THE HARDY-LITTLEWOOD MAXIMAL FUNCTION

AND SOBOLEV SPACES ON A METRIC SPACE

Olli Martio

1. Introduction

The first order Sobolev space W^{1,p}(Ω), Ω open in ℝ^{n} and 1 ≤ p ≤ ∞, is based on the space of differentiable functions: The space W^{1,p}(Ω) is the closure of C^{1}(Ω) with respect to the norm

∥u∥_{W^{1,p}(Ω)} = (∥u∥_{L^{p}(Ω)}^{p} + ∥∇u∥_{L^{p}(Ω)}^{p})^{1/p}.

P. Hajłasz [H] introduced the Sobolev spaces W^{1,p}(X) on an arbitrary metric space X with a measure μ. His approach rests on a pointwise inequality

(1.1)  |u(x) − u(y)| ≤ C|x − y|(M(|∇u|)(x) + M(|∇u|)(y)),

where u ∈ W^{1,p}(ℝ^{n}), 1 < p < ∞, C = C(n), and M(f) stands for the Hardy-Littlewood maximal function of f. In this paper we show that this approach allows many variations, which can be used to generate useful first order Sobolev spaces on metric spaces.

In Section 2 we consider the Hardy-Littlewood maximal function M(u). We show that M(u) ∈ W^{1,p}(ℝ^{n}), 1 < p < ∞, provided that u ∈ W^{1,p}(ℝ^{n}). Section 3 is devoted to the pointwise inequality (1.1). Section 4 contains a general approach to the Sobolev space W^{1,p}(X) on an arbitrary metric space (X, d) with a measure μ. The starting point is a subspace of L^{p}(X, μ) with admissible “derivatives”, which take the role of the maximal function M(|∇u|) in (1.1). In Section 5 a capacity on W^{1,p}(X) is considered.

2. The Hardy-Littlewood maximal function

Let u ∈ L_{loc}^{1}(ℝ^{n}). The (Hardy-Littlewood) maximal function of f is defined as

(2.1)  M(u)(x) = sup_{r>0} \frac{1}{m(B(x, r))} ∫_{B(x,r)} |u| dm.

We have the following well-known result, see [S, p. 5]:

2.2. Theorem. (a) |u(x)| ≤ M(u)(x) a.e.

(b) u ∈ L^{p}(ℝ^{n}), 1 ≤ p ≤ ∞ ⇒ M(u)(x) < ∞ a.e.

(c) u ∈ L^{1}(ℝ^{n}) ⇒ m\{x: M(u)(x) > α\} ≤ \frac{A}{α} ∫_{ℝ^{n}} |u| dm, A = A(n).

(d) u ∈ L^{p}(ℝ^{n}), 1 < p ≤ ∞ ⇒ ∥M(u)∥_{L^{p}} ≤ A∥u∥_{L^{p}}, A = A(n, p).

Next we are going to show that M(u) ∈ W^{1,p}(ℝ^{n}) provided that u ∈ W^{1,p}(ℝ^{n}). We need a pair of real-analytic lemmas. The first is an application of the Mazur lemma.

2.3. Lemma. Suppose that (u_{i}) is a bounded sequence in W^{1,p}(ℝ^{n}), 1 < p < ∞, and u_{i} → u a.e. Then u ∈ W^{1,p}(ℝ^{n}) and u_{i} → u, D_{k}u_{i} → D_{k}u weakly in L^{p}(ℝ^{n}).

2.4. Lemma. Let u ∈ W^{1,p}(ℝ^{n}) and let f ∈ L^{∞}(ℝ^{n}) have a compact support. Then u ∗ f ∈ W^{1,p}(ℝ^{n}),

(u ∗ f)(x) = ∫_{ℝ^{n}} u(x − y)f(y) dm(y),

and D_{k}(u ∗ f) = D_{k}u ∗ f.

Proof. Let φ ∈ C_{0}^{∞}(ℝ^{n}). By the Fubini theorem

∫_{ℝ^{n}} φ(x)(D_{k}u ∗ f)(x) dm(x) = ∫_{ℝ^{n}} f(y) ∫_{ℝ^{n}} D_{k}u(x − y)φ(x) dm(x) dm(y)

= − ∫_{ℝ^{n}} (u ∗ f)D_{k}φ dx,

and since u∗f, D_{k}u∗f ∈ L^{p}(ℝ^{n}), this shows that D_{k}u∗f is the distributional derivative of u ∗ f. □

The following result is due to J. Kinnunen [K].

2.5. Theorem. Let 1 < p < ∞. If u ∈ W^{1,p}(ℝ^{n}), then M(u) ∈ W^{1,p}(ℝ^{n}). Moreover,

(2.6)  |D_{k}M(u)| ≤ M(D_{k}|u|) a.e., k = 1, …, n.

MAXIMAL FUNCTION AND SOBOLEV SPACES ON A METRIC SPACE 115

Proof. For r > 0 let χ_{r} = \frac{χ_{B(0,r)}}{m(B(0, r))}. With this notation,

(2.7) ∫−_{B(x,r)} |u| dm = \frac{1}{m(B(x, r))} ∫_{B(x,r)} |u| dm

= \frac{1}{m(B(0, r))} ∫_{B(0,r)} |u(x − y)| dm(y) = (|u| ∗ χ_{r})(x),

and write

u_{r}(x) = (|u| ∗ χ_{r})(x).

Now |u| ∈ W^{1,p}(ℝ^{n}) and by Lemma 2.4

(2.8)  D_{k}u_{r} = D_{k}|u| ∗ χ_{r},

and by (2.7)

M(u)(x) = sup_{j} u_{r_{j}}(x),

where the supremum is taken over all rational positive radii r_{j}. Write u_{j} = u_{r_{j}} and define

v_{i}(x) = max{u_{j}(x): 1 ≤ j ≤ i}, i = 1, 2, ….

Then v_{i} ≤ M(u) a.e. and

(2.9)  |D_{k}v_{i}| ≤ max_{1≤j≤i} |D_{k}u_{j}| = max_{1≤j≤i} |χ_{r_{j}} ∗ D_{k}|u||

≤ max_{1≤j≤i} χ_{r_{j}} ∗ |D_{k}|u|| ≤ sup_{j} χ_{r_{j}} ∗ |D_{k}|u||

= M(D_{k}|u|)

holds a.e. On the other hand, Theorem 2.2(d) and (2.9) yield

∥v_{i}∥_{W^{1,p}}^{p} = ∥v_{i}∥_{L^{p}}^{p} + ∥∇v_{i}∥_{L^{p}}^{p} ≤ ∥M(u)∥_{L^{p}}^{p} + ∥( ∑_{k=1}^{n} M(D_{k}|u|)^{2})^{1/2}∥_{L^{p}}^{p}
≤ A^{p}(∥u∥_{L^{p}}^{p} + ∥∇u∥_{L^{p}}^{p}) = A^{p}∥u∥_{W^{1,p}}^{p}.

Thus (v_{i}) is a bounded sequence in W^{1,p} and v_{i} → M(u) a.e. Lemma 2.3 gives M(u) ∈ W^{1,p}(ℝ^{n}) and (2.6) follows from (2.9). □

2.10. Comments. (a) By Theorem 2.2 we have ∥M(u)∥_{W^{1,p}} ≤ c∥u∥_{W^{1,p}}, c = c(n, p), 1 < p < ∞.

(b) The inequality in (a) can be used to bypass many applications of the maximal function in potential theory.

116 OLLI MARTIO

3. Sobolev spaces and the maximal function We are first going to prove an inequality due to Hedberg. We need an

inequality for C^{1}-functions. 3.1. Lemma ([GT, p. 155]). Let B(x_{0}, r) ⊂ ℝ^{n} and u ∈ C^{1}(ℝ^{n}). Then

(3.2)  |u(x) − u_{B(x_{0},r)}| ≤ C ∫_{B(x_{0},r)} \frac{|∇u(y)|}{|x − y|^{n−1}} dm(y),
u_{B(x_{0},r)} = ∫−_{B(x_{0},r)} u dm,

for all x ∈ B(x_{0}, r). Here C = C(n) < ∞. 3.3. Lemma. (Hedberg’s inequality). Let u ∈ L_{loc}^{1}(ℝ^{n}). Then

(3.4)  ∫_{B(x_{0},r)} \frac{|u(y)|}{|x − y|^{n−1}} dm(y) ≤ CrM(u)(x)

for all x ∈ B(x_{0}, r). Here C = C(n). Proof. Fix x ∈ B(x_{0}, r) and let

A_{i} = (B(x, 2r/2^{i}) \ B(x, 2r/2^{i+1})) ∩ B(x_{0}, r), i = 0, 1, …. Now

∫_{B(x_{0},r)} \frac{|u(y)|}{|x − y|^{n−1}} dm(y) = ∑_{i=0}^{∞} ∫_{A_{i}} \frac{|u(y)|}{|x − y|^{n−1}} dm(y)

≤ ∑_{i=0}^{∞} (\frac{r}{2^{i}} )^{1−n} ∫_{B(x,2^{1−i}r)} |u(y)| dm(y)

= ∑_{i=0}^{∞} 2^{i(n−1)} \frac{m(B(0, 1))r2^{(1−i)n}}{m(B(x, 2^{1−i}r))} ∫_{B(x,2^{1−i}r)} |u(y)| dm(y)

= 2^{n}m(B(0, 1)) ∑_{i=0}^{∞} 2^{−i} ∫−_{B(x,2^{1−i}r)} |u(y)| dm(y) ≤ CrM(u)(x).

□ 3.5. Remark. A slightly better inequality than (3.4) is possible because for (3.4) there is no need to compute M(u)(x) at large distances: M(u)(x) can be replaced by

M_{2r}(u)(x) = sup_{0<t≤2r} ∫−_{B(x,t)} |u| dm.

3.6. Corollary. If u ∈ W^{1,p}(ℝ^{n}), 1 < p < ∞, then (3.2) holds for a.e. x ∈ B(x_{0}, r).

Proof. There is a sequence u_{i} ∈ C^{1}(ℝ^{n}) ∩ L^{p}(ℝ^{n}) such that u_{i} → u in L^{p} and ∇u_{i} → ∇u in L^{p}. Passing to a subsequence we may assume that ∇u_{i} → ∇u a.e. Since

| ∫_{B(x_{0},r)} \frac{|∇u_{i}(y)|}{|x − y|^{n−1}} dm(y) − ∫_{B(x_{0},r)} \frac{|∇u(y)|}{|x − y|^{n−1}} dm(y)|

≤ ∫_{B(x_{0},r)} \frac{|∇u_{i}(y) − ∇u(y)|}{|x − y|^{n−1}} dm(y) ≤ CrM(|∇u_{i} − ∇u|)(x) → 0

for a.e. x ∈ B(x_{0}, r), the claim follows from (3.2). □

3.7. Theorem (Hajłasz [H]). Suppose that u ∈ W^{1,p}(ℝ^{n}), 1 < p < ∞. Then for a.e. x, y ∈ ℝ^{n},

(3.8)  |u(x) − u(y)| ≤ C|x − y|(M(∇u)(x) + M(∇u)(y)),

where C = C(n).

Proof. For a.e. x, y ∈ ℝ^{n} we have by Corollary 3.6 and by (3.4) with

B = B(\frac{x + y}{2} , |x − y|)

|u(x) − u(y)| ≤ |u(x) − u_{B}| + |u(y) − u_{B}|

≤ C[∫_{B} \frac{|∇u(z)|}{|x − z|^{n−1}} dm(z) + ∫_{B} \frac{|∇u(z)|}{|y − z|^{n−1}} dm(z)]

≤ C|x − y|(M(∇u)(x) + M(∇u)(y)).

□

3.9. Remark. A slightly better “local” result is true in view of Remark 3.5.

3.10. Remark. It is not difficult to show, see [H] or [HM], that a function u ∈ L^{p}(ℝ^{n}), 1 < p < ∞, which satisfies

|u(x) − u(y)| ≤ |x − y|(g(x) + g(y))

for a.e. x, y ∈ ℝ^{n} and some g ∈ L^{p}(ℝ^{n}), belongs to W^{1,p}(ℝ^{n}). Thus the inequality (3.8) is both necessary and sufficient for a function u to belong to the class W^{1,p}(ℝ^{n}), 1 < p < ∞. Indeed, this is the starting point to develop the theory of Sobolev spaces on a metric space.

4. The space F^{p}, 1 ≤ p ≤ ∞

Let (X, d) be a metric space and let μ be a Borel measure on X. Suppose that F is a linear space of μ-measurable functions u: X → [−∞, ∞] and that K is a positive cone of μ-measurable functions on X. A positive cone means that each g ∈ K satisfies g ≥ 0 μ-a.e. and λ_{1}g_{1} + λ_{2}g_{2} ∈ K whenever g_{1}, g_{2} ∈ K and λ_{1}, λ_{2} ≥ 0. For a measurable function u we let D(u) = D(u, K) denote the set of all g ∈ K such that

(4.1)  |u(x) − u(y)| ≤ d(x, y)(g(x) + g(y))

holds for μ-a.e. x, y ∈ X.

If 1 ≤ p ≤ ∞, then we let F_{p} be the set of all u ∈ F such that

(4.2)  ∥u∥_{F_{p}} = (∥u∥_{L^{p}}^{p} + inf_{g∈D(u,K)} ∥g∥_{L^{p}}^{p})^{1/p} < ∞.

Here we use the abbreviation L^{p} = L^{p}(X) = L^{p}(X, μ). If D(u, K) = ∅, then we put ∥u∥_{F_{p}} = ∞. Now it is easy to see that F_{p} is a linear subspace of L^{p} and (4.2) defines a norm in F_{p}. We complete the space F_{p} with respect to the norm (4.2) and obtain the space F^{p}, which consists of equivalence classes of Cauchy sequences of functions in F_{p}. Since L^{p} is complete, every equivalence class is represented by a unique function u ∈ L^{p}, and it is not difficult to see that the induced norm on F^{p} is

(4.3)  ∥u∥_{F^{p}} = (∥u∥_{L^{p}}^{p} + lim_{ε→0} inf_{g∈D_{ε}(u)} ∥g∥_{L^{p}}^{p})^{1/p}

where for ε > 0

D_{ε}(u) = D_{ε}(u, K)
= {g ∈ K: g ∈ D(v, K) for some v ∈ F with ∥u − v∥_{L^{p}} < ε}.

For the construction of F^{p} only the space F ∩ L^{p} and the cone K ∩ L^{p+} are needed; here L^{p+} is the positive cone of non-negative functions in L^{p}.

Now F^{p} is a linear space and, by construction, complete, hence F^{p} is a Banach space. From the definition of F^{p} we obtain the following characterization.

4.4. Lemma. A function u ∈ L^{p} belongs to F^{p} if and only if there are sequences of functions u_{i} ∈ F and g_{i} ∈ D(u_{i}, K) such that u_{i} → u in L^{p} and

lim inf_{i→∞} ∥g_{i}∥_{L^{p}} < ∞.

In general, it is difficult to compute ∥ · ∥_{F^{p}} from (4.3). The following lemma makes it sometimes easier to estimate ∥ · ∥_{F^{p}}.

4.5. Lemma. Let 1 < p < ∞ and u ∈ F^{p}. Then there is a unique function g ∈ D(u, L^{p+}) such that

∥u∥_{F^{p}} = (∥u∥_{L^{p}}^{p} + ∥g∥_{L^{p}}^{p})^{1/p}

and g_{i} → g in L^{p} for some sequence g_{i} ∈ K.

Proof. By the definition of the norm in F^{p}, there are sequences of functions ṽ_{i} ∈ F and g̃_{i} ∈ D(ṽ_{i}, K) such that ṽ_{i} → u in L^{p} and

(∥ṽ_{i}∥_{L^{p}}^{p} + ∥g̃_{i}∥_{L^{p}}^{p})^{1/p} → ∥u∥_{F^{p}}.

Passing to a subsequence we may assume that ṽ_{i} →u μ-a.e. Since 1 < p < ∞ and the sequence ∥g̃_{i}∥_{L^{p}} is bounded, we may assume, passing again to a subsequence, that g̃_{i} → g weakly in L^{p}. By the Banach-Saks theorem, passing to a subsequence, the sequence,

g_{j} = \frac{1}{j} ∑_{i=1}^{j} g̃_{i}

converges to g in L^{p} and hence, passing again to a subsequence if necessary, we may assume that g_{j} → g μ-a.e. By the cone property of K we have g_{j} ∈ D(v_{j}, K) where

v_{j} = \frac{1}{j} ∑_{i=1}^{j} ṽ_{j}.

Since v_{j} → u μ-a.e. and

|v_{j}(x) − v_{j}(y)| ≤ d(x, y)(g_{j}(x) + g_{j}(y))

holds for μ-a.e. in X, we obtain as j → ∞

|u(x) − u(y)| ≤ d(x, y)(g(x) + g(y))

μ-a.e. Thus g ∈ D(u, L^{p+}) and the uniqueness of g follows from the uniform convexity of the L^{p}-norm, 1 < p < ∞. □

4.6. Example. Let 1 < p < ∞. If we take F = L^{p}(X) and K = L^{p}(X)^{+}, then the norm of F^{p}, given by (4.3), is the same as

(4.7)  ∥u∥ = (∥u∥_{L^{p}}^{p} + inf_{g∈D(u,L^{p+})} ∥g∥_{L^{p}}^{p})^{1/p};

this can be easily seen by using the method of the proof of Lemma 4.5. The space F^{p} is now the space W^{1,p}(X) introduced by P. Hajłasz [H].

5. The capacity associated with the space F^{p}

In this section we show that there is a natural variational capacity based on the norm (4.3). In [KM] we introduced this capacity in the space W^{1,p}(X) of Example 4.6.

Let (X, d), μ, F, and K be as in Section 4. Furthermore, we assume that both F and K have the following lattice property: If u_{i} ∈ F (resp. K), i = 1, 2, then max(u_{1}, u_{2}) ∈ F (resp. K). Moreover, we assume that if c ≥ 0, then min(u, c) ∈ F for each u ∈ F. The last condition means that functions in F can be truncated.

The following lemma is easy to prove.

5.1. Lemma. Suppose that u_{i} ∈ F and g_{i} ∈ D(u_{i}, K), i = 1, 2. Then max(g_{1}, g_{2}) ∈ D(max(u_{1}, u_{2}), K), and max(g_{1}, g_{2}) ∈ D(min(u_{1}, u_{2}), K). Also for c ≥ 0 we have g_{1} ∈ D(min(u_{1}, c), K).

Lemma 5.1 implies that if g ∈ D(u, K), then g ∈ D(|u|, K).

For E ⊂ X and 1 < p < ∞ we introduce the p-capacity of E as

(5.2)  C_{p}(E) = C_{p}(E; F, K) = inf_{u∈A} ∥u∥_{F^{p}}^{p},

where A = A(E) is the set of admissible functions for E, i.e., A consists of functions u ∈ L^{p}(X) such that u ≥ 1 μ-a.e. in a neighbourhood of E. If A = ∅, then we set C_{p}(E) = ∞.

5.3. Remark. As the admissible set of functions in (5.2) the set A′ = {u ∈ A: 0 ≤ u ≤ 1} can be used.

The following theorem shows that the capacity C_{p} has the same basic measure theoretic property as the classical capacities.

5.4. Theorem. The capacity C_{p} is an outer measure.

Proof. Since C_{p}(∅) = 0 and since C_{p} is clearly monotone, it remains to check the subadditivity. For this let E_{i} ⊂ X, i = 1, 2, …, and ε > 0. We may assume that

(5.5)  ∑_{i=1}^{∞} C_{p}(E_{i}) < ∞.

Fix t > 0 and choose u_{i} ∈ A(E_{i}) and v_{i} ∈ F with g_{i} ∈ D(v_{i}, K) such that

(5.6)  ∥u_{i} − v_{i}∥_{L^{p}} < t/2^{i+2}

and

(5.7)  ∥u_{i}∥_{L^{p}}^{p} + ∥g_{i}∥_{L^{p}}^{p} ≤ C_{p}(E_{i}) + ε/2^{i}.

Next consider u = sup u_{i}; clearly u ≥ 1 in a neighbourhood of E = ⋃ E_{i}. Choose k so large that the function ṽ_{k} = max_{1≤i≤k} v_{i} satisfies

∥ṽ_{k} − u∥_{L^{p}} < t.

Indeed, since ṽ_{k} → v = sup v_{i} in L^{p} and ∥v − u∥_{L^{p}} ≤ t/2 by (5.6), this is possible. Hence

inf_{g̃∈D_{t}(u)} ∥g̃∥_{L^{p}}^{p} ≤ ∥g̃_{k}∥_{L^{p}}^{p},

where g̃_{k} = max_{1≤i≤k} g_{i}; note that g̃_{k} ∈ D(ṽ_{k}, K) by Lemma 5.1.

Thus we have

inf_{g̃∈D_{t}(u)} ∥g̃∥_{L^{p}}^{p} ≤ ∥g̃_{k}∥_{L^{p}}^{p} ≤ ∥g∥_{L^{p}}^{p},

where g = sup g_{i}. Letting t → 0 we obtain

lim_{t→0} inf_{g̃∈D_{t}(u)} ∥g̃∥_{L^{p}}^{p} ≤ ∥g∥_{L^{p}}^{p}.

On the other hand,

∥g∥_{L^{p}}^{p} ≤ ∑_{i=1}^{∞} ∥g_{i}∥_{L^{p}}^{p},

and

∥u∥_{L^{p}}^{p} ≤ ∑_{i=1}^{∞} ∥u_{i}∥_{L^{p}}^{p},

and hence

C_{p}(E) ≤ ∥u∥_{L^{p}}^{p} + lim_{t→0} inf_{g̃∈D_{t}(u)} ∥g̃∥_{L^{p}}^{p}

≤ ∑_{i=1}^{∞} (∥u_{i}∥_{L^{p}}^{p} + ∥g_{i}∥_{L^{p}}^{p})

≤ ∑_{i=1}^{∞} C_{p}(E_{i}) + ε,

and the proof is complete since ε > 0 was arbitrary. □

5.8. Remark. It is not very difficult to show the following properties of C_{p}. If C_{1} ⊃ C_{2} ⊃ … are compact sets in X and C = ⋂ C_{i}, then

(5.9)  C_{p}(C) = lim_{i→∞} C_{p}(C_{i}).

If O_{1} ⊂ O_{2} ⊂ … are open sets in X and O = ⋃ O_{i}, then

(5.10)  C_{p}(O) = lim_{i→∞} C_{p}(O_{i}).

We close this section by studying the capacity properties of functions in W^{1,p}(X). A property is said to hold p-quasieverywhere (p-q.e.) if it holds except for a set of p-capacity zero.

5.11. Theorem. For each Cauchy sequence (u_{i}) of functions u_{i} ∈ F^{p} ∩ C(X) there exists a subsequence which converges p-q.e. to a function u ∈ F^{p}. Moreover, the convergence is uniform outside a set of arbitrary small p-capacity.

Proof. Choose a subsequence, denoted again by (u_{i}), such that

∑_{i=1}^{∞} 2^{ip}∥u_{i} − u_{i+1}∥_{F^{p}}^{p} < ∞.

For i = 1, 2, …, let

E_{i} = {x ∈ X: |u_{i}(x) − u_{i+1}(x)| > 2^{−i}},

and F_{j} = ⋃_{i=j}^{∞} E_{i}. By continuity the function 2^{i}|u_{i} − u_{i+1}| is admissible for

E_{i}, which implies

C_{p}(E_{i}) ≤ 2^{ip}∥u_{i} − u_{i+1}∥_{F^{p}}^{p}.

By subadditivity we obtain

C_{p}(F_{j}) ≤ ∑_{i=j}^{∞} C_{p}(E_{i}) ≤ ∑_{i=j}^{∞} 2^{ip}∥u_{i} − u_{i+1}∥_{F^{p}}^{p}.

Next let F = ⋂ F_{j}. Then

C_{p}(F) ≤ lim_{j→∞} C_{p}(F_{j}) = 0.

Now u_{i}(x) converges at each point x ∈ X \ F. Also

|u_{j}(x) − u_{k}(x)| ≤ ∑_{i=j}^{k−1} |u_{i}(x) − u_{i+1}(x)|

≤ ∑_{i=j}^{k−1} 2^{−i} ≤ 2^{1−j}

for x ∈ X \ F_{j}, k > j, which shows that the convergence is uniform in X \ F_{j}. □

We say that a function u: X → [−∞, ∞] is p-quasicontinuous, if for every ε > 0 there is E ⊂ X such that C_{p}(E) < ε and u|_{X\E} is continuous.

5.12. Corollary. Suppose that C(X) is dense in F^{p}. Then for each u ∈ F^{p} there is a p-quasicontinuous function v ∈ F^{p} such that u = v μ-a.e.

5.13. Remark. Corollary 5.12 holds, in particular, in F^{p}, if each u ∈ F is continuous.

References

[GT] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, Second edition, Springer-Verlag, Berlin-New York, 1983.

[H] P. Hajłasz, Sobolev spaces on an arbitrary metric space, Potential Anal. (to appear).

[HKM] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear potential theory of degenerate elliptic equations, Oxford Mathematical Monographs, Oxford Science Publications, Clarendon Press, Oxford University Press, New York, 1993.

[K] J. Kinnunen, The Hardy-Littlewood maximal function of a Sobolev function, Preprint 73, Department of Mathematics, University of Helsinki, 1995.

[HM] P. Hajłasz and O. Martio, Traces of Sobolev functions on fractal type sets and characterization of extension domains, Preprint 88, Department of Mathematics, University of Helsinki, 1995.

[KM] J. Kinnunen and O. Martio, The Sobolev capacity on metric spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. (to appear).

[KJF] A. Kufner, O. John and S. Fučík, Function spaces, Noordhoff International Publishing, Leyden; Academia, Prague, 1977.

[S] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N. J., 1970.

Author’s address: Department of Mathematics University of Helsinki P.O. Box 4 FIN–00014 Helsinki Finland

E-mail: martio@cc.helsinki.fi