Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 30, 2005, 99–111

Hajłasz–Sobolev type spaces and p-energy on the Sierpinski gasket

Jiaxin Hu, Yuan Ji, and Zhiying Wen

Tsinghua University, Department of Mathematical Sciences Beijing 100084, China JXHu@math.tsinghua.edu.cn; jiyuan00@mails.tsinghua.edu.cn; wenzy@tsinghua.edu.cn

Abstract. We study Hajłasz–Sobolev type spaces on metric spaces that depend on quasidistances; in particular, we may take the quasi-distance to be the power σ of the metric with σ > 1, if the metric space is highly irregular or porous. We take the Sierpinski gasket in R^{2} as an example, and show that the Hajłasz–Sobolev type space is non-trivial for 1 < σ < β_{p}/p with β_{p} characterizing the intrinsic property of the Sierpinski gasket. This work was strongly motivated by [8], and generalizes the result in [9] to any 1 < p < ∞.

1. Hajłasz–Sobolev type spaces

Let F be a non-empty set and d be a metric on F. Let q(x, y) be a quasidistance on F (cf. [14]), that is q: F × F → [0, ∞] satisfies

(1) q(x, y) = 0 if and only if x = y;

(2) q(x, y) = q(y, x) for all x, y ∈ F;

(3) there exists a constant 1 ≤ c_{1} < ∞ such that, for all x, y, z ∈ F,

q(x, y) ≤ c_{1}(q(x, z) + q(z, y)).

Let μ be a Borel measure on the metric space (F, d). Let 1 ≤ p ≤ ∞. We denote by L^{p}(μ) := L^{p}(F, μ) the usual space of all p-integrable real-valued functions on F with respect to μ, with the norm

∥f∥_{p} := (∫_{F} |f(x)|^{p} dμ(x))^{1/p}

(with the obvious modification when p = +∞). Motivated by [5], we say that a function f ∈ L^{p}(μ) belongs to a Hajłasz–Sobolev type space M^{p}(μ), if there exists a non-negative function g ∈ L^{p}(μ), termed an upper gradient of f, such that

(1.1)  |f(x) − f(y)| ≤ q(x, y)(g(x) + g(y))

2000 Mathematics Subject Classification: Primary 43A15; Secondary 46B20.

The first author was supported in part by NSFC, Grant No. 10371062 and the Humboldt Foundation. The third author was supported by the Special Funds for Major State Basic Research Projects of China.

for μ-almost all x, y ∈ F with 0 < q(x, y) < r_{0} and some r_{0} ∈ (0, ∞]. The norm of f ∈ M^{p}(μ) is defined by

∥f∥_{M^{p}(μ)} := ∥f∥_{p} + inf_{g} ∥g∥_{p},

where the infimum is taken for all g satisfying (1.1). It is not hard to see that M^{p}(μ) is a Banach space for 1 ≤ p < ∞ (the proof is similar to that in [5] or [7]). Observe that different values on r_{0} for (1.1) holding give equivalent spaces.

Note that q(x, y) = d(x, y)^{σ} is a quasi-distance on F for any 0 < σ < ∞. The case σ = 1 was addressed in [5], and it was shown that M^{p}(μ) is the usual Sobolev space W^{1,p}(F) if F is an open domain with Lipschitz boundary in R^{n} and μ is the Lebesgue measure. In [9], it was extended to the case σ > 1 when F is a fractal in the Euclidean setting, and was demonstrated that for p = 2, M^{p}(μ) is non-trivial when 1 < σ < \frac{1}{2}β and is trivial when σ > \frac{1}{2}β, if F is the Sierpinski gasket in R^{n}, where β = log(n + 3)/ log 2 is the walk dimension of F (for Hajłasz–Sobolev spaces on fractals, see also [6], [16]). (We say that M^{p}(μ) is trivial if M^{p}(μ) contains only constant functions. In this connection, see [2], [3], [1]. Note that M^{p}(μ) is always trivial if F is an open set in R^{n} and q(x, y) = |x − y|^{σ} with σ > 1, and nothing needs to be discussed under this circumstance. But if F is irregular (e.g. highly porous), the situation is considerably different, and M^{p}(μ) may be non-trivial, see [9] and below.) Whilst in this paper we will generalize the result in [9] to the non-Euclidean setting on one hand, we mainly give an example, on the other hand, that M^{p}(μ) is non-trivial for any 1 < p < ∞ and q(x, y) = d(x, y)^{σ} with σ > 1 in a certain range. We take F to be the Sierpinski gasket in R^{2} . Our example is motivated by [8]. As a by-product, we also answer the question raised in [8] of what is the domain of the p-energy. We thank R. S. Strichartz for sending [8] to our attention.

If q(x, y) = d(x, y)^{σ} (0 < σ < ∞) and μ is a doubling measure, that is μ satisfies, for some c_{2} > 0,

(1.2)  μ(B(x, 2r)) ≤ c_{2} μ(B(x, r))

for all x ∈ F and all 0 < r < ∞, where B(x, r) = {y ∈ F : d(y, x) < r} is a ball in F, then M^{p}(μ) may be characterized as follows: for f ∈ L^{p}(μ) with 1 < p < ∞, we have that f ∈ M^{p}(μ) if and only if f̃ ∈ L^{p}(μ), where

(1.3)  f̃(x) := sup_{0<r<r_{0}} \frac{1}{μ(B(x, r))} ∫_{B(x,r)} \frac{|f(x) − f(y)|}{q(x, y)} dμ(y), x ∈ F,

see also [9] (we always assume that |f(x) − f(y)|/q(x, y) = 0 if x = y). To see this, let f ∈ M^{p}(μ). Then, we have from (1.1) that

\begin{aligned}f̃(x) & ≤ sup_{0<r<r_{0}} \frac{1}{μ(B(x, r))} ∫_{B(x,r)}(g(x) + g(y)) dμ(y) \\ & = g(x) + sup_{0<r<r_{0}} \frac{1}{μ(B(x, r))} ∫_{B(x,r)} g(y) dμ(y) ∈ L^{p}(μ), \end{aligned}

since

(1.4)  M(g)(x) := sup_{0<r<r_{0}} \frac{1}{μ(B(x, r))} ∫_{B(x,r)} g(y) dμ(y) ∈ L^{p}(μ),

due to the doubling condition (1.2) (see for example [7]). Conversely, let f̃ ∈ L^{p}(μ). Fix x, y ∈ F such that 0 < r := d(x, y) < \frac{1}{2}r_{0} . Then we see that, using (1.2),

|f(x) − f(y)| ≤ \frac{1}{μ(B(x, r))} ∫_{B(x,r)}(|f(x) − f(z)| + |f(z) − f(y)|) dμ(z)

≤ \frac{1}{μ(B(x, r))} ∫_{B(x,r)} \frac{r^{σ}}{d(z, x)^{σ}} |f(x) − f(z)| dμ(z)

+ \frac{1}{μ(B(x, r))} ∫_{B(y,2r)} \frac{(2r)^{σ}}{d(z, y)^{σ}} |f(z) − f(y)| dμ(y)

≤ r^{σ}(f̃(x) + \frac{2^{σ}μ(B(y, 2r))}{μ(B(x, r))}f̃(y))

≤ C r^{σ}(f̃(x) + f̃(y))

= C q(x, y)(f̃(x) + f̃(y)),

proving that f ∈ M^{p}(μ) if f̃ ∈ L^{p}(μ). Here and in the sequel, we denote by C the general constant whose value may be different at a different occurrence. The function f̃ defined as in (1.3) is the upper gradient of f (multiple a constant). In what follows we will focus on a class of Hajłasz–Sobolev type spaces where q(x, y) = d(x, y)^{σ} and 1 < σ < ∞, and we denote this space by M_{σ}^{p}(μ).

For 1 ≤ p < ∞ and 0 < σ < ∞, we say that f ∈ Lip(σ, p, ∞)(μ) if f ∈ L^{p}(μ) and

W_{σ,p}(f)^{p} := sup_{0<r<r_{0}} r^{−σp} ∫_{F} \{\frac{1}{μ(B(x, r))} ∫_{B(x,r)} |f(x) − f(y)|^{p} dμ(y)\} dμ(x)

< ∞. (1.5)

The norm of f ∈ Lip(σ, p, ∞)(μ) is defined by

∥f∥_{Lip(σ,p,∞)(μ)} = ∥f∥_{p} + W_{σ,p}(f).

It is easy to see that Lip(σ, p, ∞)(μ) is a Banach space for 1 ≤ p < ∞ and 0 < σ < ∞ (cf. [10], [11]). By (1.1), we see that

M_{σ}^{p}(μ) ⊂ Lip(σ, p, ∞)(μ)

if μ is a doubling measure. The converse is also true if F is a smooth domain in R^{n} and μ is the Lebesgue measure, see [9]. However, if F is irregular, the converse may not be true. But for an α-regular measure μ, the space M_{σ}^{p}(μ) is arbitrarily close to Lip(σ, p, ∞)(μ). We say that a measure is α-regular if there exists a constant c_{3} > 0 such that

(1.6)  c_{3}^{−1}r^{α} ≤ μ(B(x, r)) ≤ c_{3} r^{α}

for all x ∈ F and all 0 < r < r_{0} (some r_{0} > 0). It is not hard to see that if μ is α-regular, then

(1.7)  W_{σ,p}(f)^{p} ≃ sup_{m≥1} 2^{m(α+σp)} ∫_{F} ∫_{B(x,c_{0}2^{−m})} |f(x) − f(y)|^{p} dμ(y) dμ(x),

for any fixed c_{0} > 0.

Proposition 1.1 Let 1 < p < ∞ and 0 < σ < ∞, and let 0 < α < ∞. Assume that μ is α-regular. Then

Lip(σ, p, ∞)(μ) ⊂ M_{σ−θ}^{p}(μ)

for any 0 < θ < σ.

Proof. See [9].

Proposition 1.1 says that M_{σ}^{p}(μ) is slightly smaller than the Besov space Lip(σ, p, ∞)(μ) if μ is α-regular.

2. Examples

In this section we show that M_{σ}^{p}(μ) is non-trivial for any 1 < p < ∞ and σ(> 1) in a certain range, if the underlying metric space is irregular. We take the Sierpinski gasket in R^{2} for an example. The proof is quite technical.

Figure 1.

Let F be the Sierpinski gasket in R^{2} , that is, F is the unique non-empty compact subset of R^{2} determined by

F = ⋃_{i=1}^{3} ϕ_{i}(F),

where ϕ_{i}(x) = \frac{1}{2}(q_{i} + x), x ∈ R^{2} (1 ≤ i ≤ 3), and q_{1,}q_{2,}q_{3} are the three vertices of an equilateral triangle in R^{2} . Alternatively, we may view the Sierpinski gasket F as the closure of V_{∗} = ⋃_{m=1}^{∞} V_{m} under the Euclidean metric, where V_{m} = ⋃_{i=1}^{3} ϕ_{i}(V_{m−1}), m ≥ 1, and V_{0} = {q_{1,}q_{2,}q_{3}}, see Figure 1. For p = 2, Kigami [12] constructed a local regular Dirichlet form on F by using the difference scheme. Jonsson [10] identified the domain of this Dirichlet form with a Besov space. Recently, Herman, Peirone and Strichartz [8] have extended Kigami’s result to the case 1 < p < ∞. Here we briefly describe the main result in [8] that will lead to our example. For 1 < p < ∞, let E_{p}: R^{3} →[0, ∞] be given by

E_{p}(a_{1}, a_{2}, a_{3}) = |a_{1} − a_{2}|^{p} + |a_{2} − a_{3}|^{p} + |a_{3} − a_{1}|^{p}, a_{1}, a_{2}, a_{3} ∈ R,

and define

(2.1)  E_{p}^{(m)}(f) := ∑_{|ω|=m} E_{p}(f(ϕ_{ω}(q_{1})), f(ϕ_{ω}(q_{2})), f(ϕ_{ω}(q_{3}))), m ≥ 1,

for any f: F → R, where the summation is taken over all words ω of length m, and ϕ_{ω}(q_{1}) = ϕ_{i_{1}} ∘ ϕ_{i_{2}} ∘ ⋯ ∘ ϕ_{i_{m}}(q_{1}) for the word ω = i_{1}i_{2} ⋯ i_{m} (i_{k} ∈ {1, 2, 3} for 1 ≤ k ≤ m). Let A_{p}: R^{3} →[0, ∞] be a function satisfying (among other properties)

(2.2)  c_{4}^{−1} E_{p}(a) ≤ A_{p}(a) ≤ c_{4} E_{p}(a)

for some positive constant c_{4} and for all a := (a_{1}, a_{2}, a_{3}) ∈ R^{3} ; in particular, A_{p} solves the renormalization problem: Given a ∈ R^{3} and letting

(2.3)  \begin{aligned}A_{p}^{(2)}(a, b) & := A_{p}(a_{1}, b_{2}, b_{3}) + A_{p}(b_{1}, a_{2}, b_{3}) + A_{p}(b_{1}, b_{2}, a_{3}) \\ & for b = (b_{1}, b_{2}, b_{3}) ∈ R^{3}, \end{aligned}

we have that there exists a number r_{p} such that

(2.4)  min_{b∈R^{3}} A_{p}^{(2)}(a, b) = r_{p}A_{p}(a) for all a ∈ R^{3}.

Such a function A_{p} was shown to exist in [8]. Moreover, the number r_{p} is unique (independent of the choice of A_{p}) and satisfies

(2.5)  2^{1−p} ≤ r_{p} ≤ 2^{p−1}(1 + \sqrt{1 + 2^{3−1/(p−1)}} )^{1−p} < 3 · 2^{−p},

see Lemma 3.8 in [8]. We mention that r_{p} = \frac{3}{5} for p = 2. The function A_{p} may or may not be unique on F; what is important is that the renormalization factor r_{p} is unique which reflects the intrinsic properties of the Sierpinski gasket F. Now, for any f: F → R we define the p-energy ℰ (f) of f on F as the limit of

(2.6)  ℰ_{m}(f) = r_{p}^{−m} ∑_{|ω|=m} A_{p}(f(ϕ_{ω}(q_{1})), f(ϕ_{ω}(q_{2})), f(ϕ_{ω}(q_{3}))), m ≥ 1,

that is,

(2.7)  ℰ (f) = lim_{m→∞} ℰ_{m}(f).

Note that (2.7) makes sense since

{ℰ_{m}(f)}_{m}

is increasing in m for any function f, due to the renormalization problem. Note that

(2.8)  c_{5}^{−1} ℰ_{m}(f) ≤ r_{p}^{−m} E_{p}^{(m)} ≤ c_{5} ℰ_{m}(f)

for all m ≥ 1 and all f: F → R, where c_{5} > 0. Let

(2.9)  𝒟(ℰ) = {f ∈ C(F) : ℰ (f) < ∞},

termed the domain of the p-energy, where C(F) denotes the space of all continuous functions on F with the usual supremum norm. It was shown that 𝒟(ℰ) is dense in C(F), see [8]. The space 𝒟(ℰ) will provide a critical exponent β_{p} := log_{2}(3r_{p}^{−1}) (some r_{p} > 0) that determines whether or not a Hajłasz–Sobolev type space M_{σ}^{p}(μ) (1 < p < ∞) is non-trivial. To see this, we first identify 𝒟(ℰ) with a Besov space.

Theorem 2.1. Let μ be the α := log_{2} 3-dimensional Hausdorff measure on F. Then

(2.10)  W_{β_{p}/p,p}(f)^{p} ≃ ℰ (f)

for all f ∈ C(F), where W_{β_{p}/p,p}(f) is defined as in (1.5) and β_{p} = log_{2}(3r_{p}^{−1}). Thus

(2.11)  𝒟(ℰ) = Lip(β_{p}/p, p, ∞)(μ),

where 𝒟(ℰ) is defined as in (2.9).

Remarks. 1. When p = 2, we have that r_{p} = \frac{3}{5} and so β_{p} = log_{2} 5, the walk dimension of the Sierpinski gasket.

2. If μ is α-regular and σ > α/p, then Lip(σ, p, ∞)(μ) is embedded into the Hölder space with exponent σ − α/p on F, that is,

(2.12)  |f(x) − f(y)| ≤ C |x − y|^{σ−α/p}W_{σ,p}(f)

for all f ∈ Lip(σ, p, ∞)(μ), where C is independent of x, y and f; see for example a direct proof in [4]. Thus

Lip(β_{p}/p, p, ∞)(μ) ⊂ C(F),

since β_{p}/p > α/p (due to r_{p} < 1).

3. By Theorem 2.1, the domain of the p-energy coincides with Lip(β_{p}/p, p, ∞)(μ) if μ is the Hausdorff measure. For other measures this may not be true.

Proof. The proof given here is motivated by [10] (see also [15], [17]) but with some modifications. We first show that

(2.13)  W_{β_{p}/p,p}(f)^{p} ≤ C ℰ (f)

for all f ∈ 𝒟(ℰ). To see this, let f ∈ 𝒟(ℰ). Let (2.14)

f_{k}(x) = \{ \begin{matrix}\frac{1}{3}[f(ϕ_{ω}(q_{1})) + f(ϕ_{ω}(q_{2})) + f(ϕ_{ω}(q_{3}))], & if x ∈ ϕ_{ω}(F)\ϕ_{ω}(V_{0}), \\ f(x), & if x ∈ ϕ_{ω}(V_{0}) \end{matrix}

for |ω| = k (k ≥ 1). Since F is compact and f is continuous on F, the piecewise constant function f_{k} converges to f pointwise on F as k → ∞. If we can show that, for some c_{0} > 0 (e.g. c_{0} = \sqrt{3}/2 ),

(2.15) 2^{(α+β_{p})m} ∫_{F} ∫_{|y−x|<c_{0}2^{−m}} |f_{m+k}(y) − f_{m+k}(x)|^{p} dμ(y) dμ(x) ≤ C ℰ (f)

for all integers m, k ≥ 1, where C is independent of f, then (2.13) follows by letting k → ∞ in (2.15) and using Fatou’s lemma, and (1.7). It remains to prove (2.15). For two words ω and τ with |τ| = |ω| = m, we denote by τ ∼_{m}ω if ϕ_{τ} (F) ∩ ϕ_{ω}(F) ≠ ∅ (we allow that τ = ω). Note that

(2.16) \begin{aligned}I_{m+k}(f) & := ∫_{F} ∫_{|y−x|<c_{0}2^{−m}} |f_{m+k}(y) − f_{m+k}(x)|^{p} dμ(y) dμ(x) \\ & ≤ ∑_{|ω|=m} ∑_{τ∼_{m}ω} ∫_{ϕ_{ω}(F)} ∫_{ϕ_{τ} (F)} |f_{m+k}(y) − f_{m+k}(x)|^{p} dμ(y) dμ(x) \\ & ≤ ∑_{|ω|=m} ∑_{τ∼_{m}ω} ∫_{ϕ_{ω}(F)} ∫_{ϕ_{τ} (F)} 2^{p−1}(|f_{m+k}(y) − f(x_{ω,τ})|^{p} \\ & + |f(x_{ω,τ}) − f_{m+k}(x)|^{p}) dμ(y) dμ(x) \\ & ≤ 8 · 2^{p−1} · 3^{−m} ∑_{|ω|=m} ∫_{ϕ_{ω}(F)} |f_{m+k}(x) − f(x_{ω,τ})|^{p} dμ(x), \end{aligned}

where we have used the fact that μ(ϕ_{ω}(F)) = 3^{−m} and #{τ : τ ∼_{m}ω} ≤ 4 for |ω| = m, m ≥ 1, and where x_{ω,τ} is some point in ϕ_{ω}(V_{0}) (in fact x_{ω,τ} is the unique intersection point of two sets ϕ_{ω}(V_{0}) and ϕ_{τ} (V_{0})). Noting that μ(x) = 0 for any single point x ∈ F, it follows from (2.14) that

∫ϕ_{ω}(F)|f_{m+k}(x) − f(x_{ω,τ})|^{p} dμ(x)

= ∑_{|τ|=k} ∫_{ϕ_{ω·τ} (F)}|\frac{1}{3} ∑_{j=1}^{3} f(ϕ_{ω·τ} (q_{j})) − f(x_{ω,τ})|^{p} dμ(x)

= 3^{−(m+k)} ∑_{|τ|=k}|\frac{1}{3} ∑_{j=1}^{3}(f(ϕ_{ω·τ} (q_{j})) − f(x_{ω,τ}))|^{p}

≤ 3^{−(m+k)−1} ∑_{j=1}^{3} ∑_{|τ|=k}|f(ϕ_{ω·τ} (q_{j})) − f(x_{ω,τ} )|^{p}

which combines with (2.16) to give that (2.17)

I_{m+k}(f) ≤ 2^{p+2} · 3^{−(2m+k)−1} ∑_{j=1}^{3} ∑_{|ω|=m, x_{ω,τ} ∈ϕ_{ω}(V_{0})} ∑_{|τ|=k}|f(ϕ_{ω·τ} (q_{j})) − f(x_{ω,τ})|^{p}.

Let q_{j} and τ := i_{1}i_{2} ⋯ i_{k} be fixed, and set x_{k} = ϕ_{ω·τ} (q_{j}) and x_{0} := x_{ω,τ} = ϕ_{ω}(q_{0}) for some q_{0} ∈ V_{0} . We let x_{l} = ϕ_{ω·i_{1}i_{2}···i_{l}}(q_{0}), 1 ≤ l ≤ k − 1, and obtain a sequence of points {x_{l}}_{l=0}^{k} (some of points may be the same). Repeatedly using the elementary inequality |a+b|^{p} ≤ 2^{p−1}(|a|^{p} +|b|^{p}) for any a, b ∈ R and 1 ≤ p < ∞, we see that

|f(ϕ_{ω·τ} (q_{j})) − f(x_{ω,τ})|^{p} = |f(x_{k}) − f(x_{0})|^{p}

≤ ∑_{l=1}^{k} 2^{(p−1)l}|f(x_{l}) − f(x_{l−1})|^{p}

≤ ∑_{l=1}^{k} 2^{(p−1)l}Ê_{i_{1},...,i_{l−1}}^{(ω,p)}(f),

where

Ê_{i_{1},...,i_{l−1}}^{(ω,p)}(f) = ∑_{i=1}^{3} E_{p}(f∘ϕ_{ω·i_{1}i_{2}···i_{l−1}·i}(q_{1}), f∘ϕ_{ω·i_{1}i_{2}···i_{l−1}·i}(q_{2}), f∘ϕ_{ω·i_{1}i_{2}···i_{l−1}·i}(q_{3})).

Thus

∑_{|τ|=k}|f(ϕ_{ω·τ} (q_{j})) − f(x_{ω,τ})|^{p} ≤ ∑_{i_{1},i_{2},...,i_{k}} ∑_{l=1}^{k} 2^{(p−1)l}Ê_{i_{1},...,i_{l−1}}^{(ω,p)}(f)

= ∑_{l=1}^{k} 2^{(p−1)l} · 3^{k−(l−1)} ∑_{i_{1},...,i_{l−1}} Ê_{i_{1},...,i_{l−1}}^{(ω,p)}(f). (2.18)

Observe that

∑_{|ω|=m} ∑_{i_{1},...,i_{l−1}} Ê_{i_{1},...,i_{l−1}}^{(ω,p)}(f) = ∑_{|ω|=m+l} E_{p}(f(ϕ_{ω}(q_{1})), f(ϕ_{ω}(q_{2})), f(ϕ_{ω}(q_{3})))

= E_{p}^{(m+l)}(f) (2.19)

≤ c_{5} r_{p}^{m+l} ℰ_{m+l}(f)
≤ c_{5} r_{p}^{m+l} ℰ (f)

by using (2.8) and the monotonicity of ℰ_{m}(f) in m. Combining (2.17), (2.18) and (2.19), we have that

I_{m+k}(f) ≤ C 3^{−2m} ∑_{l=1}^{k} 2^{(p−1)l} · 3^{−l}r_{p}^{m+l}ℰ (f)

≤ Cr_{p}^{m} · 3^{−2m}ℰ (f) ∑_{l=1}^{∞} 2^{(p−1)l}3^{−l}r_{p}^{l}

≤ Cr_{p}^{m} · 3^{−2m}ℰ (f) = C 2^{−m(α+β_{p})}ℰ (f), where we have used the fact that

∑_{l=1}^{∞} 2^{(p−1)l}3^{−l}r_{p}^{l} < ∞

since 2^{p−1}3^{−1}r_{p} < 1 by virtue of (2.5). Therefore, (2.15) follows.

We next show that

(2.20)  ℰ (f) ≤ CW_{β_{p}/p,p}(f)^{p}

for all f ∈ Lip(β_{p}/p, p, ∞)(μ). By (2.8), it is sufficient to show that

(2.21)  ℰ_{p}^{(m)}(f) := r_{p}^{−m} ∑_{|ω|=m} ∑_{u,v∈ϕ_{ω}(V_{0})} |f(u) − f(v)|^{p} ≤ CW_{β_{p}/p,p}(f)^{p}

for all f ∈ Lip(β_{p}/p, p, ∞)(μ) and all m ≥ 1. Let f ∈ Lip(β_{p}/p, p, ∞)(μ). By Remark 2 above we see that f is continuous on F. Noting that

|f(u) − f(v)|^{p} ≤ 2^{p−1}(|f(u) − f(x)|^{p} + |f(x) − f(v)|^{p}),

we have that

|f(u) − f(v)|^{p} ≤ \frac{2^{p−1}}{μ(ϕ_{ω}(F))} ∫_{ϕ_{ω}(F)}(|f(u) − f(x)|^{p} + |f(x) − f(v)|^{p}) dμ(x).

It follows from (2.21) that (2.22)

ℰ_{p}^{(m)}(f) ≤ 6 · 2^{p−1} r_{p}^{−m} ∑_{|ω|=m} ∑_{u∈ϕ_{ω}(V_{0})} \frac{1}{μ(ϕ_{ω}(F))} ∫_{ϕ_{ω}(F)} |f(u) − f(x)|^{p} dμ(x).

Now let x ∈ ϕ_{ω}(F) and u ∈ ϕ_{ω}(V_{0}) be fixed. There exists a point p_{0} ∈ V_{0} such that u = ϕ_{ω}(p_{0}). We take i_{0} such that ϕ_{i_{0}}(p_{0}) = p_{0} . Set

S_{0} = ϕ_{ω}(F), S_{1} = ϕ_{ω·\underbrace{i_{0} · i_{0} ⋯ i_{0}}_{k times}}(F), S_{2} = ϕ_{ω·\underbrace{i_{0} · i_{0} ⋯ i_{0}}_{2k times}}(F), …,

where k is an integer to be determined below. It is easy to see that u ∈ S_{j} for each j ≥ 0, and the sequence of the sets {S_{j}} shrinks to the single point u. For each x := x_{ω,τ} ∈ S_{0} , x_{j} ∈ S_{j} and each l ≥ 1,

\begin{aligned}|f(u) − f(x)|^{p} & ≤ 2^{p−1}(|f(u) − f(x_{l})|^{p} + |f(x_{l}) − f(x_{ω,τ})|^{p}) \\ & ≤ 2^{p−1}|f(u) − f(x_{l})|^{p} + ∑_{j=1}^{l} 2^{(p−1)(j+1)}|f(x_{j}) − f(x_{j−1})|^{p}. \end{aligned}

Integrating the above inequality with respect to each x_{j} ∈ S_{j} (0 ≤ j ≤ l) and then dividing by μ(S_{0})μ(S_{1}) ⋯ μ(S_{l}), we obtain that

(2.23) \begin{aligned}\frac{1}{μ(ϕ_{ω}(F))} & ∫_{ϕ_{ω}(F)} |f(u) − f(x)|^{p} dμ(x) ≤ \frac{2^{p−1}}{μ(S_{l})} ∫_{S_{l}} |f(u) − f(x_{l})|^{p} dμ(x_{l}) \\ & + ∑_{j=1}^{l} 2^{(p−1)(j+1)} \frac{1}{μ(S_{j−1})μ(S_{j})} \\ & × ∫_{S_{j−1}} ∫_{S_{j}} |f(x_{j}) − f(x_{j−1})|^{p} dμ(x_{j}) dμ(x_{j−1}). \end{aligned}

Noting that μ(S_{j}) = 3^{−(m+kj)} for each j ≥ 0 and

\begin{aligned}_{∫S_{j−1} ∫S_{j}} & |f(x_{j}) − f(x_{j−1})|^{p} dμ(x_{j}) dμ(x_{j−1}) \\ & ≤ ∫_{S_{0}} ∫_{|ξ−η|≤2^{−(m+(j−1)k)}} |f(ξ) − f(η)|^{p} dμ(ξ) dμ(η), \end{aligned}

we have from (2.22) and (2.23) that

(2.24) \begin{aligned}ℰ_{p}^{(m)}(f) & ≤ 6 · 2^{p−1}r_{p}^{−m} ∑_{|ω|=m} ∑_{u∈ϕ_{ω}(V_{0})}\{\frac{2^{p−1}}{μ(S_{l})} ∫_{S_{l}} |f(u) − f(x_{l})|^{p} dμ(x_{l}) \\ & + ∑_{j=1}^{l} 2^{(p−1)(j+1)}3^{(2m+(2j−1)k)} \\ & × ∫_{ϕ_{ω}(F)} ∫_{|ξ−η|≤2^{−(m+(j−1)k)}} |f(ξ) − f(η)|^{p} dμ(ξ) dμ(η)\}. \end{aligned}

Letting l → ∞, we have that the first term on the right-hand side in (2.24) tends to zero since f is continuous and

\frac{1}{μ(S_{l})} ∫_{S_{l}} |f(u) − f(x_{l})|^{p} dμ(x_{l}) → 0 as l → ∞,

and the second term is less than

Cr_{p}^{−m} ∑_{j=1}^{∞} 2^{(p−1)(j+1)}3^{(2m+(2j−1)k)} ∫_{F} ∫_{|ξ−η|≤c_{0}2^{−(m+jk)}} |f(ξ) − f(η)|^{p} dμ(ξ) dμ(η)

≤ C 3^{2m} r_{p}^{−m} ∑_{j=1}^{∞} 2^{(p−1)j}3^{2jk}2^{−(m+jk)(α+β_{p})}W_{β_{p}/p,p}(f)^{p}

= C 3^{2m}r_{p}^{−m} W_{β_{p}/p,p}(f)^{p} ∑_{j=1}^{∞} 2^{(p−1)j}3^{2jk} (3^{−2}r_{p})^{m+jk} (2.25)

= CW_{β_{p}/p,p}(f)^{p} ∑_{j=1}^{∞} 2^{(p−1)j}r_{p}^{jk},

since 2^{−(α+β_{p})} = r_{p} · 3^{−2} . Since r_{p} < 1, we take k so large that r_{p}^{k} < 2^{−(p−1)} , and so

∑_{j=1}^{∞} 2^{(p−1)j}r_{p}^{jk} < ∞.

Therefore,

ℰ_{p}^{(m)}(f) ≤ CW_{β_{p}/p,p}(f)^{p}

for all f ∈ Lip(β_{p}/p, p, ∞)(μ) and all m ≥ 1, proving (2.21). The other statement is obvious.

Corollary 2.1. Let β_{p} = log_{2}(3r_{p}^{−1}) as above. Then the space Lip(β̄/p, p, ∞)(μ) defined on the Sierpinski gasket in R^{2} contains only constant functions if β̄ > β_{p}.

Proof. By (2.24), (2.25), we see that

ℰ_{p}^{(m)}(f) ≤ CW_{β̄/p,p}(f)^{p}2^{−m(β̄−β_{p})}

for all m ≥ 1 and all f ∈ Lip(β̄/p, p, ∞)(μ). Thus we have that

ℰ (f) = lim_{m→∞} ℰ_{m}(f) ≤ C lim_{m→∞} ℰ_{p}^{(m)}(f) = 0,

giving that f =const.

Theorem 2.2. Let F be the Sierpinski gasket in R^{2} and μ be the αdimensional Hausdorff measure on F, where α = log_{2} 3. Let 1 < p < ∞. Then there exists some r_{p} ∈ [2^{1−p}, 3 · 2^{−p}) such that the Hajłasz–Sobolev space M_{σ}^{p}(μ) is dense in C(F) for all σ < p^{−1} log_{2}(3r_{p}^{−1}); in particular, the space M_{σ}^{p}(μ) is nontrivial if 1 < σ < log_{4} 5 when p = 2. Moreover, M_{σ}^{p}(μ) is trivial if σ ≥ 1 + 1/p.

Proof. By Theorem 2.1, the space Lip(σ, p, ∞)(μ) is dense in C(F) if

0 < σ ≤ \frac{β_{p}}{p} = p^{−1} log_{2}(3r_{p}^{−1}).

Since μ is α-regular, we see from Proposition 1.1 and (2.5) that there exists some r_{p} ∈ [2^{1−p}, 3 · 2^{−p}) such that M_{σ}^{p}(μ) is dense in C(F) if 0 < σ < p^{−1} log_{2}(3r_{p}^{−1}). By Corollary 2.1, the space Lip(σ, p, ∞)(μ) is trivial if σ ≥ 1 + 1/p > β_{p}/p (due to r_{p} ≥ 2^{1−p}). Thus the fact that M_{σ}^{p} ⊂ Lip(σ, p, ∞) implies that M_{σ}^{p}(μ) is also trivial if σ ≥ 1 + 1/p.

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Received 21 July 2003