The Local Regular Dirichlet Form on the Sierpi\'nski Gasket via the ...

The Local Regular Dirichlet Form on the Sierpi´nski Gasket via the Averaging Method Meng Yang

arXiv:1706.04998v1 [math.FA] 15 Jun 2017

Abstract We construct a self-similar local regular Dirichlet form on the Sierpi´ nski gasket using the averaging method which has direct corollaries that the local Dirichlet form has a domain of some Besov space and non-local Dirichlet forms can approximate the local Dirichlet form. Our construction is different from those of Kigami [14] and Strichartz [21].

1

Introduction

Sierpi´ nski gasket (SG) is the simplest self-similar set in some sense, see Figure 1.

Figure 1: Sierpi´ nski Gasket SG can be obtained as follows. Given an equilateral triangle with sides of length 1. Divide the triangle into four congruent small triangles, each with sides of length 1/2, remove the central one. Divide each of the three remaining small triangles into four congruent triangles, each with sides of length 1/4, remove the central ones, see Figure 2. Repeat above procedure infinitely many times, SG is the compact connect set K that remains. SG usually serves as a basic example of singular spaces for analysis and probability. Dirichlet form theory is a powerful tool in this approach. In general, local regular Dirichlet forms are in one-to-one correspondence to Brownian motions (BM). The construction of a BM on SG was given by Barlow and Perkins [2]. The construction of a local regular Dirichlet form on SG was given by Kigami [12] using difference quotients method which was generalized to p.c.f. (post critically finite) self-similar sets in [13, 14]. Subsequently, Strichartz [21] gave an alternative construction using the averaging method but the points Date: June 19, 2017 MSC2010: 28A80 Keywords: Sierpi´ nski gasket, local regular Dirichlet form, Γ-convergence, the averaging method The author was supported by SFB701 of the German Research Council (DFG). The author is very grateful to Professor Alexander Grigor’yan for very helpful discussions.

1

Figure 2: The Construction of Sierpi´ nski Gasket are the energy of harmonic functions and the density of piecewise harmonic functions which go back to Kigami’s construction. The local regular Dirichlet form Eloc on SG admits a heat kernel pt (x, y) satisfying   ∗   β∗β−1 |x − y| C  (1) pt (x, y)  α/β ∗ exp −c t t1/β ∗ for all x, y ∈ K, t ∈ (0, 1), where α = log 3/ log 2 is the Hausdorff dimension of SG and β ∗ :=

log 5 log 2

is the walk dimension of BM which is frequently denoted also by dw . The estimates (1) were obtained by Barlow and Perkins [2]. The domain Floc of Eloc is some Besov space. This was given by Jonsson [11]. Later on, this kind of characterization was generalized to simple nested fractals by Pietruska-Paluba [17] and p.c.f. self-similar sets by Hu and Wang [10]. This kind of characterization was also given by Pietruska-Paluba [18], Grigor’yan, Hu and Lau [7], Kumagai and Sturm [15] if local regular Dirichlet forms on metric measure spaces admit sub-Gaussian heat kernel estimates. Here, we reprove this characterization as a direct corollary of our construction. Consider the following stable-like non-local quadratic form Z Z (u(x) − u(y))2 Eβ (u, u) = ν(dx)ν(dy), |x − y|α+β K K  Fβ = u ∈ L2 (K; ν) : Eβ (u, u) < +∞ , where α = dimH K as above, ν is the normalized Hausdorff measure on K of dimension α and β > 0 is so far arbitrary. Using the estimates (1) and subordination technique, it was proved by Pietruska-Paluba [19] that lim (β ∗ − β)Eβ (u, u)  Eloc (u, u)  lim∗ (β ∗ − β)Eβ (u, u) (2) β↑β

β↑β ∗

for all u ∈ Floc . This is similar to the following classical result Z Z Z (u(x) − u(y))2 lim(2 − β) dxdy = C(n) |∇u(x)|2 dx β↑2 |x − y|n+β Rn Rn Rn for all u ∈ W 1,2 (Rn ), where C(n) is some positive constant (see [6, Example 1.4.1]). Recently, the author [22] gave an alternative proof of (2) using discretization method. Here, we reprove (2) as a direct corollary of our construction. The main purpose of this paper is to give a construction of a local regular Dirichlet form on SG totally using the averaging method. The local regular Dirichlet form given here coincides with that given by Kigami and Strichartz due to the uniqueness result given by Sabot [20]. Kusuoka and Zhou [16] also gave a construction using the averaging method and approximation of Markov chains.

2

Recently, Grigor’yan and the author [8] gave a purely analytic construction of a local regular Dirichlet form on the Sierpi´ nski carpet. This paper can be regarded as a realization of this method on SG. The ultimate purpose of this paper and [8] is to provide a new unified method of construction of local regular Dirichlet forms on a wide class of fractals that uses only self-similar property and ideally should be independent of other specific properties, in particular, p.c.f. property. The idea of our construction of Eloc is as follows. First, we use the averaging method to construct another quadratic form Eβ , equivalent to Eβ , which turns out to be a regular closed form for all β ∈ (α, β ∗ ). Second, we construct a regular closed form E as a Γ-limit of a sequence {(β ∗ − βn )Eβn } with βn ↑ β ∗ . Third, we use a standard method from [16] to construct Eloc from E. The main difficulty in the averaging method is that we do not have monotonicity result as in Kigami’s construction. But we still have weak monotonicity result and regularity result. These are obtained by estimating resistances and constructing functions with energy property and separation property which are called good functions. On the Sierpi´ nski carpet, these procedures are highly non-trivial but our construction still follows, see [8].

2

Statement of the Main Results

√ Consider the following points in R2 : p0 = (0, 0), p1 = (1, 0), p2 = (1/2, 3/2). Let fi (x) = (x + pi )/2, x ∈ R2 . Then the Sierpi´ nski gasket (SG) is the unique non-empty compact set K satisfying K = f0 (K) ∪ f1 (K) ∪ f2 (K). Let ν be the normalized Hausdorff measure on K of dimension α = log 3/ log 2. Let V0 = {p0 , p1 , p2 } , Vn+1 = f0 (Vn ) ∪ f1 (Vn ) ∪ f2 (Vn ) for all n ≥ 0. Then {Vn } is an increasing sequence of finite sets and K is the closure of V ∗ = ∪∞ n=0 Vn . The classical construction of a self-similar local regular Dirichlet form on SG is as follows. Let  n X 5 (u(x) − u(y))2 , n ≥ 0, u ∈ l(K), En (u, u) = 3 x,y∈Vn |x−y|=2−n

where l(S) is the set of all real-valued functions on the set S. Then En (u, u) is monotone increasing in n for all u ∈ l(K). Let  n X 5 Eloc (u, u) = lim En (u, u) = lim (u(x) − u(y))2 , n→+∞ n→+∞ 3 x,y∈Vn |x−y|=2−n

Floc = {u ∈ C(K) : Eloc (u, u) < +∞} , then (Eloc , Floc ) is a self-similar local regular Dirichlet form on L2 (K; ν), see [12, 13, 14]. Let W0 = {∅} and Wn = {w = w1 . . . wn : wi = 0, 1, 2, i = 1, . . . , n} for all n ≥ 1. (1)

(1)

(2)

(2)

For all w(1) = w1 . . . wm ∈ Wm , w(2) = w1 . . . wn ∈ Wn , denote w(1) w(2) as w = (1) (2) w1 . . . wm+n ∈ Wm+n with wi = wi for all i = 1, . . . , m and wm+i = wi for all i = 1, . . . , n. For all i = 0, 1, 2, denote in as w = w1 . . . wn ∈ Wn with wk = i for all k = 1, . . . , n. For all w = w1 . . . wn−1 wn ∈ Wn , denote w− = w1 . . . wn−1 ∈ Wn−1 . For all w = w1 . . . wn ∈ Wn , let fw = fw1 ◦ . . . ◦ fwn , Vw = fw1 ◦ . . . ◦ fwn (V0 ), Kw = fw1 ◦ . . . ◦ fwn (K), Pw = fw1 ◦ . . . ◦ fwn−1 (pwn ), where f∅ = id is the identity map.

3

For all n ≥ 1, let Xn be the graph with vertex set Wn and edge set Hn given by n o Hn = (w(1) , w(2) ) : w(1) , w(2) ∈ Wn , w(1) 6= w(2) , Kw(1) ∩ Kw(2) 6= ∅ . For example, we have the figure of X3 in Figure 3. Denote w(1) ∼n w(2) if (w(1) , w(2) ) ∈ Hn . If w(1) ∼n w(2) satisfies Pw(1) 6= Pw(2) , we say that w(1) ∼n w(2) is of type I. If w(1) ∼n w(2) satisfies Pw(1) = Pw(2) , we say that w(1) ∼n w(2) is of type II. For example, 000 ∼3 001 is of type I, 001 ∼3 010 is of type II.

222 220

221

202

212

200

201

211

210

022 020

122 021

002 000

001

010

120

012

102

011

100

121 112

101

110

111

Figure 3: X3 For all n ≥ 1, u ∈ L2 (K; ν), let Pn u : Wn → R be given by Z Z 1 u(x)ν(dx) = (u ◦ fw )(x)ν(dx), w ∈ Wn . Pn u(w) = ν(Kw ) Kw K Our main result is as follows. Theorem 2.1. There exists a self-similar strongly local regular Dirichlet form (Eloc , Floc ) on L2 (K; ν) satisfying  n  2 X 5 Eloc (u, u)  sup Pn u(w(1) ) − Pn u(w(2) ) , n≥1 3 w(1) ∼n w(2)    n     X 2 5 Floc = u ∈ L2 (K; ν) : sup Pn u(w(1) ) − Pn u(w(2) ) < +∞ .   n≥1 3 (1) (2) w

∼n w

Let us introduce the notion of Besov spaces. Let (M, d, µ) be a metric measure space and α, β > 0 two parameters. Define [u]B 2,2 (M ) =

∞ X

2(α+β)n

Z

Z

(u(x) − u(y))2 µ(dy)µ(dx),

α,β

n=1

[u]B 2,∞ (M ) = sup 2(α+β)n α,β

M d(x,y) 0, let Eβ (u, u) =

∞ X

2(β−α)n

n=1

Fβ =

  

u ∈ L2 (K; ν) :

X



2 Pn u(w(1) ) − Pn u(w(2) ) ,

w(1) ∼n w(2) ∞ X n=1

2(β−α)n

X w(1) ∼n w(2)



Pn u(w(1) ) − Pn u(w(2) )

2

< +∞

 

,



denote Eβ (u, u) = [u]B 2,2 (K) for simplicity. α,β In what follows, K is a locally compact separable metric space and ν is a Radon measure on K with full support. We say that (E, F) is a closed form on L2 (K; ν) in the wide sense

14

if F is complete under the inner product E1 but F is not necessary to be dense in L2 (K; ν). If (E, F) is a closed form on L2 (K; ν) in the wide sense, we extend E to be +∞ outside F, hence the information of F is encoded in E. In what follows, K is SG in R2 and ν is the normalized Hausdorff measure on K. We obtain non-local regular closed forms and Dirichlet forms as follows. Theorem 5.3. For all β ∈ (α, β ∗ ), (Eβ , Fβ ) is a non-local regular closed form on L2 (K; ν), (Eβ , Fβ ), (Eβ , Fβ ) are non-local regular Dirichlet forms on L2 (K; ν). For all β ∈ [β ∗ , +∞), Fβ consists of constant functions. Remark 5.4. Eβ does not have Markovian property but Eβ , Eβ do have Markovian property. An interesting problem in non-local analysis is for which value β > 0, (Eβ , Fβ ) is a regular Dirichlet form on L2 (K; ν). The critical value  β∗ := sup β > 0 : (Eβ , Fβ ) is a regular Dirichlet form on L2 (K; ν) is called the walk dimension of SG with Euclidean metric and Hausdorff measure. A classical approach to determine β∗ is using the estimates (1) and subordination technique to have β∗ = β ∗ =

log 5 , log 2

see [18]. The proof of above theorem provides an alternative approach without using BM. Proof. By Fatou’s lemma, it is obvious that (Eβ , Fβ ) is a closed form on L2 (K; ν) in the wide sense. For all β ∈ (α, β ∗ ). By Theorem 5.1, Fβ ⊆ C(K). We only need to show that Fβ is uniformly dense in C(K), then Fβ is dense in L2 (K; ν), hence (Eβ , Fβ ) is a regular closed form on L2 (K; ν). Indeed, by Theorem 3.6, for all U = U (x0 ,x1 ,x2 ) ∈ U, we have Eβ (U, U ) =

∞ X

2(β−α)n An (U )

n=1 ∞ X

"   2n # n
3 3 = 2 − (x0 − x1 )2 + (x1 − x2 )2 + (x0 − x2 )2 3 5 5 n=1  n ∞
X 3 2 2(β−α)n (x0 − x1 )2 + (x1 − x2 )2 + (x0 − x2 )2 < +∞, ≤ 3 5 n=1 (β−α)n 2

hence U ∈ Fβ , U ⊆ Fβ . By Proposition 3.8, Fβ separates points. It is obvious that Fβ is a sub-algebra of C(K), that is, for all u, v ∈ Fβ , c ∈ R, we have u + v, uv, cu ∈ Fβ . By Stone-Weierstrass theorem, Fβ is uniformly dense in C(K). Since Eβ , Eβ do have Markovian property, by above, (Eβ , Fβ ), (Eβ , Fβ ) are non-local regular Dirichlet forms on L2 (K; ν). For all β ∈ [β ∗ , +∞). Assume that u ∈ Fβ is not constant, then there exists some integer N ≥ 1 such that DN (u) > 0. By Theorem 4.1, we have Eβ (u, u) =

∞ X n=1

≥

1 C

 n  n ∞ X 3 3 Dn (u) ≥ 2(β−α)n Dn (u) 5 5 n=N +1  n ∞ X 3 (β−α)n 2 DN (u) = +∞, 5

2(β−α)n

n=N +1

contradiction! Hence Fβ consists of constant functions. We need some preparation about Γ-convergence. In what follows, K is a locally compact separable metric space and ν is a Radon measure on K with full support. Definition 5.5. Let E n , E be closed forms on L2 (K; ν) in the wide sense. We say that E n is Γ-convergent to E if the following conditions are satisfied.

15

(1) For all {un } ⊆ L2 (K; ν) that converges strongly to u ∈ L2 (K; ν), we have lim E n (un , un ) ≥ E(u, u). n→+∞

(2) For all u ∈ L2 (K; ν), there exists a sequence {un } ⊆ L2 (K; ν) converging strongly to u in L2 (K; ν) such that lim E n (un , un ) ≤ E(u, u). n→+∞

We have the following result about Γ-convergence. Proposition 5.6. ([4, Proposition 6.8, Theorem 8.5, Theorem 11.10, Proposition 12.16]) Let {(E n , F n )} be a sequence of closed forms on L2 (K; ν) in the wide sense, then there exist some subsequence {(E nk , F nk )} and some closed form (E, F) on L2 (K; ν) in the wide sense such that E nk is Γ-convergent to E. We need an elementary result as follows. Proposition 5.7. Let {xn } be a sequence of nonnegative real numbers. (1) ∞ X

lim xn ≤ lim(1 − λ) n→+∞

λ↑1

∞ X

λn xn ≤ lim(1 − λ) λ↑1

n=1

λn xn ≤ lim xn ≤ sup xn . n→+∞

n=1

n≥1

(2) If there exists some positive constant C such that xn ≤ Cxn+m for all n, m ≥ 1, then sup xn ≤ C lim xn . n→+∞

n≥1

Proof. The proof is elementary using ε-N argument. In what follows, K is SG in R2 and ν is the normalized Hausdorff measure on K. Take {βn } ⊆ (α, β ∗ ) with βn ↑ β ∗ . By Proposition 5.6, there exist some subsequence still denoted by {βn } and some closed form (E, F) on L2 (K; ν) in the wide sense such that (β ∗ − βn )Eβn is Γ-convergent to E. Without lose of generality, we may assume that 0 < β ∗ − βn
1/2. Since

β ∗ − βn 1 , ∗ = n→+∞ 1 − 2βn −β log 2 lim

we have E(u, u) ≥

1 2C

β ∗ − βn 1 ∗ Dn (un ) ≥ n→+∞ 1 − 2βn −β 2(log 2)C lim

lim Dn (un ).

n→+∞

Since un → u in L2 (K; ν), for all k ≥ 1, we have Dk (u) = lim Dk (un ) = n→+∞

lim

k≤n→+∞

Dk (un ) ≤ C lim Dn (un ). n→+∞

Taking supremum with respect to k ≥ 1, we have sup Dk (u) ≤ C lim Dn (un ) ≤ C lim Dn (un ) ≤ 2(log 2)C 2 E(u, u). n→+∞

n→+∞

k≥1

By Theorem 5.1, F ⊆ C(K). We only need to show that F is uniformly dense in C(K), then F is dense in L2 (K; ν), hence (E, F) is a regular closed form on L2 (K; ν). Indeed, by Theorem 3.6, for all U = U (x0 ,x1 ,x2 ) ∈ U, we have  n 5 sup Dn (U ) = sup An (U ) 3 n≥1 n≥1 "  n  n  2n #
5 2 3 3 = sup − (x0 − x1 )2 + (x1 − x2 )2 + (x0 − x2 )2 3 3 5 5 n≥1
2 ≤ (x0 − x1 )2 + (x1 − x2 )2 + (x0 − x2 )2 < +∞, 3 hence U ∈ F, U ⊆ F. By Proposition 3.8, F separates points. It is obvious that F is a sub-algebra of C(K). By Stone-Weierstrass theorem, F is uniformly dense in C(K). Now we prove Theorem 2.1 using a standard approach as follows. Proof of Theorem 2.1. For all u ∈ L2 (K; ν), n, k ≥ 1, we have  2 X Pn+k u(w(1) ) − Pn+k u(w(2) ) w(1) ∼n+k w(2)

=

X



X

Pn+k u(ww(1) ) − Pn+k u(ww(2) )

2

w∈Wn w(1) ∼k w(2)



X

+ (1)

(1)

(2)

2 Pn+k u(w(1) wn(1) . . . wn(1) ) − Pn+k u(w(2) wn(2) . . . wn(2) ) ,

(2)

w(1) =w1 ...wn ∼n w(2) =w1 ...wn

17

where for all i = 1, 2 Z Z (i) Pn+k u(ww ) = (u ◦ fww(i) )(x)ν(dx) = (u ◦ fw ◦ fw(i) )(x)ν(dx) = Pk (u ◦ fw )(w(i) ), K

K

hence X

X

Ak (u ◦ fw ) =

w∈Wn

2 Pk (u ◦ fw )(w(1) ) − Pk (u ◦ fw )(w(2) )

w∈Wn w(1) ∼k w(2)



X

≤



X

2 Pn+k u(w(1) ) − Pn+k u(w(2) ) = An+k (u),

w(1) ∼n+k w(2)

and  n X  n+k X  n+k 5 5 5 Dk (u ◦ fw ) = Ak (u ◦ fw ) ≤ An+k (u) = Dn+k (u). 3 3 3 w∈Wn

w∈Wn

For all u ∈ F, n ≥ 1, w ∈ Wn , we have X

sup Dk (u ◦ fw ) ≤ sup k≥1

k≥1 w∈W

Dk (u ◦ fw ) ≤

n

 n  n 3 3 Dn+k (u) ≤ sup Dk (u) < +∞, 5 5 k≥1

hence u ◦ fw ∈ F. For all u ∈ L2 (K; ν), n ≥ 1, let E(u, u) =

2  X

u(pi ) −

E

u(x)ν(dx)

,

K

i=0 (n)

2

Z

 n X 5 (u, u) = E(u ◦ fw , u ◦ fw ). 3 w∈Wn

By Theorem 5.1, we have E(u, u) =

≤

2 Z X

2 (u(pi ) − u(x)) ν(dx)

≤

2 Z X

2

(u(pi ) − u(x)) ν(dx)

K

i=0

K

2 Z X

  ∗ c2 |pi − x|β −α sup Dk (u) ν(dx) ≤ 3c2 sup Dk (u),

i=0

i=0

k≥1

K

k≥1

hence  n X  n X 5 5 3c2 sup Dk (u ◦ fw ) ≤ 3c2 C lim Dk (u ◦ fw ) 3 3 k≥1 w∈Wn w∈Wn k→+∞  n X 5 ≤ 3c2 C lim Dk (u ◦ fw ) ≤ 3c2 C lim Dn+k (u) ≤ 3c2 C sup Dk (u). 3 k≥1 k→+∞ k→+∞

E

(n)

(u, u) ≤

w∈Wn

On the other hand, for all u ∈ L2 (K; ν), n ≥ 1, we have Dn (u) =

 n 5 3

X w(1) ∼n w(2)

Z

Z (u ◦ fw(1) )(x)ν(dx) −

K

2 (u ◦ fw(2) )(x)ν(dx) .

K

For all w(1) ∼n w(2) , there exist i, j = 0, 1, 2 such that Kw(1) ∩ Kw(2) = {fw(1) (pi )} = {fw(2) (pj )} .

18

(6)

Hence Dn (u)   n  Z X 5 (u ◦ fw(1) )(pi ) − = (u ◦ fw(1) )(x)ν(dx) 3 K w(1) ∼n w(2)  2 Z − (u ◦ fw(2) )(pj ) − (u ◦ fw(2) )(x)ν(dx) K "  n 2 Z X 5 (u ◦ fw(1) )(pi ) − (u ◦ fw(1) )(x)ν(dx) ≤2 3 K w(1) ∼n w(2)   # Z

(7)

2

(u ◦ fw(2) )(pj ) −

+

(u ◦ fw(2) )(x)ν(dx) K

2  n X X Z 2  5 (u ◦ fw )(pi ) − (u ◦ fw )(x)ν(dx) 3 K w∈Wn i=0  n X 5 (n) =6 E(u ◦ fw , u ◦ fw ) = 6E (u, u). 3 ≤6

w∈Wn

2

For all u ∈ L (K; ν), n ≥ 1, we have E

(n+1)

(u, u) =

 n+1 5 3

X

E(u ◦ fw , u ◦ fw )

w∈Wn+1

 n+1 X 2 2 X 5 5 X (n) = E(u ◦ fi ◦ fw , u ◦ fi ◦ fw ) = E (u ◦ fi , u ◦ fi ). 3 3 i=0 i=0

(8)

w∈Wn

Let

n

1 X (l) E˜(n) (u, u) = E (u, u), u ∈ L2 (K; ν), n ≥ 1. n l=1

By Equation (6), we have E˜(n) (u, u) ≤ 3c2 C sup Dk (u)  E(u, u) for all u ∈ F, n ≥ 1. k≥1

Since (E, F) is a regular closed form on L2 (K; ν), by [3, Definition 1.3.8, Remark 1.3.9, Definition 1.3.10, Remark 1.3.11], we have (F, E1 ) is a separable Hilbert space. Let {ui }i≥1 n o be a dense subset of (F, E1 ). For all i ≥ 1, E˜(n) (ui , ui ) is a bounded sequence. n≥1 n o By diagonal argument, there exists a subsequence {nk }k≥1 such that E˜(nk ) (ui , ui ) k≥1 n o converges for all i ≥ 1. Hence E˜(nk ) (u, u) converges for all u ∈ F. Let k≥1

Eloc (u, u) = lim E˜(nk ) (u, u) for all u ∈ Floc := F. k→+∞

Then Eloc (u, u) ≤ 3c2 C sup Dk (u) for all u ∈ Floc = F. k≥1

By Equation (7), for all u ∈ Floc = F, we have 1 1 (nk ) Eloc (u, u) = lim E˜(nk ) (u, u) ≥ lim E (u, u) ≥ lim Dk (u) ≥ sup Dk (u). k→+∞ 6 6C k≥1 k→+∞ k→+∞ Hence Eloc (u, u)  sup Dk (u) for all u ∈ Floc = F. k≥1

Hence (Eloc , Floc ) is a regular closed form on L2 (K; ν). Since 1 ∈ Floc and Eloc (1, 1) = 0, by [6, Lemma 1.6.5, Theorem 1.6.3], (Eloc , Floc ) on L2 (K; ν) is conservative.

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For all u ∈ Floc = F, we have u ◦ fi ∈ F = Floc for all i = 0, 1, 2. Moreover, by Equation (8), we have 2

2

5X 5X Eloc (u ◦ fi , u ◦ fi ) = lim E˜(nk ) (u ◦ fi , u ◦ fi ) 3 i=0 3 i=0 k→+∞ # " 2 nk nk 2 5X 1 X 1 X 5 X (l) (l) = E (u ◦ fi , u ◦ fi ) = lim E (u ◦ fi , u ◦ fi ) lim k→+∞ nk 3 i=0 k→+∞ nk 3 i=0 l=1

l=1

nk nk +1 1 X 1 X (l+1) (l) = lim E (u, u) = lim E (u, u) k→+∞ nk k→+∞ nk l=2 l=1 # " nk 1 1 (1) 1 X (l) (nk +1) = lim E (u, u) + E (u, u) − E (u, u) k→+∞ nk nk nk l=1

= lim E˜(nk ) (u, u) = Eloc (u, u). k→+∞

Hence (Eloc , Floc ) on L2 (K; ν) is self-similar. For all u, v ∈ Floc satisfying supp(u), supp(v) are compact and v is constant in an open neighborhood U of supp(u), we have K\U is compact and supp(u) ∩ (K\U ) = ∅, hence δ = dist(supp(u), K\U ) > 0. Taking sufficiently large n ≥ 1 such that 21−n < δ, by self-similarity, we have  n X 5 Eloc (u ◦ fw , v ◦ fw ). Eloc (u, v) = 3 w∈Wn

For all w ∈ Wn , we have u ◦ fw = 0 or v ◦ fw is constant, hence Eloc (u ◦ fw , v ◦ fw ) = 0, hence Eloc (u, v) = 0, that is, (Eloc , Floc ) on L2 (K; ν) is strongly local. For all u ∈ Floc , it is obvious that u+ , u− , 1 − u, u = (0 ∨ u) ∧ 1 ∈ Floc and Eloc (u, u) = Eloc (1 − u, 1 − u). Since u+ u− = 0 and (Eloc , Floc ) on L2 (K; ν) is strongly local, we have Eloc (u+ , u− ) = 0. Hence Eloc (u, u) = Eloc (u+ − u− , u+ − u− ) = Eloc (u+ , u+ ) + Eloc (u− , u− ) − 2Eloc (u+ , u− ) = Eloc (u+ , u+ ) + Eloc (u− , u− ) ≥ Eloc (u+ , u+ ) = Eloc (1 − u+ , 1 − u+ ) ≥ Eloc ((1 − u+ )+ , (1 − u+ )+ ) = Eloc (1 − (1 − u+ )+ , 1 − (1 − u+ )+ ) = Eloc (u, u), that is, (Eloc , Floc ) on L2 (K; ν) is Markovian. Hence (Eloc , Floc ) is a self-similar strongly local regular Dirichlet form on L2 (K; ν). Remark 5.9. The idea of the standard approach is from [16, Section 6]. The proof of Markovian property is from the proof of [1, Theorem 2.1].

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[20] C. Sabot, Existence and uniqueness of diffusions on finitely ramified self-similar frac´ tals, Ann. Sci. Ecole Norm. Sup. (4), 30 (1997), pp. 605–673. [21] R. S. Strichartz, The Laplacian on the Sierpinski gasket via the method of averages, Pacific J. Math., 201 (2001), pp. 241–256. [22] M. Yang, Equivalent semi-norms of non-local dirichlet forms on Sierpi´ nski gasket and applications, ArXiv e-prints, (2016). ¨ t fu ¨ r Mathematik, Universita ¨ t Bielefeld, Postfach 100131, 33501 BieleFakulta feld, Germany. E-mail address: [email protected] and Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China. E-mail address: [email protected]

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