
U be a non-empty open subset of K. Noting that the non-positive self-adjoint operator ∆ν,U of (U, ν|U , E U , FU ) (the generator of {Ttν,U }t∈(0,∞) ) has discrete spectrum
and that tr Ttν,U < ∞ for t ∈ (0, ∞) by [21, Theorem 2.1.4], let {λν,Un }n∈N be thenon-decreasing enumeration of all the eigenvalues of −∆ν,U , where each eigenvalue
is repeated according to its multiplicity. The eigenvalue counting function Nν,Uand the partition function Zν,U of the Dirichlet space (U, ν|U , E U , FU ) are defined
respectively by, for λ ∈ R and t ∈ (0, ∞),Nν,U (λ) := #{n ∈ N | λν,U
≤ λ},
n
∫
∫
∑
ν,U
−λν,U
t
−λt
n
(7.2) Zν,U (t) := tr Tt =e=e dNν,U (λ) =pU
ν (t, x, x)dν(x).
(7.1)

n∈N

R

K

In the situation of Definition 7.1, Nν,U (λ) < ∞ for λ ∈ R by limn→∞ λν,U
= ∞,
n
ν,U
Nν,U (λ) = 0 for λ ∈ (−∞, 0) by λ1 ≥ 0, and Zν,U is (0, ∞)-valued and continuous.
The main result of [51] is stated as follows.
Theorem 7.2. There exist cN , cZ ∈ (0, ∞) such that for any non-empty open
subset U of K with HρdH (∂K U ) = 0,
(7.3)

lim

λ→∞

Nµ,U (λ)= cN HρdH (U )
λd/2

and

lim td/2 Zµ,U (t) = cZ HρdH (U ).
t↓0ANALYSIS AND GEOMETRY OF MEASURABLE RIEMANNIAN STRUCTURE ON S.G. 23

Recall that for the eigenvalue counting function NU and the partition function
ZU associated with the Dirichlet Laplacian on a non-empty open subset U of Rk
with volk (U ) < ∞, where volk denotes the Lebesgue measure on Rk , it holds that
NU (λ)= (2π)−k vk volk (U )λ→∞ λk/2(7.4) limand

lim tk/2 ZU (t) = (4π)−k/2 volk (U )t↓0with vk := volk ({x ∈ R | |x| < 1}). See e.g. [17, 18, 70, 71, 72, 78] and references
therein for known results concerning Weyl’s asymptotics on Euclidean domains andRiemannian manifolds. According to Theorem 7.2, the Dirichlet Laplacian ∆µ,U
on a non-empty open subset U of K satisfies Weyl’s eigenvalue asymptotics similarto (7.4) unless ∂K U is “too rough”, but the limit is given by a constant multipleof the Hausdorﬀ measure HρdH , which is, unlike (7.4), singular to the “Riemannianvolume measure” µ by Corollary 6.8.
The rest of this section is devoted to a sketch of the proof of Theorem 7.2.
The main idea of the proof is to follow the method due to Kigami and Lapidus[61] of obtaining a renewal equation for the eigenvalue counting function (or thepartition function) from the self-similarity of (E, F) to apply a suitable renewal
theorem. The problem in doing so for the present setting of (K, µ, E, F) is that thereference measure µ is not self-similar, but it can be resolved by incorporating theinformation on the( pair {h1 ◦ Fw , h2 ◦) Fw } of V0 -harmonic functions appearing inµ ◦ Fw = (5/3)|w| µhh1 ◦Fw i + µhh2 ◦Fw i as the second variable, in the following way.
( )Definition 7.3. Let M = ac db ∈ R2×2
. We define ΦM := M Φ : K → R2 ,
0
k

(7.5)

µM := µhah1 +bh2 i + µhch1 +dh2 i ,(7.6) ρM (x, y) := inf{`M (γ) | γ : [0, 1] → K, γ is continuous, γ(0) = x, γ(1) = y}for x, y ∈ K, where `M (γ) := `R2 (ΦM ◦γ) for each continuous map γ : [aγ , bγ ] → K,
aγ , bγ ∈ R, aγ ≤ bγ . Note that ΦM (x) = ((ah1 + bh2 )(x), (ch1 + dh2 )(x)) for x ∈ K.
( )Namely, we identify M = ac db ∈ R2×2
with the pair {ah1 + bh2 , ch1 + dh2 }0
of V0 -harmonic functions and define the counterpart of Φ, µ, ρH for M by replacing{h1 , h2 } with {ah1 +bh2 , ch1 +dh2 }. Then for M ∈ R2×2
with det M 6= 0, Theorems
0
3.8 and 3.10-(1) are valid with ΦM , µM , ZM := kM Zk−2 M ZM ∗( in placeof Φ, µ, Z)
(Theorem 3.10-(1) remains valid for det M = 0 if we set ZM := 10 00 for M Z = 0);note that ZM (x) is the orthogonal projection onto M (Im Zx ) for each x ∈ K.
We also define the corresponding (scale) on Σ as follows. Note that kM k2 =kah1 + bh2 k2E + kch1 + dh2 k2E for M = ac db ∈ R2×2.
0
Definition 7.4. (1) We set S2×2 := {M ∈ R2×2 | kM k = 1}.
(2) Let M ∈ S2×2 . We define SM = {ΛMscale on Σ induced bys }s∈(0,1] to be the√the gauge function lM : W∗ → (0, 1], lM (w) := kM Tw k = (3/5)|w| µM (Kw ).
Then for M ∈ S2×2 , by exactly the same proofs we still have Theorem 4.15
and Lemma 4.17 with M in place of h, and Lemma 4.22 and Proposition 4.23 withhhµM , lM , SM = {ΛMs }s∈(0,1] in place of µhhi , lh , S = {Λs }s∈(0,1] and cG , cv the same.
Consequently Theorem 4.25, Proposition 4.26, Theorem 5.2, Corollaries 5.3, 5.4, 5.5
and Proposition 5.6 hold for (ν, ρ) = (µM , ρM ), M ∈ R2×2
, with cV , cP , cL , cU , cU (n)0unchanged. Theorem 4.19 and Lemma 4.20 with ρM , `M in place of ρH , `H are also
valid for any M ∈ R2×2
with det M 6= 0 by exactly the same proofs.
024

NAOTAKA KAJINO

Remark 7.5. Here we do not exclude the case of det M = 0 where µM , ρM , `M
are equal to µhhi , ρh , `h for some h ∈ H0 \ R1 with khkE = kM k and lM , SM are
equal to lh , Sh when M ∈ S2×2 . This consideration is absolutely necessary for the
proof of Theorem 7.2, whose reason will be described at the very end of this section.
Noting that #∂K Kw < ∞ for w ∈ W∗ by ∂K Kw ⊂ Fw (V0 ), we can easily verify
the following lemma from Proposition 3.2 and the definitions of ρM and HρdM .
Lemma 7.6. HρdM ◦ Fw = HρdM Tw for any M ∈ R2×2
and any w ∈ W∗ .
0
The following proposition is a crucial step for the proof of Theorem 7.2.
Proposition 7.7. (1) There exist c7.1 , c7.2 ∈ (0, ∞) such that
(7.7)

c7.1 kM kd ≤ HρdM (K) ≤ c7.2 kM kd ,M ∈ R2×2.
0→ (0, ∞) by T(M ) := HρdM (K)1/d . Then T is continuous, and(2) Define T : R2×2
0
2×2
for any M ∈ R0 , any U ∈ O(2) and any a ∈ R \ {0},∑(7.8) T(M )d =T(M Ti )d ,T(U M ) = T(M )and
T(aM ) = |a|T(M ).
i∈SRemark 7.8. (1) Let M ∈ R2×2
, det M 6= 0 and cM := inf T ∈R2×2 kM T k/kT k,0
0
so that cM > 0. Then a monotone class argument using (6.5), Lemma 7.6 and (7.7)
−1
d
d
d
d d
d
easily shows that c−1
6.4 c7.1 cM HρH ≤ HρM ≤ c6.3 c7.2 kM k HρH . In particular, HρM
is singular to µ by Corollary 6.8.
d-Hölder continuous on(2) In fact, for each r, s ∈ (0, ∞) with r ≤ s, T is d+1
2×2{M ∈ R0 | r ≤ kM k ≤ s} with respect to the norm metric.
(3) The properties stated in Proposition 7.7-(2) characterizes T uniquely up to
constant multiples: if T0 : R2×2
→ [0, ∞) is continuous and satisfies (7.8) for any0
M ∈ R02×2 , any U ∈ O(2) and any a ∈ R \ {0}, then T0 = c0 T for some c0 ∈ [0, ∞).
Proposition 7.7-(1) follows from Proposition 6.3, Lemma 6.5 and (4.8) with M
in place of h, and (7.8) is immediate from Lemma 7.6 and the definition of T. On
the other hand, the proof of the continuity of T at M ∈ R2×2
satisfying det M = 00is quite involved and relies heavily on Theorem 4.15-(2).
By virtue of Proposition 7.7, now we can associate to the function (t, M ) 7→
ZµM ,K\V0 (t) a certain renewal equation to which Kesten’s renewal theorem [55,
Theorem 2] for functionals of Markov chains is applicable, as follows. Define(7.9) ST := {M ∈ R2×2
| T(M ) = 1} and M T := T(M )−1 M ∈ ST , M ∈ R2×2,
00
so that ST is compact by Proposition 7.7, and further define N : ST × R → N ∪ {0}and Z : ST × (0, ∞) → (0, ∞) by(7.10)

N(M, λ) := NµM ,K\V0 (λ)
−d/2

and

Z(M, t) := ZµM ,K\V0 (t).
µ,K\V2/d µ,K\V00
Let M ∈ ST . Then µM ≤ c7.1 µ by (7.7) and(hence λ1 M≥ c)7.1 λ1> 0.
T2Since NµM ,Ki \Fi (V0 ) (λ) = NµM Ti ,K\V0 (λ) = N (M Ti ) , T(M Ti ) λ for λ ∈ R andi ∈ S by (2.4) and µM ◦ Fi = (5/3)µM Ti , [57, Corollary 4.1.8] yields∑ ()
0 ≤ N(M, λ) −N (M Ti )T , T(M Ti )2 λ = NµM ,K\V0 (λ) − NµM ,K\V1 (λ) ≤ 3i∈SANALYSIS AND GEOMETRY OF MEASURABLE RIEMANNIAN STRUCTURE ON S.G. 25

for λ ∈ R, and hence for any t ∈ (0, ∞),∑ ()(7.11)
0 ≤ Z(M, t) −Z (M Ti )T , T(M Ti )−2 t ≤ 3.
i∈S

Now we define f, g : ST × R → R by f (M, s) := e−ds Z(M, e−2s ) and
)
(
∑ (
)
−ds
−2s
T
−2 −2s(7.12)g(M, s) := eZ(M, e ) −Z (M Ti ) , T(M Ti ) e,i∈Sso that for any (M, s) ∈ ST × R, 0 ≤ g(M, s) ≤ 3e−ds and∑()(7.13)f (M, s) = g(M, s) +T(M Ti )d f (M Ti )T , s + log T(M Ti ) .
i∈S∑Since i∈S T(M Ti )d = T(M )d = 1 by (7.8), we observe that the sum in (7.13)
involves a conservative Markovian kernel (see Definition 8.2 below) P on (ST , B(ST ))given by∑
(7.14)
P(M, ·) :=T(M Ti )d δ(M Ti )T ,M ∈ ST ,i∈S

where δM (A) := 1A (M ), M ∈ ST , A ⊂ ST . Then by considering a Markov chain
X = (Ω, M, {Xn }n∈N∪{0} , {PM }M ∈ST ) on ST such that PM ◦ X1−1 = P(M, ·) forM ∈ ST and another sequence {un }n∈N of real random variables on (Ω, M) suchthat un = − log T(Xn Ti ) on {Xn+1 = (Xn Ti )T }, from (7.13) we obtain[n−1]∑
g(Xk , s − Vk ) + EM [f (Xn , s − Vn )](7.15)f (M, s) = EM
k=0

∑n−1
for any (M, s) ∈ ST × R and any n ∈ N, where Vn := k=0 uk . It is not diﬃcult
to see that 0 ≤ f (Xn , s − Vn ) ≤ c7.3 eds (3/5)dn for n large enough (depending onlyon s) for some c7.3 ∈ (0, ∞), and hence letting n → ∞ in (7.15) results in[∞]∑(7.16)f (M, s) = EMg(Xn , s − Vn ) ,(M, s) ∈ ST × R.
n=0

Kesten’s renewal theorem [55, Theorem 2] asserts the convergence as s → ∞ of afunction of the form (7.16) to a finite limit which is independent of M . Therefore
once the assumptions of [55, Theorem 2] are verified, we can conclude (7.3) for
U = Kw \ Fw (V0 ), w ∈ W∗ , where the strict positivity of the limit is guaranteed byan a priori bound [48, Theorem 7.2] for Zµ,K , and then the extension to general
U with HρdH (∂K U ) = 0 is straightforward. The assumptions of [55, Theorem 2]
consist of conditions [55, I.1–I.4] on the random variables {(Xn , Vn )}n∈N∪{0} , thedirect Riemann integrability [55, Definition 1] of the function g : ST × R → R andthe continuity of g. The direct Riemann integrability of g follows from the estimates
2/d µ,K\V0µ ,K\V0≥ c7.1 λ1> 0 and 0 ≤ g(M, s) ≤ 3e−ds for (M, s) ∈ ST × R, and for
λ1 M
the sake of the continuity of g we need to consider the partition functions ZµM ,K\V0
and not the eigenvalue counting functions NµM ,K\V0 .
To verify part of the conditions [55, I.1 and I.2], we need the existence and theuniqueness of an invariant probability measure for P. The existence follows by theclassical theorem of Krylov and Bogolioubov [63] (see [32, Theorem 1.10]) since ST
is compact and P is a Feller Markovian kernel, i.e. Pu ∈ C(ST ) for any u ∈ C(ST ),by the continuity of T. The uniqueness is implied by a recent powerful result [99,26

NAOTAKA KAJINO

Theorem 6.4] of Worm and Hille (see also [98, Theorem 7.4.6]). (To be precise, theuniqueness is assured only after taking the quotient ŠT := O(2)\ST of ST by the
canonical left action O(2) × ST 3 (U, M ) 7→ U M ∈ ST of O(2) on ST . Accordingly
ST in the above argument has to be replaced by ŠT .) Note that the state space STof the Markov chain X would not be compact if S0T := {M ∈ ST | det M = 0} were
removed. Moreover, since S0T is closed in ST and P-invariant (i.e. P(M, S0T ) = 1 forany M ∈ S0T ), the invariant measure for P has to be supported on S0T (to be precise,
P needs to be regarded as a Markovian kernel on (ŠT , B(ŠT ))). For these reasons
we cannot exclude S0T from the state space of the Markov chain X.
8. Connections to general theories on metric measure spaces
In this section, we briefly mention some connections to general theories ofanalysis and geometry on metric measure spaces. In Subsection 8.1, we state somerecent results of Koskela and Zhou [62, Section 4] on connections to the theoryof diﬀerential calculus on metric measure spaces initiated by Cheeger [19] anddeveloped further by many people, e.g. Shanmugalingam [86] and Keith [52, 53,54]. Subsection 8.2 concerns the theory of Ricci curvature lower bound for metricmeasure spaces established by Lott and Villani [75, 74] and Sturm [91, 92] andpartially also by Ohta [80]. In fact, very recently there have been attempts tounify methods and ideas developed in those fields to establish diﬀerential calculus
on an even wider range of metric measure spaces, e.g. Ambrosio, Gigli and Savaré[2, 3, 4], Gigli [29] and Koskela and Zhou [62]. There are also a huge number ofother related results and it is beyond the author’s ability to review even just thecentral achievements of these fields. For further details, we refer the reader to theabove-mentioned works, monographs [34, 1, 96, 97] and references therein.
We need the following definitions for the discussions below.
Definition 8.1. Let (E, ρ) be a metric space and let γ : [a, b] → E, a, b ∈ R,a ≤ b, be continuous.
(1) The ρ-length `ρ (γ) of γ is defined by(8.1)∑n`ρ (γ) := sup{ k=1 ρ(γ(tk−1 ), γ(tk )) | n ∈ N, {tk }nk=0 ⊂ [a, b] is non-decreasing}.
(2) Suppose `ρ (γ) < ∞ and define ϕγ : [a, b] → [0, `ρ (γ)] by ϕγ (t) := `ρ (γ|[a,t] ).
Since γ(s) = γ(t) for any s, t ∈ [a, b] with ϕγ (s) = ϕγ (t), there exists a unique
map γρ : [0, `ρ (γ)] → E such that γ = γρ ◦ ϕγ , and then γρ is continuous and`ρ (γρ |[0,t] ) = t for any t ∈ [0, `ρ (γ)]. γρ is called the ρ-length parametrization of γ.
(3) Suppose a < b. Then γ is called a minimal ρ-geodesic if and only if `ρ (γ|[a,t] ) =
|t−s|
t−a
b−a ρ(γ(a), γ(b)) for any t ∈ [a, b], or equivalently ρ(γ(s), γ(t)) = b−a ρ(γ(a), γ(b))
for any s, t ∈ [a, b]. Let G(E, ρ) denote the set of minimal ρ-geodesics with domain[0, 1], which is equipped with the metric ρG(E,ρ) (γ1 , γ2 ) := supt∈[0,1] ρ(γ1 (t), γ2 (t)).
Also recall the following standard definition.
Definition 8.2. Let (E, B) be a measurable space. Then P : E × B → [0, 1] iscalled a Markovian kernel on (E, B) if and only if P(x, ·) : B → [0, 1] is a measureon (E, B) for any x ∈ E and P(·, A) : E → [0, 1] is B-measurable for any A ∈ B.
Such P is called conservative if and only if P(x, E) = 1 for any x ∈ E.

ANALYSIS AND GEOMETRY OF MEASURABLE RIEMANNIAN STRUCTURE ON S.G. 27

8.1. Identification of Dirichlet form as Cheeger energy. In [19] Cheeger
established a theory of diﬀerential calculus on a general metric measure space which
admits the volume doubling property and the weak (1, p)-Poincaré inequality (interms of upper gradients) for some p ∈ [1, ∞). Since (K, ρH , µ) and (K, ρh , µhhi ),h ∈ H0 \ R1, satisfy the volume doubling property (Theorem 4.25) and the weak
Poincaré inequality in terms of (the densities of) the E-energy measures (Proposition
4.26), it is natural to expect that Cheeger’s results in [19] are applicable to them.
Koskela and Zhou [62, Section 4] have recently proved that this is indeed the case,that Cheeger’s Rademacher theorem [19, Theorem 4.38] takes an explicit form
using Φ and h as the coordinate functions, and that the associated Cheeger 2energies coincide with the Dirichlet form (E, F). To be precise, they have proved
e and du of u ∈ F .
the following results. Recall Theorem 3.10 for the derivatives ∇udhTheorem 8.3 ([62, Theorems 4.2 and 4.3]). Let h ∈ H0 \ R1 and u ∈ F. Thenfor µ-a.e. x ∈ K, or equivalently for µhhi -a.e. x ∈ K (recall Theorem 3.6),(8.2)eu(y) − u(x) = h∇u(x),Φ(y) − Φ(x)i + o(ρH (x, y))(8.3)u(y) − u(x) =du(x)(h(y) − h(x)) + o(ρh (x, y))dh

as y → x,
as y → x.

Definition 8.4. Let (E, ρ) be a metric space. For f : E → R, we define(8.4)

Lipρ f (x) := lim

supr↓0 y∈Br (x,ρ)\{x}|f (y) − f (x)|,
ρ(x, y)x ∈ E.

Note that Lipρ f : E → [0, ∞] is Borel measurable if f ∈ C(E).
Theorem 8.5 ([62, Theorems 4.1 and 4.3]). Let h ∈ H0 \ R1 and u ∈ F. Then
(8.5)

e = Lipρ u|∇u|Hµ-a.e.

du= Lipρh udhandµhhi -a.e.

As a consequence, we obtain the identification of (E, F) as the Cheeger 2-energyand (1, 2)-Sobolev space, for which let us recall Cheeger’s definitions in [19].
Definition 8.6 ([35]). Let (E, ρ) be a metric space and let f : E → R. A
Borel measurable function g : X → [0, ∞] is called an upper ρ-gradient for f if andonly if for any continuous map γ : [0, 1] → E with `ρ (γ) < ∞,∫ `ρ (γ)(8.6)|f (γ(1)) − f (γ(0))| ≤g(γρ (s))ds.
0

Definition 8.7 ([19, Section 2]). Let (E, ρ) be a metric space and let m bea Borel measure on E such that m(Br (x, ρ)) ∈ (0, ∞) for any (r, x) ∈ (0, ∞) × E.
For p ∈ [1, ∞), we define the Cheeger p-energy Chρ,m: Lp (E, m) → [0, ∞] by
p
(8.7)


∫
gn is an upper ρ-gradient for an m-version lim infg p dm of fn for each n ∈ N for∫ some {fn }n∈N ⊂Chρ,mp (f ) := inf
 n→∞ E n
Lp (E, m) with limn→∞ E |fn −f |p dm = 0 
and the Cheeger (1, p)-Sobolev space H 1,p (E, ρ, m) over (E, ρ, m) by(8.8)

H 1,p (E, ρ, m) := {f ∈ Lp (E, m) | Chρ,mp (f ) < ∞}.

28

NAOTAKA KAJINO

Remark 8.8. (1) Chρ,mwas originally termed the “upper gradient p-energy”
pin [19]. Here we have followed the notation and terminology adopted in [2, 3, 29].
(2) Shanmugalingam [86, Definition 2.5] proposed another way of defining a (1, p)Sobolev space under the same framework, independently of Cheeger’s work [19,Section 2]. Her Sobolev space is denoted as N 1,p (E, ρ, m) and called the Newtonian
space of index p ∈ [1, ∞), and she proved in [86, Theorem 4.10] that for p ∈ (1, ∞),H 1,p (E, ρ, m) = N 1,p (E, ρ, m) and they are equipped with exactly the same norm.
ρ ,µhhiCorollary 8.9. Let h ∈ H0 \ R1. Then E(u, u) = Chρ2H ,µ (u) = Ch2h
for any u ∈ F and F = H 1,2 (K, ρH , µ) = H 1,2 (K, ρh , µhhi ).

(u)Proof. On the basis of Theorem 4.25 and Proposition 4.26, the first assertion
follows from (8.5), [52, Theorem 2], [19, Theorem 6.1] and [62, Theorem 2.2-(i)],whereas [62, Theorem 2.2-(i)] and [86, Theorem 4.10] yield the latter.
Remark 8.10. Let M ∈ R2×2
and let ΦM , µM , ρM be as in Definition 7.3. Then0
Theorems 8.3, 8.5 and Corollary 8.9 with ΦM , µM , ρM in place of Φ, µ, ρH are still
e is given
valid with exactly the same proofs as those in [62, Section 4], where ∇uby the version of Theorem 3.10-(1)( for) ΦM , µM with Z and kT[ω]m k replaced by
ZM := kM Zk−2 M ZM ∗ (ZM := 10 00 when M Z = 0) and kM T[ω]m k, respectively.
8.2. Invalidity of Ricci curvature lower bound. In the last decade, Lottand Villani [75, 74] and Sturm [91, 92] formulated Ricci curvature lower boundfor general metric measure spaces and showed that for a complete Riemannian
manifold equipped with Riemannian distance and volume, their formulations areequivalent to the usual lower bound of its Ricci curvature tensor. Around the same
period, Ohta [80] and Sturm [92] proposed another formulation of Ricci curvature
lower bound which is in principle weaker but easier to handle. Their main idea
was to make use of notions from optimal transport theory, and they also derived
various analytic and geometric consequences of their formulations.
The purpose of this subsection is to show that (K, ρH , µ) does not satisfy anyof those conditions for Ricci curvature lower bound. We need to introduce several
notions from optimal transport theory to state the Ricci bound conditions precisely.
Throughout this subsection, we fix a complete separable∪ metric space (E, ρ) anda Borel measure m on E such that m(E) > 0 and E = U ⊂E open in E, m(U )<∞ U .
Definition 8.11 (Wasserstein space). Let P(E) denote the set of all Borel
probability measures on E. Let p ∈ [1, ∞) and define{}∫
Pp (E, ρ) := ν ∈ P(E) E ρ(x, y)p dν(y) < ∞ for some (any) x ∈ E ,
(8.9)
{∫
}
(8.10) Wpρ (ν1 , ν2 )p := inf E×E ρp d$ $ ∈ P(E × E), $ ◦ pr−1
j = νj , j = 1, 2
for ν1 , ν2 ∈ P(E), where prj : E × E → E is given by prj (x1 , x2 ) := xj . Then Wpρ isindeed a metric on Pp (E, ρ) by [96, Theorem 7.3-(i)] and called the p-Wasserstein
distance over (E, ρ). (Pp (E, ρ), Wpρ ) is called the p-Wasserstein space over (E, ρ),which is a complete separable metric space by [97, Theorem 6.18].
Definition 8.12 (Relative entropy). For ν ∈ P(E), we write ν  m if andonly if ν is absolutely continuous with respect to m. We define the relative entropy
Entm : P2 (E, ρ) → [−∞, ∞] with respect to m by{∫)
∫ ( dνdνdνdν +log dmdm if ν  m and E dm
log dmdm < ∞,
dm
E(8.11) Entm (ν) :=∞otherwiseANALYSIS AND GEOMETRY OF MEASURABLE RIEMANNIAN STRUCTURE ON S.G. 29

for ν ∈ P2 (E, ρ), and we set P2∗ (E, ρ, m) := {v ∈ P2 (E, ρ) | Entm (ν) < ∞}.
Definition 8.13 (Curvature-dimension condition, [91, Definition 4.5-(i)]). Let
k ∈ R. We say that (E, ρ, m) satisfies the curvature-dimension condition CD(k, ∞)or (E, ρ, m) has curvature ≥ k if and only if for any ν0 , ν1 ∈ P2∗ (E, ρ, m) there exists
a minimal W2ρ -geodesic α : [0, 1] → P2 (E, ρ) such that α(0) = ν0 , α(1) = ν1 andk(8.12) Entm (α(t)) ≤ (1 − t) Entm (α(0)) + t Entm (α(1)) − t(1 − t)W2ρ (α(0), α(1))2
2∗
for any t ∈ [0, 1]. Note that α(t) ∈ P2 (E, ρ, m) for any t ∈ [0, 1] for such α.
The curvature-dimension condition CD(k, ∞) is a generalization of the notionof Ricci curvature lower bound adapted to the setting of a general metric measurespace. Indeed, the following equivalence holds for complete Riemannian manifolds.
Theorem 8.14 ([85, Theorem 1.1], [91, Theorem 4.9]). Let (M, g) be a complete Riemannian manifold with Riemannian geodesic distance ρg and Riemannianvolume measure mg . Let k ∈ R. Then (M, ρg , mg ) satisfies CD(k, ∞) if and only if
Ricg ≥ kg, i.e. Ricg (v, v) ≥ kg(v, v) for any x ∈ M and any v ∈ Tx M , where Tx M
denotes the tangent space of M at x and Ricg the Ricci curvature tensor of (M, g).
Definition 8.15. For A, B ⊂ E and t ∈ [0, 1], we define}{ρ(x, z) = tρ(x, y) and ρ(z, y) = (1−t)ρ(x, y).
(8.13)[A, B]ρt := z ∈ E
for some (x, y) ∈ A × B∩
νNote that [A, B]ρt ∈ ν∈P(E) B(E) if A, B ∈ B(E) by [22, III.13, III.17 and III.33],
ν

where B(E) denotes the completion of B(E) by ν. We also set [x, y]ρt := [{x}, {y}]ρt
for x, y ∈ E and t ∈ [0, 1], so that [x, y]ρt is a closed subset of E.
Let A, B ∈ B(E) be such that m(A), m(B) ∈ (0, ∞) and set ν0 := m(A)−1 1A m
and ν1 := m(B)−1 1B m. By [91, Lemma 2.11-(ii)], if ν0 , ν1 ∈ P2 (E, ρ) and α :[0, 1] → P2 (E, ρ) is a minimal W2ρ -geodesic with α(0) = ν0 and α(1) = ν1 , thenα(t)([A, B]ρt ) = 1 for any t ∈ [0, 1]. Thus an application of Jensen’s inequality to theleft-hand side of (8.12), together with the lower semicontinuity of W2ρ under weak
convergence ([97, Remark 6.12]), yields the following generalized Brunn-Minkowski
inequality, which was new even for complete Riemannian manifolds.
Proposition 8.16 (Brunn-Minkowski inequality, cf. [92, Proposition 2.1]). Let
k ∈ R and suppose that (E, ρ, m) satisfies CD(k, ∞). Then for any A, B ∈ B(E)with m(A), m(B) ∈ (0, ∞) and any t ∈ [0, 1],(8.14)
(
)2
k
1A
1B
ρ
ρ
log m([A, B]t ) ≥ (1 − t) log m(A) + t log m(B) + t(1 − t)W2m,m .
2m(A) m(B)Remark 8.17. (1) Some time after [91], Sturm introduced in [92, Definition 1.3] a variant of the curvature-dimension condition for (E, ρ, m) denoted as
CD(k, N ), where k ∈ R and N ∈ [1, ∞). Roughly speaking, (E, ρ, m) satisfying
CD(k, N ) has Ricci curvature bounded from below by k and dimension at mostN . Indeed, according to [92, Theorem 1.7], for (M, g), ρg , mg as in Theorem 8.14,(M, ρg , mg ) satisfies CD(k, N ) if and only if Ricg ≥ kg and dim M ≤ N . As naturally expected from its meaning, if (E, ρ, m) satisfies CD(k, N ) then it also satisfies
CD(k 0 , N 0 ) for any k 0 ∈ (−∞, k] and any N 0 ∈ [N, ∞) by [92, Proposition 1.6-(i)].

30

NAOTAKA KAJINO

Moreover by [92, Proposition 1.6-(ii)], if k ∈ R, (E, ρ, m) satisfies CD(k, N ) forsome N ∈ [1, ∞) and m(E) < ∞, then (E, ρ, m) also satisfies CD(k, ∞).
(2) Independently of Sturm’s work [91, 92], Lott and Villani [74, Definition 4.7]
(see also [75, Definitions 5.12 and 5.13]) also defined the notion of (E, ρ, m) having
N -Ricci curvature bounded below by k for k ∈ R and N ∈ [1, ∞], which is essentiallythe same as the curvature-dimension condition CD(k, N ). As noted in [74, Remark
4.11], this notion gets weaker as K decreases and N increases. (In the main texts of[75, 74], the compactness of the metric space is assumed for the sake of simplicity,
but many results there can be extended to the non-compact case; see [75, Appendix
E] and [97, Chapters 29 and 30].) As remarked in [74, discussions before and after
Lemma 4.14], for k ∈ R, N ∈ [1, ∞] and (E, ρ, m) with E compact and m(E) < ∞,the condition of N -Ricci curvature bounded below by k implies CD(k, N ).
(3) An important feature of the notions of CD(k, N ) and of N -Ricci curvaturebounded below by k is that they are preserved under measured Gromov-Hausdorﬀ
limits; see [91, Subsection 4.5], [92, Section 3], [75, Subsections 4.2, 5.3 and E.5]
and [97, Chapter 29] for details. They also give rise to a lot of fundamental analytic
and geometric consequences; see e.g. [92, Section 2], [75, Section 6], [74, Section
5] and [97, Chapter 30].
There is yet another formulation of Ricci curvature lower bound for (E, ρ, m)
due to Ohta [80] and Sturm [92], which is given as follows.
Definition 8.18 (Measure contraction property, [92, Section 5], cf. [80]). Let
k ∈ R and N ∈ [1, ∞). (E, ρ, m) is said to satisfy the measure contraction property
MCP(k, N ) if and only if for each t ∈ (0, 1), there exists for m-a.e. x ∈ E a Markovian
kernel Pt,x on (E, B(E)) such that Pt,x (y, [x, y]ρt ) = 1 for m-a.e. y ∈ E and∫(t)(8.15)
ςk,N (ρ(x, y))Pt,x (y, ·)dm(y) ≤ m,E(t)where ςk,N : [0, ∞) → (0, ∞] is defined by(8.16)∞) )N −1( ( √

t sin (tθ√ k/(N − 1))sin θ k/(N − 1)(t)ςk,N (θ) :=
 tN


( √
))
(


sinh tθ −k/(N − 1) N −1



( √
)
t
sinh θ −k/(N − 1)

if kθ2 ≥ (N − 1)π 2 ,if 0 < kθ2 < (N − 1)π 2 ,if kθ2 = 0,
if kθ2 < 0(t)

for N > 1 and ςk,1 (θ) := ∞1(0,∞) (kθ2 ) + t1(−∞,0] (kθ2 ).
Remark 8.19. (1) Let t ∈ (0, 1), k, k 0 ∈ R, k ≥ k 0 and N, N 0 ∈ [1, ∞), N ≤ N 0 .
(t)(t)
Then it is easy to see that ςk,N ≥ ςk0 ,N 0 , so that MCP(k, N ) implies MCP(k 0 , N 0 ).
(2) Sturm’s original version [92, Definition 5.1] of MCP(k, N ) is slightly stronger
than as in Definition 8.18, requiring additionally the Borel measurability of Pt,x (y, ·)∫ (1−t)in (x, y) ∈ E × E and E ςk,N (ρ(x, y))Pt,x (y, ·)dm(x) ≤ m for m-a.e. y ∈ E.
(3) MCP(k, N ) as in Definition 8.18 is implied by Ohta’s definition [80, Definition
2.1] of MCP(k, N ) under his standing assumptions in [80] that m(Br (x, ρ)) ∈ (0, ∞)for any (r, x) ∈ (0, ∞) × E and that (E, ρ) is a length space, i.e. for any x, y ∈ E,
ρ(x, y) = inf{`ρ (γ) | γ : [0, 1] → E, γ is continuous, γ(0) = x, γ(1) = y}. Indeed,ANALYSIS AND GEOMETRY OF MEASURABLE RIEMANNIAN STRUCTURE ON S.G. 31

this( observation is easily) verified by using [23, Theorems 10.2.1 and 10.2.2] (andm E \ Bπ√(N −1)/k (x, ρ) = 0 for any x ∈ E when k(N − 1) > 0, which follows from[80, Theorem 4.3 and Lemma 4.4-(i)]).
In principle, MCP(k, N ) is weaker than CD(k, N ) but still means Ricci curvature
lower bound by k and dimension upper bound by N . Indeed, by [80, Corollary 2.7]
or [92, Remark 5.3], if (E, ρ, m) satisfies MCP(k, N ) then dimH (suppE [m], ρ) ≤ N ,where suppE [m] := {x ∈ E | m(U ) > 0 for any open subset U of E with x ∈ U }.
Furthermore CD(k, N ) implies MCP(k, N ) under mild conditions on (E, ρ, m), andfor a complete Riemannian manifold (M, g), MCP(k, N ) corresponds to Ricg ≥ kg
and dim M ≤ N in a weaker sense than CD(k, N ) does, as follows.
Theorem 8.20 ([92, Lemma 4.1 and Theorem 5.4]). Let k ∈ R, N ∈ [1, ∞)and suppose that (E, ρ, m) satisfies CD(k, N ).
(1) If (E, ρ) is non-branching, i.e. x1 = x2 for any z, x0 , x1 , x2 ∈ E with 2ρ(z, xi ) =ρ(x0 , x1 ) = ρ(x0 , x2 ), i ∈ {0, 1, 2}, then there exists a Borel measurable map γ :E × E → G(E, ρ) such that for m × m-a.e. (x, y) ∈ E × E, γ(x, y) is the uniqueminimal ρ-geodesic with γ(x, y)(0) = x and γ(x, y)(1) = y.
(2) If a map γ : E×E → G(E, ρ) as in (1) exists, then (E, ρ, m) satisfies MCP(k, N ).
Theorem 8.21 ([80, Theorem 3.2 and Corollary 3.3], [92, Corollary 5.5]). Let(M, g) be a complete Riemannian manifold with Riemannian geodesic distance ρgand Riemannian volume measure mg . (Note that (M, ρg ) is non-branching.) Letk ∈ R. Then (M, ρg , mg ) satisfies MCP(k, dim M ) if and only if Ricg ≥ kg.
In fact, Rajala [82] has recently proved the implication CD(k, N ) ⇒ MCP(k, N )without the additional geometric conditions on (E, ρ, m) assumed in Theorem 8.20.
Theorem 8.22 ([82, Theorem 1.4]). Let k ∈ R, N ∈ (1, ∞) and suppose that(E, ρ) is a locally compact length space. If (E, ρ, m) satisfies CD(k, N ), then it alsosatisfies MCP(k, N ).
Remark 8.23. To be precise, the formulation [82, Definition 2.1] of CD(k, N )in [82] is slightly stronger than that in [92], requiring the convexity of the entropy
functionals also for measures which are not absolutely continuous with respect tom. This formulation is in fact essentially the same as Sturm’s original one in [92];see [75, Proposition 3.21 and Lemma 3.24] for a detailed discussion in this regard
when E is a compact length space and N = ∞, and see [97, Corollary 29.23] forthe general case.
Remark 8.24. (1) In principle, MCP(k, N ) should hold for a wider range of
metric measure spaces than CD(k, N ). Indeed, for (M, g), ρg , mg as in Theorem
8.21, MCP(k, N ) with dim M < N does not imply CD(k, N ) nor Ricg ≥ kg in
general, as noted in [92, Remark 5.6]. Furthermore for the (2n + 1)-dimensional
Heisenberg group Hn (R) := R2n+1 equipped with the Carnot-Carathéodory metricand the Lebesgue measure, where n ∈ N, MCP(k, N ) holds if and only (k, N ) ∈(−∞, 0]×[2n+3, ∞) by [46, Theorem 2.3 and Remark 3.3], whereas CD(k, N ) does
not hold for any (k, N ) ∈ R × [1, ∞] by [46, Theorem 3.2 and Remark 3.3].
Another advantage of MCP(k, N ) is that it is easier to handle than CD(k, N ).
For example, MCP(k, N ) is preserved by the operations of taking products andcones of metric measure spaces by [81, Proposition 3.3 and Theorem 4.2], butsimilar properties for CD(k, N ) are not known except for products of non-branchingcompact CD(k, ∞)-spaces due to [91, Proposition 4.16].

32

NAOTAKA KAJINO(2) Similarly to CD(k, N ), MCP(k, N ) is preserved by measured Gromov-Hausdorﬀ
limits; see [80, Section 6] and [92, Theorem 5.9 and Corollary 5.10]. MCP(k, N )also admits almost the same analytic and geometric consequences as CD(k, N ); see[80, Sections 2, 4 and 5], [92, Sections 5 and 6] and [84, Section 3] for details.
Now we turn to the case of the Sierpiński gasket K equipped with the harmonic
geodesic metric ρH and the Kusuoka measure µ. We have the following result.
Theorem 8.25. Let k ∈ R and N ∈ [1, ∞). Then (K, ρH , µ) does not satisfy
any one of CD(k, ∞), CD(k, N ), MCP(k, N ).
The rest of this section is devoted to the proof of Theorem 8.25. We start withthe following easy lemma. Recall that we set xy := {(1 − t)x + ty | t ∈ [0, 1]} forx, y ∈ R2 (Definition 4.11-(1)).
∪
Lemma 8.26. Set 4∗ := w∈W∗ Fw (q1 q2 ∪ q2 q3 ∪ q3 q1 ). Then µ(4∗ ) = 0.
Proof. Let h ∈ H0 , w ∈ W∗ , i ∈ S and S \ {i} = {j, k}. By [45, Lemma 4.2],14µhhi (Kv ),v ∈ W∗(8.17)µhhi (Kvj ∪ Kvk ) = µhhi (Kv ) − µhhi (Kvi ) ≤15(see [42, Proposition 3.8] and [39, Proposition 5.27] for similar estimates in moregeneral settings), and then an inductive use of (8.17) easily shows that( 14 )m(∪)µhhi (Fw (qj qk )) = lim µhhi v∈{j,k}m Kwv ≤ limµhhi (Kw ) = 0.
m→∞ 15m→∞Thus µhhi (4∗ ) = 0 and hence µ(4∗ ) = µhh1 i (4∗ ) + µhh2 i (4∗ ) = 0.
The key to the proof of Theorem 8.25 is the following proposition, which is aneasy consequence of Theorem 4.19.
Proposition 8.27. Let A, B ⊂ K and t ∈ (0, 1). Then [A, B]ρt H ⊂ 4∗ ∪(A∩B).
Proof. Let z ∈ [A, B]ρt H . Take (x, y) ∈ A × B such that ρH (x, z) = tρH (x, y)and ρH (z, y) = (1 − t)ρH (x, y). If x = y, then ρH (x, z) = 0 and z = x = y ∈ A ∩ B.
Assume x 6= y, so that z 6∈ {x, y}. By Theorem 4.15-(2), there exist harmonic
geodesics γxz : [0, t] → K and γzy : [t, 1] → K such that γxz (0) = x, γxz (t) =z = γzy (t), γzy (1) = y, `H (γxz ) = ρH (x, z) = tρH (x, y) and `H (γzy ) = ρH (z, y) =(1 − t)ρH (x, y). Define γ : [0, 1] → K by γ|[0,t] := γxz and γ|[t,1] := γzy . Thenγ is continuous, γ(0) = x, γ(t) = z, γ(1) = y and `H (γ) = `H (γxz ) + `H (γzy ) =ρH (x, y). Now Theorem 4.19 implies that γ = γb ◦ ϕγ for some harmonic geodesicγb : [0, 1] → K and a non-decreasing continuous surjection ϕγ : [0, 1] → [0, 1], andthen z = γb(ϕγ (t)) ∈ 4∗ by z 6∈ {x, y} = {bγ (0), γb(1)} and the definition of γb beinga harmonic geodesic (Definition 4.13-(2)). Thus [A, B]ρt H ⊂ 4∗ ∪ (A ∩ B).

Proof of Theorem 8.25. Let A, B ∈ B(K) be such that µ(A)µ(B) > 0 andµ(A∩B) = 0. Let t ∈ (0, 1). Then µ([A, B]ρt H ) = 0 by Proposition 8.27 and Lemma
8.26, so that (8.14) does not hold and hence neither does CD(k, ∞) by Proposition
8.16. [92, Proposition 1.6-(ii)] (see Remark 8.17-(1) above) further implies that(K, ρH , µ) does not satisfy CD(k, N ), either.
For MCP(k, N ), let t ∈ (0, 1), x ∈ K and suppose that there exists a Markovian
kernel Pt,x on (K, B(K)) satisfying (8.15) and Pt,x (y, [x, y]ρt H ) = 1, µ-a.e. y ∈ K.
Then since [x, y]ρt H ⊂ 4∗ for any y ∈ K \ {x} by Proposition 8.27, Pt,x (y, 4∗ ) = 1∫ (t)for µ-a.e. y ∈ K and hence (8.15) yields µ(4∗ ) ≥ K ςk,N (ρH (x, y))dµ(y) > 0,
contradicting Lemma 8.26. Therefore (K, ρH , µ) does not satisfy MCP(k, N ).


ANALYSIS AND GEOMETRY OF MEASURABLE RIEMANNIAN STRUCTURE ON S.G. 33

Remark 8.28. Unfortunately, the above proof of Theorem 8.25 does not workfor (K, ρh , µhhi ), h ∈ H0 \ R1, since the assertion of Theorem 4.19 is not validfor `h and ρh by Remark 4.21. Theorem 8.25 is quite likely to be true also for(K, ρh , µhhi ), but the author has no idea at this moment how to manage this case.
On the other hand, for any M ∈ R2×2
with det M 6= 0 and with µM , ρM , `M as
0
in Definition 7.3, Theorem 4.19 with ρM , `M in place of ρH , `H is valid and hence
Theorem 8.25 and Proposition 8.27 hold also for (K, ρM , µM ).
9. Possible generalizations to other self-similar fractals
We conclude this paper with some remarks on possible generalizations to otherself-similar fractals. In this paper, we have restricted our attention to the particularcase of the 2-dimensional standard Sierpiński gasket. In fact, it is almost the only
self-similar fractal that possesses all the required properties for the framework andthe results of this paper. Extension to other self-similar fractals would be only
partially possible and involve essential diﬃculties which would vary depending oneach fractal. Below we illustrate the actual (complicated) situation by mentioning
each of the concrete examples in Figure 3 separately.
9.1. Sierpiński gaskets. Let k ∈ N, k ≥ 3. The k-dimensional standard
Sierpiński gasket is the direct k-dimensional analogue of the Sierpiński gasket. The
results in [67, 56, 38] except those in [56, Lemma 4.2 and Appendix] are statedand proved including this case and therefore Section 3 is immediately extended
to this case (with obvious changes). As shown in [58, Section 3] for the Kusuoka
measure µ, Lemma 4.22 and Proposition 4.23 can be verified similarly with suitablechanges of the constants. On the other hand, the proof of Lemma 4.17 for the
Sierpiński gasket heavily relied on the 2-dimensionality of the space, as is observed
from its dependence on Theorem 4.18, and extension to the k-dimensional case is
not straightforward at all. Since the proof of Theorem 4.15 was based on Lemma
4.17, it is still unclear how we can verify Theorem 4.15 for the k-dimensional case.
Recall that Lemma 4.17 has two assertions (4.9) and (4.10). It seems possible
to prove (4.9) by taking the projection of the harmonic k-dimensional Sierpiński
gasket onto a suitable 2-dimensional subspace and then applying Theorem 4.18. To
the contrary, the lower inequality of (4.10) is not valid for the k-dimensional case.
Indeed, if h is a V0 -harmonic function taking 1 at x ∈ V0 , −1 at y ∈ V0 and 0 on
V0 \ {x, y}, then h = 0 on the hyperplane containing V0 \ {x, y}, from which we
can easily show that the lower inequality of (4.10) does not hold. This degeneracy
causes a lot of troubles in the proofs of various geometric inequalities and therefore
extension to the k-dimensional case should require significant eﬀort, although mostof the results in this paper are quite likely to hold also for the k-dimensional case.
Another possible extension is the case of the 2-dimensional level-l Sierpiński
gasket with l ≥ 3 (see Figure 3 for a picture of the level-3 case; the Sierpiński gasket
is regarded as the level-2 case). For simplicity we consider here the level-3 caseonly. Then the measurable Riemannian structure can be introduced in exactly thesame manner, and by virtue of the 2-dimensionality and Theorem 4.18 we can prove
Lemma 4.17, thereby Theorem 4.15, and also Theorems 4.19 and 8.25. Interestingly,
however, it is also possible to show that Lemma 4.22-(2) does not hold, on the basisof the denseness of vertices from which the space spreads away in three directions.
Consequently Proposition 4.23 is not valid by [59, Theorems 1.3.5 and 1.4.3] andhence neither is the volume doubling property (Theorem 4.25). Therefore by [59,34

NAOTAKA KAJINO

t

t

t
t

Figure 3. Typical self-similar fractals. From the upper left, threedimensional standard (level-2) Sierpiński gasket, two-dimensional
level-3 Sierpiński gasket, pentagasket (5-polygasket), hexagasket(6-polygasket), nonagasket (9-polygasket), snowflake, the Vicsek
set and Sierpiński carpet. In each fractal except the last one, theset V0 of its boundary points is marked by solid circles.
()
Theorem 3.2.3], even the on-diagonal upper bound pµ (t, x, x) ≤ cU /µ B√t (x, ρH )of the heat kernel pµ is false in this case.
It is an interesting challenging problem to establish a sharp two-sided estimate
for the heat kernel pµ in this case, but the actual behavior of pµ is expected to bevery wild and this problem should be diﬃcult. On the other hand, it is still likelythat we can extend Theorem 5.2 and the results in Sections 6 and 7 to this case, but
the actual proofs will be much more involved. In the case of the (level-2) Sierpiński
gasket, Lemma 4.22-(2) is used especially in the proof of the continuity of thefunction T : R2×2
→ (0, ∞) defined in Proposition 7.7-(2), and the extension to the0level-3 case, where Lemma 4.22-(2) is invalid, will require a non-trivial improvementin the proof.
We remark that Hino [41, Theorem 2.3] has recently proved the equality ρ = ρνasserted in Theorem 5.2 for a class of post-critically finite self-similar fractals with
#V0 = 3 under the assumption that the harmonic structure is non-degenerate, i.e.
Ai ∈ R(#V0 )×(#V0 ) defined by (2.11) is invertible for any i ∈ S. This result inparticular applies to the 2-dimensional level-l Sierpiński gasket with 2 ≤ l ≤ 50;
see [41, Example 2.4] for details.
Lack of the volume doubling property also aﬀects the validity of the assertionsin Subsection 8.1. In fact, Bate [12, Theorem 10.4] and Gong [31, Theorem 1.6
and Proposition 6.1] have recently given a simple equivalent condition for a metricmeasure space (E, ρ, m) to admit a measurable diﬀerentiable structure in the senseof Cheeger’s Rademacher theorem [19, Theorem 4.38] for Lipschitz functions. Theirequivalent condition contains the pointwise doubling property(9.1)

lim supr↓0m(B2r (x, ρ))<∞m(Br (x, ρ))for m-a.e. x ∈ E,

ANALYSIS AND GEOMETRY OF MEASURABLE RIEMANNIAN STRUCTURE ON S.G. 35

and in particular (9.1) is necessary for the validity of Cheeger’s Rademacher theorem[19, Theorem 4.38], which was proved first by Bate and Speight [13, Corollary 2.6].
In the case of the measurable Riemannian structure on the 2-dimensional level-3
Sierpiński gasket, it is possible to prove even that the lim sup in (9.1) is equal to ∞µ-a.e., which implies that Theorem 8.3 is not valid, whereas interestingly Theorem
3.10 still holds by [38, Theorem 5.4]. In view of these facts it seems interesting toe of u ∈ F given by Theorem 3.10
ask in this case how the “gradient vector field” ∇uis related to upper gradients for u and how the canonical Dirichlet form (E, F) isrelated to the Cheeger (1, 2)-Sobolev space and the Cheeger 2-energy.
9.2. Other nested fractals and Sierpiński carpets. For most other nested
fractals such as the N -polygasket with N ≥ 5, N/4 6∈ N (see [49, Subsection 5.2]
for its precise definition), the snowflake and the Vicsek set shown in Figure 3, thesituation is much worse than in the case of Sierpiński gaskets. The problem is thatthe harmonic structure is degenerate, i.e. non-constant V0 -harmonic functions canbe constant on some Kw , or in other words, the family {Ai }i∈S ⊂ R(#V0 )×(#V0 ) of
matrices defined by (2.11) contains non-invertible ones. In such cases it is highly∑#V −1
0 −1
non-trivial whether the Kusuoka measure µ := i=10 µhhi i , where {hi }#V⊂
i=1
H0 is arbitrarily chosen to be orthonormal in (H0 /R1, E), is energy-dominant in
the sense that µhui is absolutely continuous with respect to µ for any u ∈ F. The
method of introducing a measurable Riemannian structure on the basis of µ makes
sense only if µ is energy-dominant, which may or may not be the case dependingon each self-similar fractal.
For example, in the case of the Vicsek set, any V0 -harmonic function is constanton each connected component of the complement of the two diagonals, so that the
Kusuoka measure µ is supported only on the union of the diagonals, which is muchsmaller than the whole Vicsek set. For the N -polygasket with N 6= 6, 9 and thesnowflake, it is still not known whether the Kusuoka measure µ is energy-dominant.
On the other hand, in the case of the hexagasket (6-polygasket) and the nonagasket (9-polygasket), by virtue of their dihedral symmetry we can prove that the
Kusuoka measure µ is energy-dominant, which was essentially stated and provedin [93, Section 7]. Therefore µ gives rise to a measurable Riemannian structure.
We can extend Lemma 4.17 and thereby Theorem 4.15 to this case with a suitable
notion of harmonic geodesics, whereas the degeneracy of the harmonic structure
easily implies that Lemma 4.22-(2) and the lower inequalities in Lemma 4.22-(1)are not true. Consequently Proposition 4.23 and Theorem 4.25 do not( hold and)hence neither does the on-diagonal upper bound pµ (t, x, x) ≤ cU /µ B√t (x, ρH )of the heat kernel pµ by [59, Theorem 3.2.3]. (To be precise, [59, Theorem 3.2.3]
is not directly applicable to this case due to the lack of the lower inequalities inLemma 4.22-(1), but we can still verify [59, Lemmas 3.5.4 and 3.5.5] by using certain specific properties of the hexagasket and the nonagasket.) Similarly to thecase of the 2-dimensional level-3 Sierpiński gasket, it is likely that we can extend
Theorem 5.2 and the results in Sections 6 and 7. In fact, since the linear map
Fw∗ : H0 /R1 → H0 /R1, Fw∗ h := h ◦ Fw , has rank one with the same image (sameup to symmetry of the space) for “most” w ∈ W∗ , just the usual renewal theorem[25, Section XI.1, Renewal theorem (Alternative form)] should suﬃce for the proofof Theorem 7.2 and therefore Proposition 7.7-(2) should not be required. For extension of the results in Subsection 8.1, the last paragraph of the previous subsection
verbatim applies to this case as well.

36

NAOTAKA KAJINO

Finally, for the Sierpiński carpet, and its generalizations called generalized
Sierpiński carpets, nothing is known about non-degeneracy of V0 -harmonic functions and possibility of introducing a measurable Riemannian structure by using
the energy measures of V0 -harmonic functions. Note that the set V0 of boundary
points is an infinite set in this case; for example, V0 = ∂R2 ([0, 1]2 ) for the Sierpiński
carpet. On one hand, this property gives us plenty of choices of V0 -harmonic functions and it is very likely that some choice of V0 -harmonic functions should workfor the purpose of introducing a measurable Riemannian structure. On the other
hand, the infiniteness of V0 makes any kind of explicit calculations for the canonical
Dirichlet form (E, F) impossible, so that non-degeneracy of V0 -harmonic functions
is very diﬃcult to verify despite plentifulness of V0 -harmonic functions.
In any of the above cases, we could use instead of V0 -harmonic functions a∑Ngeneral family g = {gn }N
n=1 ⊂ F of functions such that µhgi :=n=1 µhgn i isenergy dominant, as is done in [38, 40], but then it would become much morediﬃcult to establish fundamental geometric properties like those in Section 4.
Appendix A. Case of the standard Laplacian on the Sierpiński gasket
We follow the notation introduced in Section 2 throughout this appendix. Here
we briefly review some important results for the so-called standard Laplacian ∆0 on
K, which is the non-positive self-adjoint operator of the Dirichlet space (K, µ0 , E, F)
with µ0 the self-similar measure on K with weight (1/3, 1/3, 1/3). Namely, µ0 is
the unique Borel measure on K such that µ0 (Kw ) = 3−|w| for any w ∈ W∗ . µ0
is in fact a constant multiple of the df -dimensional Hausdorﬀ measure on K withrespect to the Euclidean metric, where df := log2 3 is the Hausdorﬀ dimension of
K with respect to the Euclidean metric; for details see e.g. [57, Section 1.5].
The Brownian motion on the Sierpiński gasket, which is the diﬀusion process
corresponding to (K, µ0 , E, F), was first constructed by Goldstein [30] and Kusuoka[66] and then intensively studied in a seminal work [11] by Barlow and Perkins. The
most important result in [11] is the following sub-Gaussian bound for the transition
density of the Brownian motion on K with respect to µ0 , which is nothing but theheat kernel pµ0 associated with (K, µ0 , E, F) in our notation. Let(A.1) ρ0 (x, y) := inf{`R2 (γ) | γ : [0, 1] → K, γ is continuous, γ(0) = x, γ(1) = y}for x, y ∈ K, so that ρ0 is a metric on K comparable to the Euclidean metric on
K.
Theorem A.1 ([11, Theorem 1.5]). Let dw := log2 5 and set ds := df /dw .
Then there exist cA.1 , cA.2 ∈ (0, ∞) such that for any (t, x, y) ∈ (0, 1] × K × K,
(A.2)
( (
)
( (
)
cA.1
ρ0 (x, y)dw ) dw1−1
cA.2
ρ0 (x, y)dw ) dw1−1
exp
−
≤
p
(t,
x,
y)
≤
exp
−
.
µ0
cA.1 tcA.2 ttds /2
tds /2
Note that dw > 2, which is why an estimates of the form (A.2) is called subGaussian. Roughly speaking, (A.2) says that heat on K diﬀuses up to the distancecomparable to t1/dw at time t on average, which is not the case (at least for small t)
for the heat kernels on Riemannian manifolds. The following theorem and proposition are part of the reasons for such a non-classical behavior of pµ0 .
Theorem A.2 ([67, §6, Example 1]). µhui is singular to µ0 for any u ∈ F .
Proposition A.3. ρ0 (x, ·) 6∈ F for any x ∈ K.

ANALYSIS AND GEOMETRY OF MEASURABLE RIEMANNIAN STRUCTURE ON S.G. 37

Theorems A.1 and A.2 have been extended to a wide range of self-similar fractals including (aﬃne) nested fractals, a class of post-critically finite self-similar
fractals with certain good symmetry, and generalized Sierpiński carpets, a naturalgeneralization of the Sierpiński carpet. See e.g. [64, 26, 7, 8, 9, 59, 60] for extensions of Theorem A.1 and [14, 36, 43] for those of Theorem A.2. (See Figure 3
above for pictures of several typical nested fractals and the Sierpiński carpet.) On
the other hand, the author is not sure whether a proof of Proposition A.3 is given
in any reference and to what extent it can be generalized, although it is essentiallyknown to experts at least in simple cases. For the reader’s convenience, we give
here a proof of Proposition A.3 in the present setting of the Sierpiński gasket.
Proof of Proposition A.3. We first prove the assertion for x ∈ V0 . Recall
that V0 = {q1√, q2 , q3 }. Without loss
√ of generality we may assume that q1 = (0, 0),q2 = (1, −1/ 3) and q3 = (1, 1/ 3), and by the rotational symmetry of K√ and(E, F) it suﬃces to show for x = q1 . Define f : R2 → R by f (y, z) := (2/ 3)y.
Then an induction in m easily shows that ρ0 (q1 , y) = f (y) for any m ∈ N ∪ {0} andany y ∈ Vm and hence ρ0 (q1 , ·) = f |K by the denseness√ of V∗ in K. Now suppose
f |K ∈ F. Then f |K ◦ Fi =(1/2)f|+(1(i)/3)1 ∈ F for any i ∈ S and
K
{2,3}
∑
hence E(f |K , f |K ) = (5/3) i∈S E(f |K /2, f |K /2) = (5/4)E(f |K , f |K ) by (2.4), sothat E(f |K , f |K ) = 0 which contradicts f |K 6∈ R1. Thus ρ0 (q1 , ·) = f |K 6∈ F .
Next for general x ∈ K, choose i ∈ S so that x ∈ Ki and let j, k ∈ S \ {i}be such that j 6= k and ρ0 (x, Fi (qj )) ≤ ρ0 (x, Fi (qk )). Then we easily see that
ρ0 (x, Fj (·)) = (1/2)ρ0 (qi , ·) + ρ0 (x, Fi (qj ))1 6∈ F and hence ρ0 (x, ·) 6∈ F.
Also as opposed to the case of the heat kernels on Riemannian manifolds, pµ0
is known to exhibit various oscillatory asymptotic behavior, as follows.
Theorem A.4 ([65, Theorem 1.2-a)]). There exists a continuous log(5/2)periodic non-constant function G : R → (0, ∞) such that()() d 1−1
(
)
dw
s
n
n
w
d
−1
w
(A.3) lim (2/5) s
log pµ0 (2/5) s, x, y = −ρ0 (x, y)
G log
n→∞
ρ0 (x, y)for any (s, x, y) ∈ (0, ∞) × K × K with x 6= y. In particular, for any x, y ∈ K with
1
x 6= y, the limit limt↓0 t dw −1 log pµ0 (t, x, y) does not exist.
Theorem A.5 ([49, Corollary 6.2]). The limit limt↓0 tds /2 pµ0 (t, x, x) does notexist for any x ∈ K.
A similar oscillation is observed also in the Laplacian eigenvalue asymptotics.
Theorem A.6 ([61, 10], cf. [28], [57, Theorems 4.1.5, 4.3.4, 4.4.10 and B.4.3]).
Let {λ0n }n∈N be the non-decreasing enumeration of all the eigenvalues of −∆0 , where
each eigenvalue is repeated according to its multiplicity, and define the eigenvalue
counting function N0 of (K, µ0 , E, F) by N0 (λ) := #{n ∈ N | λ0n ≤ λ}, λ ∈ R. Then
there exists a right-continuous log 5-periodic discontinuous function G0 : R → R
with 0 < inf s∈R G0 (s) < sups∈R G0 (s) < ∞, such that
(A.4)

N0 (λ) = λds /2 G0 (log λ) + O(1)as λ → ∞.

Roughly speaking, the asymptotic log-periodicity stated in Theorems A.4 andA.6 is more or less implied by the self-similarity of µ0 and (E, F), whereas it ishighly non-trivial to prove that there does exist oscillation in the asymptotics as in
Theorems A.4, A.5 and A.6. Theorem A.4 was proved by utilizing a very detailed

38

NAOTAKA KAJINO

description of the behavior of the Brownian motion on K provided in [11], and no
essential extension to other fractals is known for this result because such detailed
information of the Brownian motion is not available for most fractals.
The existence of G0 in Theorem A.6 except its discontinuity was proved by
Kigami and Lapidus in [61, Theorem 2.4 and Corollary 2.5] in the general framework of a self-similar regular Dirichlet form on a post-critically finite self-similar
set equipped with a self-similar measure. In [61] they established the method of
obtaining a certain renewal equation for the eigenvalue counting function N0 (λ) toapply the renewal theorem [57, Theorems B.4.2 and B.4.3] (see also [25, Section
XI.1]) for Borel probability measures on (0, ∞). Then the same method has beenused by many authors in the context of analysis on fractals; see e.g. [65, 33, 47, 51].
(A variant of the method of [61] is described in some detail in Section 7 above.)
The discontinuity of G0 in Theorem A.6 was proved by Barlow and Kigami[10] for aﬃne nested fractals by showing the existence of localized eigenfunctionsof the Laplacian. For example, in our present setting of the Sierpiński gasket, thedihedral symmetry of (K, µ0 , E, F) implies that there exists an eigenfunction ϕ of−∆0 with eigenvalue λ ∈ (0, ∞) such that ϕ|V0 = 0, and then for any w ∈ W∗ ,{ϕ ◦ Fw−1 on Kw ,
(A.5)
ϕw :=
0
on K \ Kw
is also an eigenfunction of −∆0 with eigenvalue 5|w| λ. This fact immediately implies
the discontinuity of G0 , and Theorem A.5 for “generic” x ∈ K is also proved on thebasis of this fact and the eigenfunction expansion [21, (2.1.4)] of the heat kernel
pµ0 , by noticing that ϕw in (A.5) has amplitude comparable to 3|w|/2 after it isnormalized in L2 (K, µ0 ). Some more work is required to show the non-existence ofthe limit limt↓0 tds /2 pµ0 (t, x, x) for any x ∈ K and it has been done in [49] only fora limited class of concrete nested fractals, whereas the non-existence of this limit
for “generic” points can be still verified for general aﬃne nested fractals, as statedin [49, Theorem 4.5]. In fact, very recently, the author has proved in [50] thattds /2 pµ0 (t, x, x) is neither regularly varying at 0 nor asymptotically log-periodic ast ↓ 0 for “generic” points x, for a wide range of self-similar fractals including most(but not all) nested fractals and all generalized Sierpiński carpets.
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http://www.math.leidenuniv.nl/nl/reports/1185/321/ (accessed February 5, 2013)
Department of Mathematics, University of Bielefeld, Postfach 10 01 31, 33501
Bielefeld, Germany
Current address: Department of Mathematics, Graduate School of Science, Kobe University,
Rokkodai-cho 1-1, Nada-ku, Kobe 657-8501, Japan
E-mail address: nkajino@math.kobe-u.ac.jp
URL: http://www.math.kobe-u.ac.jp/HOME/nkajino/