DOMAINS OF DIRICHLET FORMS AND EFFECTIVE RESISTANCE ESTIMATES ON P.C.F. FRACTALS. 1. Introduction

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1 DOMAINS OF DIRICHLET FORMS AND EFFECTIVE RESISTANCE ESTIMATES ON P.C.F. FRACTALS JIAXIN HU 1 AND XINGSHENG WANG Abstract. In this paper we consider post-critically finite self-similar fractals with regular harmonic structures. We first obtain effective resistance estimates in terms of the Euclidean metric, which particularly imply the embedding theorem for the domains of the Dirichlet forms associated with the harmonic structures. We then characterize the domains of the Dirichlet forms. 1. Introduction Let K, {F i } M i=1) be a post-critically finite p.c.f.) self-similar fractal in R n n 1) with a regular harmonic structure H, r), and let E, D) be the Dirichlet form associated with H, r). Let R be the effective resistance determined by the form E, D). In this paper, we are concerned with the following problems: 1) What is the relationship between R and the Euclidean metric? 2) How to characterize the domain D of E? These two problems are important in studying the dynamical aspects of fractals, such as PDE s, Brownian motions, heat kernels, and function spaces on fractals. Recall that the first problem above is obvious, if K is a bounded open interval of R and E is the classical energy form with respect to the Lebesgue measure 1.1) Ef, g) = 1 f gdx. 2 As a matter of fact, there exists some c > 0 such that, for all x, y K R, 1.2) K c 1 x y Rx, y) c x y. The second inequality in 1.2) follows by using the definition of R, see 2.9) below, and the Sobolev embedding theorem: 1.3) fx) fy) c 1/2 x y 1/2 Ef, f) 1/2, Date: October 13, Revised: July, Mathematics Subject Classification. Primary 28A80; Secondary 46E30. Key words and phrases. p.c.f fractal, Dirichlet form, effective resistance, partition, domain, Sobolev-type space. 1 Research supported in part by NSFC No ). 1

2 2 J.X. HU AND X.S. WANG see for example [1, Formula 9), p.98]. The first inequality in 1.2) also follows; this is because for any x 0 < y 0 in K, letting 0, if x x 0, x x f 0 x) = 0 y 0 x 0, if x 0 x y 0, 1, if x y 0, we obtain that Ef 0, f 0 ) = 1 2 K f 0x) 2 dx = 1 2 y 0 x 0 1, and then use the definition 2.8) below cf. [18, Sect 1.6]). Note that for the higherdimensional case, there does not exist such an elegant estimate for R as in 1.2). The second problem above is also clear for the classical case: if K is an open domain of R n n 1) and E is as in 1.1), then the domain D of E is just W 1,2 K), the usual Sobolev space on K. However, for the fractal case, the above two problems are non-trivial. Recall that for the problem 1), if K is a nested fractal, there exists a geodesic metric d on K, and Barlow [2, Lemma 8.17] obtained the relationship between R and d as follows Rx, y) dx, y) θ, where θ = log ρ/ log γ, and ρ, γ are the resistance and shortest path scaling factors, respectively. If K is a Sirepinski gasket in R 2, Strichartz [18, Sect 1.6] obtained a relationship between R and the Euclidean metric 1.4) Rx, y) x y dw d f, where d f = ln 3 and d ln 2 w = ln 5 are, respectively, the Hausdorff and walk dimensions ln 2 of the Sierpinski gasket. Under certain mild conditions, we shall obtain in this paper the relationship between R and the Euclidean metric for p.c.f. fractals with regular harmonic structures, see Section 3. If the harmonic structure is not regular, then Rx, y) may be infinite for some points x and y, but x y < for any x, y K since K is bounded. So R can not be controlled from above by the Euclidean metric. Therefore, the estimate 1.4) fails.) In particular, we show that 1.4) holds with different exponents for a certain class of nested fractals. One may use the heat kernel estimates in [12] to derive 1.4) for some nested fractals. This is another story.) As for the problem 2), the first result was obtained by Jonsson [9] for the Sierpinski gasket K in R n. It was shown that the domain D of the energy form E is equivalent to a Sobolev-type space on K, that is, 1.5) D W β/2,2 µ) := {f L 2 K, µ) : W β/2,2 f) < },

3 DOMAINS OF DIRICHLET FORMS AND EFFECTIVE RESISTANCE ESTIMATES 3 where µ is the α := logn+1) log 2 is the walk dimension, and 1.6) W β/2,2 f) = sup r α+β) 0<r<1 K -dimensional Hausdorff measure on K, and β = logn+3) log 2 Bx,r) fy) fx) 2 dµy)dµx). Here Bx, r) = {y K : y x < r} is a ball of radius r and center x in K under the Euclidean metric. Note that Jonsson used a different notion Lipβ/2, 2, )K, µ) to denote the space W β/2,2 µ).) Pietruska-Pa luba generalized Jonsson s result to a certain class of nested fractals in R n, see [16]. On the other hand, one can characterize the domain D of the energy E with the help of heat kernel estimates. Assume that the heat kernel or the transition density) pt, x, y) exists on K, and satisfies 1.7) t α/β Φ 1 t 1/β x y ) pt, x, y) t α/β Φ 2 t 1/β x y ) for all x, y K and 0 < t < 1, where α, β > 0 and Φ i 0 is continuous and decreasing on [0, ) for i = 1, 2. Under certain mild assumptions on Φ 1 and Φ 2, one can obtain that the domain D of the Dirichlet form E, D) associated with the heat kernel pt, x, y) is equivalent to W β/2,2 µ), where µ is the α-measure on K that is, µbx, r)) r α ), see [17] for the Euclidean case and [4] for metric spaces. However, it is much complicated to obtain heat kernel estimates like 1.7), see [7, 13] for p.c.f fractals with regular harmonic structures. In Section 4, we characterize the domain D of the energy E on p.c.f. fractals with regular harmonic structures. We avoid using the heat kernel estimates, and present a direct proof. We do follow the technique in [9], but there are some new twists in our proof. The effective metric R and the self-similar measure µ with the standard weights will be used in defining the functions spaces W β/2,2 µ). We mention in passing here that a closely related problem was studies in [11], where the domain of the Laplacian on p.c.f self-similar sets was characterized. Notation. The constants in this paper sometimes change from line to line while they are all denoted by a single letter c. The integers M, M i and constants c i are fixed for i 0. For two non-negative functions f, g, by f g we mean that there is some c > 0 such that c 1 f g c f. 2. Preliminaries 2.1. p.c.f. fractals and Dirichlet forms. We first recall the concept of p.c.f. fractals introduced by Kigami [10]. Let M 2 be an integer, and set S = {1, 2,, M}. Let W = m 0 Sm be the collection of all finite words. Let X, d) be a complete metric space, and let {F i } M i=1 be a family of strict contractions on X, d). Then there exists a unique non-empty

4 4 J.X. HU AND X.S. WANG compact subset K of X such that 2.1) K = M F i K), i=1 see [8] or [3]. For any word w = i 1 i 2 i m S m, any sequence of positive numbers {p i } M i=1, and any function f : K R, denote by w = m the length of w, and set F w = F i1 F i2 F im, p w = p i1 p i2 p im, f w = f F w. K w = F w K), For the empty word w, set p w = 1 and F w = id. Denote by Bx 0, r) = {y K : y x 0 < r} for x 0 K and r > 0. Define a continuous surjection π : S N K by {πw)} = m 1 F i1 i m K) for any infinite word w = i 1 i 2 i m S N. Let C = i j K i K j ), Γ = π 1 C), P = n 1 σ n Γ), where σ : S N S N is the shift map defined by σi 1 i 2 i 3 ) = i 2 i 3. If P is finite, the triple K, S, {F i } i S ) is termed a post-critically finite self-similar set, see [10, Definition , p.23]. Let V 0 = πp), V m = 2.2) F w V 0 ) m 1), V = V m. w S m m 0 If K, {F i } M i=1) is a p.c.f. fractal, then V m V m+1 m 0). From now on we assume that K, {F i } M i=1), or simply K, is a p.c.f. fractal. We now recall how to construct a Dirichlet form on a p.c.f. fractal K. Let V 0 be as in 2.2), and l V 0 ) = {f f : V 0 R} be the collection of all real functions on V 0. Let H be a Laplace matrix, or simply a Laplace, on V 0, that is H satisfies that, for any f, g lv 0 ), H pq = H qp 0 for any p q V 0, where H pq are the entries of H; f, Hg) := ) p V 0 fp) q V H 0 pqgq) 0; Hf = 0 if and only if f is constant.

5 DOMAINS OF DIRICHLET FORMS AND EFFECTIVE RESISTANCE ESTIMATES 5 Given a Laplace H on V 0, and a family of positive numbers r = {r i } M i=1, we define an energy form E m on V m for m 0 by 2.3) E 0 f, g) = f, Hg), E m f, g) = w S m r 1 w E 0 f w, g w ) m 1), for f, g : V m R. In the sequel, we write E m f) := E m f, f) for simplicity. If there exists a pair H, r) such that the following variational problem 2.4) min{e 1 g) : g V0 = f} = E 0 f) is solvable for any f lv 0 ), then we say that K possesses a harmonic structure. If in addition r i < 1 for all 1 i M, then the harmonic structure is said to be regular, see [10, Definition , p.69]. Form now on we assume that K possesses a regular harmonic structure H, {r i } M i=1). Note that 2.4) implies that the sequence {E m f)} m 0 is non-decreasing in m for any f : V R. Let 2.5) 2.6) Ef) := lim E mf), m D = {f CK) : Ef) < }, where CK) is the space of all continuous functions on K. It is known that E, D) defined as in 2.5) and 2.6) is a local, regular, irreducible Dirichlet form on L 2 K, µ) for any Borel measure µ which charges every set of the form K w for w S m, see [2, Theorem 7.14, p.99] or [10, Theorem 3.4.6, p.92]. Clearly E is self-similar: for any f D, we have that f F i D for each i, and 2.7) Ef) = i S r 1 i Ef F i ). We call {r i } M i=1 the weights of the energy E. In order to characterize the domain D of the form E, D), we need the effective resistance R on K. Let R : K K [0, ] be defined by Rx, x) = 0 for x K, and 2.8) Rx, y) 1 = inf{ef) : fx) = 0, fy) = 1} for any x y K. Note that 2.8) is equivalent to { } fx) fy) 2 2.9) Rx, y) = sup : Ef) > 0. Ef) for x y K. It turns out that R is a metric on K, and the topology induced by R is equal to the original topology on K, see [10, Theorem 3.3.4, p.85] or [2, Proposition 7.18, p.101].

6 6 J.X. HU AND X.S. WANG 2.2. Partition. The partition will be useful in our analysis the idea of partition on p.c.f. goes back to Hambly [6]). Let a = {a i } M i=1 be a family of numbers with 0 < a i < 1 for each i. For 0 < λ < 1, define 2.10) Λ a λ) = {w = i 1 i 2 i m : a w λ < a i1 a i2 a im 1 } with the convention that a = 1. We call Λ a λ) the partition with respect to a and λ. Note that the set Λ a λ) is finite; this is easily seen since 0 < λ < 1 and 0 < a i < 1 for each i. For simplicity, denote by τ w for τ, w Λ r λ) if K τ K w. For x, y K, we write x y, if x, y K w for some w Λ a λ). Clearly, by 2.1) and 2.7), we see that, for any partition Λ a λ), K = K w, 2.11) Ef) = w Λ aλ) w Λ aλ) For f : V R and 0 < λ < 1, define 2.12) E λ f) := r 1 w Ef w ) f D). r 1 w E 0 f w ). Proposition 2.1. Let K, {F i } M i=1) be a p.c.f. fractal with a regular harmonic structure H, {r i } M i=1). Then {Eλ f)} is increasing as λ 0 for any f, and 2.13) lim λ 0 E λ f) = Ef), f D. Proof. Let 0 < λ 1 < λ 2 < 1. Then we have two partitions Λ r λ 1 ) and Λ r λ 2 ). And Λ r λ 2 ) is a father of Λ r λ 1 ), that is, any word w Λ r λ 1 ) can be written as w = τw with τ Λ r λ 2 ); w W being possibly an empty word. Indeed, let w = i 1 i 2 i m Λ r λ 1 )\Λ r λ 2 ) m 1). Then we have λ 2 r i1 r im 1 ; otherwise we would see that r i1 r im λ 1 < λ 2 < r i1 r im 1, and so w = i 1 i 2 i m Λ r λ 2 ) by the definition, a contradiction. Let 1 k m 1 be an integer such that r i1 r ik λ 2 < r i1 r ik 1. This implies τ := i 1 i 2 i k Λ r λ 2 ). Setting w := i k+1 i m, we see that w = τw with τ Λ r λ 2 ). This shows that Λ r λ 2 ) is a father of Λ r λ 1 ). Therefore, for f D, E λ1 f) = rw 1 E 0 f w ) rτ 1 E 0 f τ ) = E λ2 f), w Λ rλ 1 ) τ Λ rλ 2 )

7 DOMAINS OF DIRICHLET FORMS AND EFFECTIVE RESISTANCE ESTIMATES 7 proving that {E λ f)} is decreasing in λ for any f. Here, the above inequality follows from both the harmonic structure and the post-critical finiteness of K. Finally, for 0 < λ < 1, letting we see that S m 1 Thus m 1 = m 1 r, λ) = min { w : w Λ r λ)}, m 2 = m 2 r, λ) = max { w : w Λ r λ)}, is a father of Λ r λ), and Λ r λ) a father of S m 2. Hence, E m1 f) E λ f) E m2 f) f D). Ef) = lim E m 1 f) lim E λ f) lim E m 2 f) = Ef). m 1 λ 0 m 2 This finishes the proof. 3. Effective resistance estimates In this section we will give two-sided estimates of the effective resistance R in terms of the Euclidean metric. Two exponents appearing in the two-sided estimates of R are calculated for some nested fractals and some non-nested fractals. Theorem 3.1. Let K, {F i } M i=1) be a p.c.f. fractal in R n with a regular harmonic structure H, {r i } M i=1). Assume that si < 1 is the contraction ratio of F i, that is F i x) F i y) s i x y for x, y R n. Then there exists some c > 0 such that, for all x, y K, 3.1) c 1 x y α 1 Rx, y) { ln r where α 1 = max i 1 i M ln s i }. Proof. Let x 0 y 0 K. Without loss of generality, assume that Rx 0, y 0 ) < 2c 1 ) 1 where c 1 > 0 will be determined below. Set λ = 2c 1 Rx 0, y 0 ) < 1. Then Λ r λ) is a partition. There exist two words w 1, w 2 Λ r λ) such that x 0 K w1 and y 0 K w2. We claim that K w1 K w2. Otherwise there would exist a Λ r λ)-harmonic function f satisfying 3.2) f Vw1 = 1 and f Vλ \V w1 = 0, where V w := F w V 0 ) for w W, and V λ = w Λrλ)F w V 0 ). We say that a function f on K is Λ r λ)-harmonic if Ef, ϕ) = 0

8 8 J.X. HU AND X.S. WANG for any ϕ D with ϕ Vλ = 0.) Note that fx 0 ) = 1 and fy 0 ) = 0. Since f is Λ r λ)-harmonic, we have that Ef) = E λ f) = rw 1 E 0 f w ) 3.3) = r 1 w 1 2 p,q V 0 H pq ff w p)) ff w q))) 2 Using 3.2), we see that the right-hand side of 3.3) is actually equal to the summation of the terms r 1 w H pq ff w p)) ff w q))) 2 = r 1 w H pq, with w w 1 ) being taken over all the words in Λ r λ) with w w 1, and p and q being taken over V 0 such that F w p) V w1 so that ff w p)) = 1 and ff w q)) = 0); all the other terms are equal to zero. Hence, noting that r w λ r min, the right-hand side of 3.3) is bounded by H max M 0 M 0 1)r 1 w H max rmin ) 1M0 M 0 1)λ 1 := c 1 λ 1, where H max = max p q V0 {H pq }, M 0 = #V 0 ) and r min = min{r i }. Therefore, by 2.8), it follows that Rx 0, y 0 ) 1 c 1 λ 1, and so Rx 0, y 0 ) c 1 1 λ = 2Rx 0, y 0 ), yielding a contradiction. So the claim holds. Now let z 0 K w1 K w2. Since x 0, z 0 K w1, we see that, writing x 0 = F w1 x 0) and z 0 = F w1 z 0) for some x 0, z 0 K, x 0 z 0 = F w1 x 0) F w1 z 0) s w1 diamk) r w1 ) 1/α 1 diamk) λ 1/α 1 diamk). Similarly, noting that y 0, z 0 K w2, we see that Therefore, y 0 z 0 λ 1/α 1 diamk). x 0 y 0 x 0 z 0 + z 0 y 0 2λ 1/α 1 diamk) = c Rx 0, y 0 ) 1/α 1, giving that Rx 0, y 0 ) c 1 x 0 y 0 α 1. In order to bound R from above, we need the following separation property: C1): There exist a family of numbers b = {b i } M i=1 with 0 < b i < 1 for every i, and a constant c 2 > 0 such that, for any 0 < λ < 1, distk w, K τ ) c 2 λ, ).

9 DOMAINS OF DIRICHLET FORMS AND EFFECTIVE RESISTANCE ESTIMATES 9 if K w K τ = for w, τ Λ b λ). We remark here that c 2 is independent of λ, but may depend on {b i } M i=1. Condition C1) says that any two disjoint components obtained from any partition Λ b λ) with 0 < λ < 1 are apart away by distance c 2 λ. Theorem 3.2. Let K, {F i } M i=1) be a p.c.f. fractal in R n n 1) with a regular harmonic structure H, {r i } M 1=1). Assume that condition C1) holds for some b = {b i } M i=1. Then 3.4) Rx, y) c x y α 2 { } ln r for all x, y K, where α 2 = min i 1 i M ln b i and c > 0. Proof. First note that, for all x, y K, Rx, y) c <, since the harmonic structure is regular, see [10, Theorem 3.3.4, p. 85]. This implies that fx) fy) 2 Rx, y) Ef) c Ef) for any f D. In particular, for x, y K w w W ), writing x = F w x ) and y = F w y ) for some x, y K, we have that 3.5) fx) fy) 2 = f w x ) f w y ) 2 c Ef w ). Now let x 0 y 0 K. Without loss of generality, we assume that x 0 y 0 < c 2 2, where c 2 is the same as in condition C1). Let λ = 2 c 2 x 0 y 0 < 1. There are two words w 1, w 2 Λ b λ) such that x 0 K w1 and y 0 K w2. Then K w1 K w2 ; otherwise, we would have from condition C1) that x 0 y 0 distk w1, K w2 ) c 2 λ = 2 x 0 y 0, a contradiction. Let z 0 K w1 K w2. For f D, using the fact that x 0, z 0 K w1, we see from 3.5) and 2.11) that 3.6) fx 0 ) fz 0 ) 2 c Ef w1 ) = c r w1 r w1 ) 1 Ef w1 ) Similarly, since z 0, y 0 K w2, we have that c r w1 Ef) c b w ) α 2 Ef) c λ α 2 Ef). fz 0 ) fy 0 ) 2 c λ α 2 Ef).

10 10 J.X. HU AND X.S. WANG Therefore, fx 0 ) fy 0 ) 2 2 fx 0 ) fz 0 ) 2 + fz 0 ) fy 0 ) 2) c λ α 2 Ef) = c x 0 y 0 α 2 Ef), which gives that Rx 0, y 0 ) c x 0 y 0 α 2. Thus 3.4) follows. Condition C1) may be replaced by the following connectivity property: C2): There exist b } M = { bi i=1 with 0 < b i < 1 for each i, and a small) constant c 3 > 0 and an integer M 1 such that, for any 0 < λ < 1 and for any x 0 K, any point y Bx 0, c 3 λ) can be connected to x 0 by a sequence of points {x k } n 0 k=0 in K with 1 n 0 M 1, where x n0 = y and x k 1 x k for 1 k n 0. For a partition Λ bλ) with 0 < λ < 1, the condition C2) means that any point y in any ball B x 0, c 3 λ) can be connected to its center x 0 by at most M 1 components obtained from the partition Λ bλ). Theorem 3.3. Let K, {F i } M i=1) be a p.c.f. fractal in R n n 1) with a regular harmonic structure H, {r i } 1=1) M. Assume that condition C2) holds for some b = } M { bi. Then, for all x, y K, i=1 3.7) Rx, y) c x y α 3, { } ln r where α 3 = min i 1 i M and c > 0. ln b i Proof. Let x 0 y 0 K. Without loss of generality, we assume that x 0 y 0 < c 3 2 where c 3 is the same as in C2). Set λ := 2c 1 3 x 0 y 0. Let Λ bλ) be the partition with respect to b and λ. Note that y 0 Bx 0, c 3 λ). Then, by condition C2), there exists a sequence of points {x k } n 0 k=0 with 1 n 0 M 1 satisfying that x n0 = y, and x k 1, x k K wk for some w k Λ bλ) k = 1,, n 0 ). For f D, as in 3.6), we have that ) fx k ) fx k 1 ) 2 α3 c r wk Ef) c bwk Ef) cλ α 3 Ef), k = 1,, n 0. Therefore, fx 0 ) fy 0 ) 2 = n0 ) 2 n 0 fx k ) fx k 1 )) n 0 fx k ) fx k 1 )) 2 k=1 c M 2 1 λ α 3 Ef) = c x 0 y 0 α 3 Ef), k=1

11 DOMAINS OF DIRICHLET FORMS AND EFFECTIVE RESISTANCE ESTIMATES 11 which implies that Thus 3.7) follows. Rx 0, y 0 ) c x 0 y 0 α 3. We remark that Theorem 3.2 or 3.3 implies the Morrey-Sobolev embedding of the function space D: 3.8) fx) fy) c x y β Ef) for all x, y K and all f D, for some c, β > 0. We now give some examples of p.c.f. fractals where condition C1) or C2) holds so that Theorem 3.2 or 3.3 is true. Nested fractals. The nested fractal K, {F i } M i=1) was introduced by Lindstrøm [14]. It belongs to the class of p.c.f. fractals in R n with the same contraction ratio 0 < ρ < 1, that is F i x) F i y) = ρ x y x, y R n ). It is known that K possesses a regular harmonic structure H, {r i } M i=1) with ri := r < 1 for 1 i M, see for example [15]. And condition C2) holds for b i = ρ for a certain class of nested fractals, see [12, Lemma 5.4]. Thus, by Theorems 3.1 and 3.3, we see that there exists some c > 0 such that, for all x, y K, c 1 x y ln r ln ρ Rx, y) c x y ln r ln ρ. As a typical representative of nested fractals, the Sierpinski gasket K in R n admits an effective resistance R satisfying c 1 x y ln n+1) n+3 ln 2 Rx, y) c x y ln n+3 n+1) ln 2, by taking r i = n+1 in constructing the Dirichlet form. Note that the exponent n+3 ln ) n+3 n+1 lnn + 3) lnn + 1) = := d w d f, ln 2 ln 2 ln 2 is the difference between the walk dimension d w and Hausdorff dimension d f of the Sierpinski gasket, see also [18] for n = 2. Vicsek sets. Let p 1 = 0, 0), p 2 = 1, 0), p 3 = 1, 1), p 4 = 0, 1), p 5 = 1 2, 1 ) 2 be the four corners and center of the unit square in the plane. Define F i x) = 1 4 x p i) + p i 1 i 4), F 5 = 1 2 x p 5) + p 5 x R 2 ). The Vicsek set K is determined by K = 5 i=1 F ik). It is a p.c.f. fractal but not a nested fractal, and the boundary is V 0 = {p 1, p 2, p 3, p 4 }.

12 12 J.X. HU AND X.S. WANG Let b = {b i } 5 i=1 where b i = 1 each i. Then the Vicsek set satisfies the condition 4 C1). Indeed, for 0 < λ < 1, let m 1 be an integer such that 4 m λ < 4 m 1). Then Λ b λ) = S m, and distk w, K τ ) m 1) > 1 2 λ if K w K τ = for w, τ S m. It is not hard to construct a regular harmonic structure H, {r i } 5 i=1) on the Vicsek set. In fact, let H = , and r 1 = r 2 = r 3 = r 4 = r) and r 5 = r with 0 < r < 1. One can verify that H, {r i } 5 i=1) is a regular harmonic structure on K for any 0 < r < 1. Thus we see from Theorems 3.1 and 3.2 that, for all x, y K, 3.9) { where α 1 = max c 1 x y α 1 Rx, y) c x y α 2, } {, ln r and α ln 2 2 = min ln r)) ln 4 ln r)) ln 4 }, ln r. ln 4 Note that the effective resistance R here can not be controlled by any powered Euclidean metric, that is, the following 3.10) Rx, y) x y θ x, y K) fails for any θ > 0. In fact, let m 1 be any integer and set w 1 = 11 1 S m. Choose a family of points {x m, y m )} m 1 in K, where x m = 0, 0) = F w1 p 1 ) and y m = 4 m, 0 ) = F w1 p 2 ). Clearly x m, y m ) F w1 V 0 ) with x m y m = 4 m. Let f be the S m -harmonic function on K satisfying fx m ) = 1 and f Vm\{x m} = 0. Then we have which gives that Ef) = E m f) = r) ) m, Rx m, y m ) 1 = inf {Eu) : u D and ux m ) = 1, uy m ) = 0} Ef) = r) ) m = 3 4 mθ 1 = 3 x m y m θ 1, where θ 1 = ln r)). Therefore, Rx ln 4 m, y m ) 3 1 x m y m θ 1. On the other hand, for any u D, we see that Eu) E m u) r 1 w 1 ux m ) uy m )) 2 = r) ) m uxm ) uy m )) 2 = x m y m θ 1 ux m ) uy m )) 2,

13 DOMAINS OF DIRICHLET FORMS AND EFFECTIVE RESISTANCE ESTIMATES 13 which implies that Rx m, y m ) x m y m θ 1 by using 2.9). Therefore, 3.11) 3 1 x m y m θ 1 Rx m, y m ) x m y m θ 1. Similarly, let w 2 = 55 5 S m, and take x m = F w2 p 1 ) and y m = F w2 p 2 ). Clearly x m y m = 2 m. By the same calculation as above, we can obtain that 3.12) 1 r 31 + r) x m y m θ 2 R x m, y m) x m y m θ 2, where θ 2 = ln r ln 2. If r 1 2, we see that θ 1 θ 2. It follows from 3.11) and 3.12) that 3.10) can not hold for any θ > 0, provided that r Domains of Dirichlet forms Let K, {F i } M i=1) be a p.c.f. fractal with a regular harmonic structure H, {r i } M i=1), and let E, D) be the associated Dirichlet form defined as in 2.5) and 2.6). In this section, we give a characterization of the domain D. In order to characterize D, we need to introduce a measure µ. We choose µ to be the normalized self-similar measure with the standard weights {p i } M i=1, that is 4.1) where p i = r α i, with α given by µ = M i=1 p i µ F 1 i 4.2) For any w τ W, we have that M ri α = 1. i=1 4.3) µk w ) = r w ) α and µk w K τ ) = 0. Observe that there exists two constants 0 < c 4 c 5 independent of x and λ such that for any x K and 0 < λ < 1, 4.4) B R x, c 4 λ) N λ x) B R x, c 5 λ), where B R x, λ) = {y K : Ry, x) < λ} is a ball under the metric R, and N λ x) = {K w : x K w and w Λ r λ)}, is the union of all components K w w Λ r λ)) to which x belongs. Indeed, the first inclusion in 4.4) follows since, for x K and y / N λ x), one can find a function f such that fx) = 1 and fy) = 0, and Ef) c 4 λ) 1 for some c 4 > 0, see the proof of Theorem 3.1. So Rx, y) c 4 λ by using 2.8). Therefore B R x, c 4 λ) N λ x). The second inclusion follows from [2], Prop. 8.9, p.110).

14 14 J.X. HU AND X.S. WANG Theorem ) 4.1. Let K, {F i } M i=1) be a p.c.f. fractal with a regular harmonic structure H, {ri } M i=1, and let E, D) be the associated Dirichlet form defined as in 2.5) and 2.6). Let µ be a self-similar measure with standard weights. Then there exists some c > 0 such that 4.5) c 1 W α f) Ef) c W α f) for all f CK), where 4.6) W α f) := sup λ 2α+1) 0<λ<1 K B R x,c 4 λ) fx) fy) 2 dµy)dµx), and the constants α and c 4 are the same as in 4.2) and in 4.4), respectively. In particular, we have that D = {f CK) : W α f) < }. We decompose Theorem 4.1 into Lemmas 4.3 and 4.4 below. In order to prove Lemma 4.3, we need the following proposition. Proposition 4.2. Let K, {F i } i=1) M, E, D) and µ be as in Theorem 4.1. Then, for 0 < λ < 1 and f CK), 4.7) fx) f w x 0 ) 2 dµx) c λ α+1 Ef). K w where x 0 is any point in V 0, and c is independent of λ, f and x 0. Proof. The proof is motivated by [5]. Without loss of generality, assume that f D and x 0 V 0. Since Λ r λ) is a partition, we see that the set { wτ : w Λr λ) and τ S k} is also a partition for any k 1. Therefore, for µ-almost all x K, there is exactly one τ S k such that x K wτ. We define f k x) := f wτ x 0 ) if x K wτ. Obviously the function f k is defined µ-almost everywhere on K, and is constant on each component of the form K wτ where w Λ r λ) and τ S k. Since f is continuous, we see that f k x) fx) for µ-almost all x K as k. In order to derive 4.7), it is enough to show that 4.8) f k x) f w x 0 ) 2 dµx) c λ α+1 Ef). K w In fact, if 4.8) holds, letting k in 4.8) and using the dominated convergence theorem, we then obtain 4.7). Fix w Λ r λ) and τ := i 1 i 2 i k for k 1 temporally. Let x l = F wi1 i 2 i l x 0 ), 1 l k.

15 Note that DOMAINS OF DIRICHLET FORMS AND EFFECTIVE RESISTANCE ESTIMATES ) fx k ) fx 0 )) 2 = k 1 k 1 ) 2 a 1/2 l a 1/2 l fx l+1 ) fx l )) ) k 1 a 1 l a l fx l+1 ) fx l )) 2 ) k 1 c a l fx l+1 ) fx l )) 2, where {a l } is a sequence of positive numbers satisfying a 1 l <, which will be specified later on. Observing that we see from 4.9) that fx l+1 ) fx l )) 2 = f wi1 i l F il+1 x 0 )) f wi1 i l x 0 ) ) 2 c E 1 f wi1 i l ) c Ef wi1 i l ), k 1 fx k ) fx 0 )) 2 c a l Ef wi1 i l ). Therefore, using the fact that µk wτ ) = r w ) α µk τ ) λ α µk τ ), K wτ f k x) f w x 0 )) 2 dµx) = µk wτ ) fx k ) fx 0 )) 2 k 1 c λ α µk i1 i 2 i k ) a l Ef wi1 i l ) for any w Λ r λ) and any τ := i 1 i 2 i k S k k 1). Hence, 4.10) K w f k x) f w x 0 )) 2 dµx) = τ S k K wτ f k x) f w x 0 )) 2 dµx) c λ k 1 α µk i1 i 2 i k ) a l Ef wi1 i l ) i 1,,i k k 1 c λ α a l i 1,,i l Ef wi1 i l ). In the last inequality above, we have exchanged the order of the summations, and then used the fact that i l+1,,i k S µk i l+1 i k ) = 1 and µk i1 i l ) 1 l 1). On

16 16 J.X. HU AND X.S. WANG the other hand, we have that, for any l 0, Ef wi1 i l ) = r wi1 i l ) r wi1 i l ) 1 Ef wi1 i l ) i 1,,i l i 1,,i l 4.11) λ r max ) l = λ r max ) l Ef), i 1,,i l r wi1 i l ) 1 Ef wi1 i l ) since r wi1 i l = r w r i1 i l λ r max ) l, where r max := max i {r i } < 1. Therefore, we obtain from 4.10) and 4.11) that k 1 ) f k x) f w x 0 )) 2 dµx) c λ α a l Ef wi1 i l ) K w i 1,,i l c λ α+1 Ef) a l r max ) l c λ α+1 Ef), where we have chosen a l := r max ) l/2 that satisfies a 1 l <. Thus 4.8) follows. This finishes the proof. Lemma 4.3. Let K, {F i } M i=1), E, D) and µ be as in Theorem 4.1. Then there exists some c > 0 such that, for all f CK), 4.12) where W α f) is defined as in 4.6). W α f) c Ef) Proof. Assume that f D; otherwise 4.12) automatically holds. Let 0 < λ < 1. Note that µk w K τ ) = 0 for any distinct w, τ Λ r λ). Set I λ f) := fx) fy) 2 dµy)dµx). Then we have that, using 4.4), I λ f) = K B R x,c 4 λ) K w τ w B R x,c 4 λ) K w fx) fy) 2 dµy)dµx) K τ fx) fy) 2 dµy)dµx). For x K w, y K τ, let z 0 K w K τ = F w V 0 ) F τ V 0 ) if w = τ, we simply take any point z 0 F w V 0 ) and run the same proof as below. So we only consider the case w τ). Using the elementary inequality fx) fy) 2 2 fx) fz 0 ) 2 + fz 0 ) fy) 2 ),

17 DOMAINS OF DIRICHLET FORMS AND EFFECTIVE RESISTANCE ESTIMATES 17 and the fact that {τ : τ w} M 2 for an integer M 2 independent of w and λ cf. [10, Lemma 4.2.3, p.139]), we obtain that I λ f) c λ α fx) fz 0 ) 2 dµx). K w w Λrλ) z 0 FwV 0 ) Let z 0 = F w x 0 ) for some x 0 V 0. By 4.7) and the fact that V 0 <, it immediately follows that I λ f) c λ 2α+1 Ef), proving 4.12). Lemma 4.4. Let K, {F i } M i=1), E, D) and µ be as in Theorem 4.1. Then 4.13) for all f CK), where c > 0. Ef) c W α f) Proof. Let 0 < λ < c 4 /c 5 1. Let f CK). Without loss of generality, we assume that W α f) <. We have that E λ f) = ) ) rw 1 E 0 f w ) c rw ) fp) fq). p,q F wv 0 ) Noting that, for any x 0 K w, fp) fq) 2 2 fp) fx 0 )) 2 + fx 0 ) fq)) 2), we see that fp) fq) 2 2 fp) fx0 ) ) 2 dµx0 ) + µk w ) K w Hence, 4.15) p,q F wv 0 ) fp) fq) ) 2 M3 p F wv 0 ) fx0 ) fq) ) ) 2 dµx0 ). K w 1 fp) fx0 ) ) 2 dµx0 ), µk w ) K w where M 3 = 4#F w V 0 )) = 4#V 0 ). Let w Λ r λ) and p F w V 0 ) be fixed. We now estimate the integral K w fp) fx0 ) ) 2 dµx0 ). Let 0 < a < 1 be any fixed number for example, a = 1 ). For each integer l 0, let 2 Λλa l ) := Λ r λa l ) be a partition. We choose a sequence of subsets of K w : K w, K wτ1, K wτ2,, K wτl, such that wτ l Λλa l ) and p K wτl for each l 0. Note that, for any l > i 0, K wτl K wτi ;

18 18 J.X. HU AND X.S. WANG this is because λa l < λa i, and so the partition Λλa i ) is a father of Λλa l ). For simplicity, we denote by Note that, for any x l K l l 0), K 0 = K w and K l = K wτl l 1). fp) fp) fx 0 )) 2 = fxk ) ) k 1 + a 1/2 l a 1/2 l fxl+1 ) fx l ) )) 2 2fp) fx k )) a 1 l ) k 1 where {a l } is a sequence of positive numbers satisfying a 1 l <, a l fxl+1 ) fx l ) ) ) 2, which will be determined below. Integrating the above inequality with respect to each x l K l for 0 l k, and then dividing by µk 0) µk k ), we obtain that 1 fp) fx 0 )) 2 dµx 0 ) 2 µk w ) K w µk k ) fp) fx k )) 2 dµx k ) K k k 1 a l + c µk l+1 )µk l ) fxl+1 ) fx l ) ) ) dµxl+1 )dµx l ). K l Note that the first term on the right-hand side of 4.16) tends to zero as k, since {K k } shrinks to p as k and f is continuous. In order to estimate the second term, we denote by 4.17) A w,k f) := k 1 K l+1 a l µk l+1 )µk l ) fxl+1 ) fx l ) ) 2 dµxl+1 )dµx l ). K l K l+1 By 4.4), we have that K l N λa lx l ) B R x l, c 5 λa l ) for any x l K l and l 0. Using the fact that K l+1 K l K w, we obtain that fxl+1 ) fx l ) ) 2 dµxl+1 )dµx l ) K l K l+1 fxl+1 ) fx l ) ) 2 dµxl+1 )dµx l ). K w B R x l,c 5 λa l ) Note that, using 4.3) and the fact that wτ l Λ r λa l ), µ K l) = r wτl ) α λa l) α for any l 0.

19 DOMAINS OF DIRICHLET FORMS AND EFFECTIVE RESISTANCE ESTIMATES 19 Therefore, it follows from 4.17) that k ) A w,k f) c a l λa l ) 2α K w B R x,c 5 λa l ) fy) fx) ) 2dµy)dµx). Therefore, combining 4.14), 4.15) and 4.16), we obtain that, for any k 0, E λ f) c rw 1 A w,k f) + rw 1 c 4.19) µk k ) fp) fz)) 2 dµz). w Λrλ) p FwV 0 ) On the other hand, noting that r w λ for w Λ r λ), it follows from 4.18) that rw 1 A w,k f) c λ 1 A w,k f) 4.20) k 1 c λ 1 a l λa l ) 2α k 1 { = c a l a l c 5 λa l ) 2α+1) k 1 c W α f) a l a l c W α f). K B R x,c 5 λa l ) K K k fy) fx) ) 2dµy)dµx) B R x,c 5 λa l ) fy) fx) ) 2dµy)dµx) } Here we have chosen a l := a l/2 that satisfies l 0 a 1 l <. Therefore, by 4.19) and 4.20), E λ f) c W α f) + w Λrλ) p FwV 0 ) r 1 w c µk k ) K k fp) fz)) 2 dµz) k 0), where c is independent of k and λ. Letting k, we see that E λ f) c W α f). This gives 4.13) by letting λ 0 and using 2.13). Finally we remark that Theorem 4.1 directly follows from Lemmas 4.3 and 4.4. Acknowledgement. The authors thank the referee for the helpful comments. References [1] R. A. Adams, Sobolev spaces, Academic Press, New York-London, [2] M.T. Barlow, Diffusions on fractals, Lect. Notes Math. 1690, Springer, 1998, [3] K.J. Falconer, Fractal Geometry- Mathematical Foundation and Applications, John Wiley, [4] A. Grigor yan, J. Hu and K.-S. Lau, Heat kernels on metric-measure spaces and an application to semilinear elliptic equations, Trans. Amer. Math. Soc ),

20 20 J.X. HU AND X.S. WANG [5] A. Grigor yan, J. Hu and K.-S. Lau, Equivalence conditions for on-diagonal upper bounds of heat kernels on self-similar spaces, J. Func. Anal ), [6] B.M. Hambly, Brownian motion on a random recursive Sierpinski gasket, Ann. Probab ), [7] B.M. Hambly and T. Kumagai, Transition density estimates for diffusion processes on post critically finite self-similar fractals, Proc. London Math. Soc. 3) ), [8] J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J ), [9] A. Jonsson, Brownian motion on fractals and function spaces, Math. Zeit ), [10] J. Kigami, Analysis on Fractals, Cambridge University Press, [11] J. Kigami, Harmonic analysis for resistance forms, J. Funct. Anal ), [12] T. Kumagai, Estimates of the transition densities for Brownian motion on nested fractals, Probab. Theory and Related Fields, ), [13] T. Kumagai and K.T. Sturm, Construction of diffusion processes on fractals, d-sets, and general metric measure spaces, J. Math. Kyoto Univ ), [14] T. Lindstrøm, Brownian motion on nested fractals, Mem. Amer. Math. Soc ). [15] V. Metz, Renormalization contracts on nested fractals, J. Reine Angew. Math ), [16] K. Pietruska-Pa luba, Some function spaces related to the Brownian motion on simple nested fractals, Stochastics and Stochastics Reports, ), [17] K. Pietruska-Pa luba, On function spaces related to fractional diffusion on d-sets, Stochastics and Stochastics Reports, ), [18] R. Strichartz, Differential equations on fractals: A tutorial, scheduled to be published by Princeton University Press. Department of Mathematical Sciences, Tsinghua University, Beijing , China address: hujiaxin@mail.tsinghua.edu.cn address: wang-xs04@mails.tsinghua.edu.cn