Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 26, 2001, 455–464

MODULUS AND CONTINUOUS CAPACITY

Sari Kallunki and Nageswari Shanmugalingam

University of Jyväskylä, Department of Mathematics P.O. Box 35, FI-40351 Jyväskylä, Finland; sakallun@math.jyu.fi University of Texas at Austin, Department of Mathematics Austin, TX 78712-1082 U.S.A.; nageswari@math.utexas.edu

Abstract. It is shown that if Ω is a domain in a metric measure space X such that X is proper, doubling, supports a (1, p)-Poincaré inequality, and is φ-convex, then Mod_{p}(E, F, Ω) is equal to the locally Lipschitz p-capacity of the triple (E, F, Ω).

1. Introduction

Let X be a metric measure space equipped with a Borel measure, and Ω be a domain in X, that is, Ω is an open connected set in X. Assume also that for each ball B ⊂ X, μ(B) is finite and non-zero. If Γ is any collection of paths in Ω, for 1 ≤ p < ∞ the p-modulus of the collection is defined to be the (possibly infinite) number

Mod_{p}(Γ) := inf_{ϱ} ∥ϱ∥_{L^{p}(Ω)}^{p},

where the infimum is taken over all non-negative Borel-measurable functions ϱ such that for each locally rectifiable path γ in Γ, the integral ∫_{γ} ϱ ds is not less than 1. If E and F are disjoint non-empty compact sets in Ω, let Mod_{p}(E, F, Ω) denote the p-modulus of the collection of all rectifiable paths γ in Ω with one endpoint in E and the other endpoint in F.

Following [HeK1], a non-negative Borel-measurable function ϱ is said to be an upper gradient of a real-valued function u if for all rectifiable paths γ

(1)  |u(x) − u(y)| ≤ ∫_{γ} ϱ ds,

where x and y denote the endpoints of γ. If inequality (1) holds true only for the paths that are not in a fixed collection of p-modulus zero, then ϱ is said to be a p-weak upper gradient of u. In the rest of the paper let p be a fixed number such that 1 < p < ∞, and p-weak upper gradients are refered to as weak upper gradients. By [KM, Lemma 2.4], the existence of a weak upper gradient in L^{p}(Ω) is equivalent to the existence of upper gradients in L^{p}(Ω) that converge in L^{p}(Ω) to the weak upper gradient. Hence in the following definitions of capacity one can consider weak upper gradients instead of only upper gradients to obtain the same number. The p-capacity of the triple (E, F, Ω) is defined to be the (possibly infinite) number

2000 Mathematics Subject Classification: Primary 31C15, 46E35.

The second author was partly supported by an N.S.F. research assistantship. This research was done while the second author was visiting University of Jyväskylä.

Cap_{p}(E, F, Ω) = inf_{ϱ} ∥ϱ∥_{L^{p}(Ω)}^{p},

where the infimum is taken over all non-negative Borel-measurable functions ϱ that are upper gradients (or weak upper gradients) of some function u with the property that u|_{E} ≥ 1 and u|_{F} ≤ 0. It is easily seen that the same number is obtained if the above definition is modified to say that u|_{E} = 1, u|_{F} = 0, and 0 ≤ u ≤ 1. This definition does not assume any regularity on the functions u; it is not even required of u to be measurable. A more sensitive capacity is obtained if in the above definition it is also required that the functions u be continuous; denote this number Cont-Cap_{p}(E, F, Ω). If u is also required to be locally Lipschitz, then the corresponding number obtained is denoted locLip-Cap_{p}(E, F, Ω). It is immediate that

Mod_{p}(E, F, Ω) ≤ Cap_{p}(E, F, Ω) ≤ Cont-Cap_{p}(E, F, Ω) ≤ locLip-Cap_{p}(E, F, Ω).

In [HeK1, Proposition 2.15] Heinonen and Koskela show that in arbitrary metric spaces Mod_{p}(E, F, Ω) = Cap_{p}(E, F, Ω). Furthermore, they also prove that if Ω is a compact φ-convex metric measure space, then Mod_{p}(E, F, Ω) = Cont-Cap_{p}(E, F, Ω). A metric space X is said to be φ-convex if there is a cover of X by open sets {U_{α}} together with homeomorphisms {φ_{α}: [0, ∞) → [0, ∞)} such that each pair of distinct points x and y in U_{α} can be joined by a curve whose length does not exceed φ_{α}(d(x, y)). For domains in R^{n} equipped with the Lebesgue measure Hesse, Shlyk, and Ziemer proved that Mod_{p}(E, F, Ω) = Cont-Cap_{p}(E, F, Ω), while Shlyk, Aikawa and Ohtsuka prove this for bounded Euclidean domains with strong A_{∞} weights, see [Z1], [Z2], [Shl1], [Shl2], [AO], [H], and [F]. In [HeK1, Remark 2.17] Heinonen and Koskela pose the question whether it is true that Mod_{p}(E, F, Ω) = Cont-Cap_{p}(E, F, Ω) if Ω is a general domain in a metric measure space, not necessarily bounded. This paper answers this question in the affirmative in the case that Ω is a domain in a φ-convex metric measure space X that is proper, doubling, and supports a (1, p)-Poincaré inequality. The arguments used in this paper are different from those in the above citations. The machinery of Newtonian spaces developed in [Sh] is used here. Properness means that closed balls are compact and the measure μ is doubling if there is a constant C_{μ} ≥ 1 such that μ(2B) ≤ C_{μ}μ(B) for all balls B in the space; 2B is the ball with the same center as B but with twice the radius of B. Following [HeK1], a metric measure space X is said to support a (1, p)-Poincaré inequality if there are constants C > 0 and τ ≥ 1 so that for each ball B in X and each continuous function u in τB with upper gradient ϱ in τB the following inequality holds:

(2)  ∫−_{B} |u − u_{B}| ≤ C(diam(B))(∫−_{τB} ϱ^{p})^{1/p},

where f_{B} denotes the mean value integral of f over B. In the rest of the paper it is assumed that τ = 1. Minor modifications would yield the results when τ > 1.

The main theorem of this paper is:

Theorem 1.1. If X is a proper φ-convex metric measure space equipped with a doubling measure and supporting a (1, p)-Poincaré inequality with 1 < p < ∞, and Ω is a domain in X, then for all disjoint compact non-empty subsets E and F of Ω,

Mod_{p}(E, F, Ω) = Cont-Cap_{p}(E, F, Ω) = locLip-Cap_{p}(E, F, Ω),

where locLip-Cap_{p}(E, F, Ω) is defined similarly to Cont-Cap_{p}(E, F, Ω), with the test functions being required to be locally Lipschitz.

The authors do not know whether the (1, p)-Poincaré inequality assumption is essential for the conclusion in Theorem 1.1. This theorem has a wide range of applications. For example, in [KST, Corollary C], Koskela, Shanmugalingam and Tuominen show that certain types of porous sets are removable for the Loewner condition by using this theorem. The idea here is to show that these porous sets are removable for continuous capacitary estimates, and then as a consequence of Theorem 1.1 they note that the Loewner property is preserved when such porous sets are removed.

Following the notation in [Sh], the space N_{loc}^{1,p}(Ω) is defined to be the collection of all functions in L_{loc}^{p}(Ω) which have upper gradients in L^{p}(Ω). Note that the upper gradients are required to be in L^{p}(Ω), not merely in L_{loc}^{p}(Ω). The collection L_{loc}^{p}(Ω) is the collection of all functions in Ω that are p-integrable on every bounded subset of Ω. The proof of Theorem 1.1 uses the fact that in the setting of Theorem 1.1 locally Lipschitz functions are dense in N_{loc}^{1,p}(Ω).

Proposition 1.2. If X is a metric measure space equipped with a doubling measure and supporting a (1, p)-Poincaré inequality with 1 < p < ∞, and Ω is a domain in X, then there exists a constant C > 0 such that for each function u in N_{loc}^{1,p}(Ω) and for all ε > 0 there is a locally Lipschitz function u_{ε} in N_{loc}^{1,p}(Ω) with the property that there exists an upper gradient g_{ε} of u_{ε} − u so that

∥g_{ε}∥_{L^{p}(Ω)} < ε and ∥u_{ε} − u∥_{L^{p}(Ω)} < Cε. Proposition 1.2 is interesting in its own right as it proves a local analogue of

the classical Sobolev space theory result H^{1,p} = W^{1,p}. Using an idea of Semmes [S1], the result is proved for the case Ω = X in [Sh, Theorem 4.1].

Section 2 is devoted to proving Proposition 1.2, and in Section 3 Theorem 1.1 is proved.

Acknowledgement. The authors wish to thank Pekka Koskela, Juha Heinonen and Olli Martio for numerous helpful suggestions.

2. Proof of Proposition 1.2

The idea behind the proof is to approximate u in balls in Ω by applying [Sh, Theorem 4.1] in these balls, and then paste the approximations together using a partition of unity to obtain an approximation in Ω.

The following covering lemma is needed for the proof of Proposition 1.2. For a proof of this lemma see [S2] and [KST].

If B is a ball of radius r, then for k > 0 the ball with the same center as B and radius kr is denoted kB.

Lemma 2.1. Let X be a metric measure space equipped with a doubling measure, and Ω a domain in X. Then there is a collection {B_{i}}_{i∈N} of balls in X such that

(i) ⋃_{i} B_{i} = Ω, (ii) there exists a constant C_{1} ≥ 1 such that if 2B_{i} ∩ 2B_{j} ≠ ∅, then

\frac{1}{C_{1}} rad (B_{i}) ≤ rad (B_{j}) ≤ C_{1}rad (B_{i})

where rad (B_{i}) is a pre-assigned radius of B_{i}, (iii) with C_{2} = 12(C_{1} + 1), C_{2}B_{i} ⋐ Ω, and (iv) there exists a constant D ≥ 1 so that ∑_{i} χ_{2B_{i}} ≤ D.

Note that a ball in X may have more than one center and more than one radius. Hence in this lemma it may be necessary to consider the balls B_{i} to have pre-assigned centers and radii.

If {B_{i}}_{i∈N} is the collection obtained by Lemma 2.1, then for each positive integer i there exists a (C/diam (B_{i}))-Lipschitz function φ_{i} so that 0 < φ_{i} ≤ 1, ∑_{i∈N} φ_{i} = χ_{Ω} , and φ_{i}|_{Ω\2B_{i}} = 0. The constant C in the Lipschitz constant estimate depends only on the constants C_{1} and D of Lemma 2.1.

The following lemma also is needed in the proof of Proposition 1.2. Its proof is a modification of a technique in [Ha, Proposition 1], and will be omitted here. Let u be a real-valued function on a metric measure space Y . Following the notation in [Sh], the function u is said to be ACC_{p} in Y if on all rectifiable paths γ ⊂ Y outside a family of p-modulus zero the function u∘γ is absolutely continuous. By [Sh, Proposition 3.1], functions in N_{loc}^{1,p}(Ω) are ACC_{p} in Ω.

Lemma 2.2. Let Y be a metric measure space, and u be a real-valued ACC_{p} function on Y . If there exist two non-negative Borel measurable functions g and h on Y such that for all rectifiable paths γ connecting x to y it is true that

|u(x) − u(y)| ≤ ∫_{γ} g + d(x, y)(h(x) + h(y)),

then g + 4h is a weak upper gradient of u.

Proof of Proposition 1.2. By [HeK2, Theorem A], the (1, p)-Poincaré inequality (2) is satisfied by any real-valued function u in L_{loc}^{1}(X) and its upper gradient g, since X is proper, φ-convex, doubling, and supports a (1, p)-Poincaré inequality.

Let {B_{i}}_{i∈N} be as in Lemma 2.1, and {φ_{i}}_{i∈N} be the corresponding partition of unity constructed above. For each i the proof of [Sh, Theorem 4.1] can be applied on the space N^{1,p}(C_{2}B_{i}) to obtain Lipschitz approximations to u|_{C_{2}B_{i}}. The proof of [Sh, Theorem 4.1] uses the doubling property of the measure and the Poincaré inequality. In this general situation of an arbitrary domain Ω, we can still use these two properties on the balls B_{i} because of the properties from Lemma 2.1. Hence we can apply the proof of [Sh, Theorem 4.1] here.

Let v_{i,λ} be such a Cλ-Lipschitz approximation of u|_{C_{2}B_{i}}. Fix ε > 0. Choosing λ sufficiently large, an approximation v_{i} = v_{i,λ} can be obtained for each ball 2C_{2}B_{i} so that in addition,

(3)  ∥v_{i} − u∥_{N^{1,p}(2C_{2}B_{i})} < 2^{−i}ε,

and

(4)  ∥v_{i} − u∥_{L^{p}((2C_{2}B_{i}))} < 2^{−i}diam (B_{i})ε.

By (3), we have an upper gradient g_{i} of v_{i} − u with ∥g_{i}∥_{L^{p}((2C_{2}B_{i})} < 2^{−i}ε. Now let u_{ε} = ∑_{i} v_{i}φ_{i}. While v_{i}φ_{i} is defined in all of Ω, since it has its non-zero values only inside 2B_{i}, by (iv) of Lemma 2.1 the above sum is a finite sum on Ω. Furthermore, it is easily seen that u_{ε} is Lipschitz on each ball B_{i}, and hence is locally Lipschitz. Since u = ∑_{i} uφ_{i}, we have u_{ε} − u = ∑_{i}(v_{i} − u)φ_{i}. Just as in the papers [S2] and [KST] it can be shown that the function

g_{ε} = ∑_{i}(g_{i} + \frac{4C}{diam (B_{i})}|v_{i} − u|)χ_{2B_{i}}

is an upper gradient of u_{ε} − u. Now just as in [KST], it can be easily seen by the bounded overlap property (iv) of Lemma 2.1 and by inequalities (3) and (4) that ∫_{Ω} g_{ε}(x)^{p} ≤ Cε^{p}. Therefore ∥g_{ε}∥_{L^{p}(Ω)} ≤ Cε, and the function u_{ε} has g + g_{ε} ∈ L^{p}(Ω) as an upper gradient. Furthermore, by inequality (4),

∥u_{ε} − u∥_{L^{p}(Ω)} ≤ ∑_{i} ∥v_{i} − u∥_{L^{p}((4C_{1}+2)B_{i})} ≤ (∑_{i} 2^{−i})ε.

Hence u_{ε} is a locally Lipschitz function that approximates u as required.

3. Proof of Theorem 1.1

The idea behind the proof of Theorem 1.1 is as follows. It is first shown that if A, B are non-empty compact subsets of Ω, then

Mod_{p}(A, B, Ω) = N_{loc}^{1,p} − Cap_{p}(A, B, Ω)

where N_{loc}^{1,p} − Cap_{p}(A, B, Ω) is defined similarly to the other capacity definitions, with the test functions u being required to be in N_{loc}^{1,p}(Ω). Theorem 1.1 is then proved if N_{loc}^{1,p} − Cap_{p}(A, B, Ω) can be replaced by Cont-Cap_{p}(E, F, Ω). This is done as follows: If E and F are non-empty, disjoint compact subsets of Ω, compact neighbourhoods A_{ε} and B_{ε} of E and F are considered. The test functions u used to calculate the number N_{loc}^{1,p} − Cap_{p}(A_{ε}, B_{ε}, Ω) are “smoothed out” by locally Lipschitz approximations. These approximations take on the same value as u at the “centers” E and F of A_{ε} and B_{ε}, and hence are functions that can be used to calculate upper bounds for Cont-Cap_{p}(E, F, Ω). The argument is completed by using a continuity property of N_{loc}^{1,p} − Cap_{p}(A_{ε}, B_{ε}, Ω), that is, as the sets A_{ε} and B_{ε} shrink to the sets E and F respectively N_{loc}^{1,p} − Cap_{p}(A_{ε}, B_{ε}, Ω) tends to the number N_{loc}^{1,p} − Cap_{p}(E, F, Ω).

By the proof in [HeK1, Proposition 2.15], it is true that if A and B are nonempty disjoint compact subsets of Ω with X φ-convex and Ω_{j} is a subdomain of Ω such that A ∪ B ⊂ Ω_{j} ⋐ Ω, then

(5)  Cont-Cap_{p}(A, B, \overline{Ω}_{j}) ≤ Cap_{p}(A, B, Ω) = Mod_{p}(A, B, Ω).

Here Cont-Cap_{p}(A, B, \overline{Ω}_{j}) is defined similarly to Cont-Cap_{p}(A, B, Ω). In order to prove that Mod_{p}(A, B, Ω) is equal to N_{loc}^{1,p} − Cap_{p}(A, B, Ω), the domain Ω is exhausted by such subdomains Ω_{j} (which is possible since X is proper), and it is shown that

lim_{j→∞} Cont-Cap_{p}(A, B, \overline{Ω}_{j}) ≥ N_{loc}^{1,p} − Cap_{p}(A, B, Ω).

In order to do so, it is necessary to build up N_{loc}^{1,p}(Ω)-test functions admissible for calculating N_{loc}^{1,p} − Cap_{p}(A, B, Ω) from the test functions used in calculating Cont-Cap_{p}(A, B, Ω_{j}). To overcome the restrictions imposed by not knowing whether N^{1,p}(Ω_{j}) is reflexive or not in this general setting, it is necessary to obtain the N_{loc}^{1,p} -test function via Mazur’s lemma applied to convex combinations of functions and upper gradients in L^{p}(Ω).

Lemma 3.1. Let Y be a metric measure space. If {f_{j}}_{j∈N} is a sequence of functions in L^{p}(Y) with upper gradients {g_{j}}_{j∈N} in L^{p}(Y), such that f_{j} weakly converges to f in L^{p} and g_{j} weakly converges to g in L^{p}, then g is a weak upper gradient of f after modifying f on a set of measure zero, and there is a convex combination sequence f̃_{j} = ∑_{k=j}^{n_{j}} λ_{kj}f_{k} and g̃_{j} = ∑_{k=j}^{n_{j}} λ_{kj}g_{k} with ∑_{k=j}^{n_{j}} λ_{kj} = 1, λ_{kj} > 0, so that f̃_{j} converges in L^{p} to f and g̃_{j} converges in L^{p} to g.

Proof. Applying Mazur’s lemma (see [Y]) to each sequence {f_{j}}_{j=1}^{∞} and {g_{j}}_{j=1}^{∞} simultaneously, a sequence of convex combinations of f_{j} ’s and g_{j} ’s that converge in the L^{p}(Y)-norm to f and some function g can be formed. Denote these convex combination sequences {f̃_{i}} and {g̃_{i}}. It is easy to see that g̃_{i} is a weak upper gradient of f̃_{i}.

To see that g is an upper gradient of f in Y , note by a theorem of Fuglede [F] that if g̃_{i} is a sequence of Borel-measurable non-negative functions in L^{p}(Y) converging in L^{p}(Y) to a function g, then there exists a subsequence of {g̃_{i}}, also denoted {g̃_{i}} for brevity, and a collection Γ of rectifiable paths in Y with Mod_{p}(Γ) = 0, so that whenever γ is a rectifiable path not in Γ, then ∫_{γ} g̃_{i} → ∫_{γ} g, and ∫_{γ} g < ∞. One can redefine f on a set of measure zero in Y so that

(6)  f(x) = \frac{1}{2}\{lim sup_{i→∞} f̃_{i}(x) + lim inf_{i→∞} f̃_{i}(x)\},

wherever it makes sense; see [Sh]. Now just as in [Sh], it can be shown that g is a weak upper gradient of f.

Lemma 3.2. Let Ω_{j} be a sequence of bounded domains such that Ω_{j} ⊂ Ω_{j+1} , Ω_{j} ⋐ Ω, Ω = ⋃_{j} Ω_{j}, and E, F are compact subsets of Ω_{j} for each j. Then

lim_{j→∞} Cont-Cap_{p}(E, F, \overline{Ω}_{j}) ≥ N_{loc}^{1,p} − Cap_{p}(E, F, Ω).

Proof. If lim_{j→∞} Cont-Cap_{p}(E, F, \overline{Ω}_{j}) = ∞, then there is nothing to prove, and because Cont-Cap_{p}(E, F, \overline{Ω}_{j}) is an increasing function of j, it can be assumed that the limit of Cont-Cap_{p}(E, F, \overline{Ω}_{j}) exists and is equal to M < ∞. Now Cont-Cap_{p}(E, F, \overline{Ω}_{j}) ≤ M for each j. For each positive integer j, a continuous function f_{j} can be chosen, together with its upper gradient g_{j} in \overline{Ω}_{j}, so that 0 ≤ f_{j} ≤ 1, ∥g_{j}∥_{L^{p}(Ω)}^{p} ≤ Cont-Cap_{p}(E, F, \overline{Ω}_{j}) + ε, and f_{j}|_{A} = 1, f_{j}|_{B} = 0.

Fix j ∈ N. For each integer k ≥ j, f_{k} is defined on \overline{Ω}_{j} and has g_{k} as an upper gradient on Ω_{j}, ∥f_{k}∥_{L^{p}(\overline{Ω}_{j})} ≤ μ(\overline{Ω}_{j})^{1/p}, and ∥g_{k}∥_{L^{p}(\overline{Ω}_{j})} ≤ ∥g_{k}∥_{L^{p}(\overline{Ω}_{k})} ≤ (M + ε)^{1/p}. Thus both sequences {f_{k}}_{k≥j} and {g_{k}}_{k≥j} are bounded sequences in L^{p}(\overline{Ω}_{j}), and by the weak compactness property of L^{p}(\overline{Ω}_{j}), there exist functions f^{j} and g^{j} in L^{p}(\overline{Ω}_{j}) to which subsequences of the two sequences weakly converge respectively. By Lemma 3.1, there is a convex combination f̃_{k,j} and the corresponding convex combination g̃_{k,j} converging in L^{p}(\overline{Ω}_{j}) to f^{j} and g^{j} respectively, and g^{j} is a weak upper gradient of f^{j} in \overline{Ω}_{j}. Moreover, by definition (6), f^{j}|_{E} = 1, f^{j}|_{F} = 0, and 0 ≤ f^{j} ≤ 1. Now for every k ∈ N, by a diagonalization argument it can be ensured that f^{k}|_{Ω_{k−1}} = f^{k−1} outside of a set of zero p-capacity in Ω_{k−1} (see [Sh, Corollary 3.3]), and g^{k}|_{Ω_{k−1}} = g^{k−1} almost everywhere in Ω_{k−1} . Hence the function f can be defined by f(x) = f^{k}(x) whenever x ∈ \overline{Ω}_{k} and g can be defined by g(x) = g^{k}(x) whenever x ∈ \overline{Ω}_{k}\\overline{Ω}_{k−1} . Then f ∈ L_{loc}^{p}(Ω), and g is a weak upper gradient of f since every rectifiable curve in Ω lies in some Ω_{k} and g = g^{k} almost everywhere in Ω_{k}. Since ∥g_{k}∥_{L^{p}(Ω_{j})}^{p} ≤ M + ε, extending each g_{i} by zero to all of Ω,

(∫_{Ω} |g^{k}|^{p})^{1/p} = (∫_{Ω}(∑_{s=j}^{n_{j}} λ_{js}g_{s})^{p})^{1/p} ≤ (M + ε)^{1/p}.

Hence ∥g∥_{L^{p}(Ω)}^{p} ≤ M + ε and therefore f ∈ N_{loc}^{1,p}(Ω) is an admissible function in calculating N_{loc}^{1,p} − Cap_{p}(E, F, Ω). Hence,

M + ε ≥ N_{loc}^{1,p} − Cap_{p}(E, F, Ω),

where we use the fact that M = lim_{j→∞} Cont-Cap_{p}(E, F, \overline{Ω}_{j}).

Remark 3.3. By Lemma 3.2 and by inequality (5), if p > 1 then

Mod_{p}(E, F, Ω) = Cap_{p}(E, F, Ω)
≥ Cont-Cap_{p}(E, F, \overline{Ω}_{j}) → M ≥ N_{loc}^{1,p} − Cap_{p}(E, F, Ω)

as j → ∞. Since N_{loc}^{1,p} − Cap_{p}(E, F, Ω) ≥ Cap_{p}(E, F, Ω), it follows that

(7)  Mod_{p}(E, F, Ω) = N_{loc}^{1,p} − Cap_{p}(E, F, Ω) = Cap_{p}(E, F, Ω)

whenever E and F are non-empty disjoint compact subsets of a domain Ω in a proper φ-convex metric measure space.

Lemma 3.4. If E and F are non-empty disjoint compact subsets of Ω, there exist for each 0 < ε < \frac{1}{2} disjoint compact sets A_{ε}, B_{ε} of Ω such that for some δ > 0

(i) \overline{⋃_{x∈E} B(x, δ)} = A_{ε}, (ii) \overline{⋃_{x∈F} B(x, δ)} = B_{ε}, and (iii) N_{loc}^{1,p} − Cap_{p}(A_{ε}, B_{ε}, Ω) ≤ (1 − 2ε)^{−p}N_{loc}^{1,p} − Cap_{p}(E, F, Ω) + ε(1 − 2ε)^{−p} + 2ε.

Proof. We just give a brief sketch of the proof here. The idea is to use the test functions considered in calculating N_{loc}^{1,p} − Cap_{p}(E, F, Ω) to construct test functions for calculating N_{loc}^{1,p} − Cap_{p}(A_{ε}, B_{ε}, Ω). A priori we only know that the test functions u used in calculating N_{loc}^{1,p} − Cap_{p}(E, F, Ω) take on the value of 1 on E and the value of 0 on F. We need test functions that take on the value of 1 in a neighborhood of E and the value of 0 in a neighborhood of F. This is done by noting that by Proposition 1.2 we have that u is p-quasi-continuous, and hence by modifying u on a set of small p-capacity we obtain the required test function for calculating N_{loc}^{1,p} − Cap_{p}(A_{ε}, B_{ε}, Ω).

By Lemma 3.4 and the equality (7), for each 0 < ε < \frac{1}{2} there exist non-empty disjoint compact sets A_{ε}, B_{ε} as in Lemma 3.4 so that

(8)  \frac{1}{(1 − 2ε)^{p}} Mod_{p}(E, F, Ω) + η(ε) ≥ N_{loc}^{1,p} − Cap_{p}(A_{ε}, B_{ε}, Ω),

where lim_{ε→0} η(ε) = 0.

Lemma 3.5. With X and Ω as in Theorem 1.1 and ε, δ, A_{ε}, B_{ε}, E, F, and Ω as in Lemma 3.4, it is true that

N_{loc}^{1,p} − Cap_{p}(A_{ε}, B_{ε}, Ω) ≥ locLip-Cap_{p}(E, F, Ω). Here the definition of locLip-Cap_{p}(E, F, Ω) is the same as the definition of

Cont-Cap_{p}(E, F, Ω), but with functions u being required to be locally Lipschitz in Ω rather than merely continuous.

Proof. If u is an admissible function for calculating N_{loc}^{1,p} − Cap_{p}(A_{ε}, B_{ε}, Ω), then u|\overline{⋃_{x∈E} B(x,δ)} = 1 and u|\overline{⋃_{x∈F} B(x,δ)} = 0. Therefore if g is an upper gradient

of u, then so is

g̃ = gχ_{Ω\\overline{⋃_{x∈E∪F} B(x,δ)}}.

Then if x ∈ E ∪ F, denoting U = Ω \ \overline{⋃_{x∈E∪F} B(x, δ)},

Mg̃^{p}(x) = sup_{r>0} ∫−_{B(x,r)} g^{p}χ_{U} = sup_{r≥δ} ∫−_{B(x,r)} g^{p}χ_{U} ≤ \frac{1}{μ(B(x, δ))}∥g∥_{L^{p}(Ω)}^{p}.

Since the measure μ is doubling, for all 0 < r ≤ R and for all x_{1} ∈ X and x_{0} ∈ B(x_{1}, R),

\frac{μ(B(x_{0}, r))}{μ(B(x_{1}, R))} ≥ C(\frac{r}{R})^{s}

where C is a constant independent of x_{0} , x_{1} , r, and R, and s = log_{2} C_{2} (C_{2} being the doubling constant associated with μ). As E ∪F is compact, there exists a positive number ϱ > δ > 0 and x_{1} ∈ X so that E ∪ F ⊂ B(x_{1}, ϱ). Hence if x_{0} is a point in E ∪ F, then

μ(B(x_{0}, δ)) ≥ C(\frac{δ}{ϱ})^{s}μ(B(x_{1}, ϱ)) =: C_{E∪F} .

Therefore for x ∈ E ∪ F,

Mg̃^{p}(x) ≤ \frac{1}{C_{E∪F}} ∥g∥_{L^{p}(Ω)}^{p}.

In the use of [Sh, Theorem 4.1] in the proof of Proposition 1.2, we can choose the Lipschitz constant λ for the approximating function to be sufficiently large, namely, choose λ so that λ > ∥g∥_{L^{p}(Ω)}^{p}/C_{E∪F} , so that the approximating function u_{ε} agrees with the function u being approximated on E ∪ F. Hence u_{ε} is an admissible function for calculating locLip-Cap_{p}(E, F, Ω):

locLip-Cap_{p}(E, F, Ω) ≤ ∥g + g_{ε}∥_{L^{p}(Ω)}^{p} ≤ ∥g∥_{L^{p}(Ω)}^{p} + ε.

Thus locLip-Cap_{p}(E, F, Ω) ≤ N_{loc}^{1,p} − Cap_{p}(A_{ε}, B_{ε}, Ω).

464 Sari Kallunki and Nageswari Shanmugalingam

Now combining inequalities (8) and Lemma 3.5 and then letting ε tend to zero, we obtain a proof of Theorem 1.1.

Remark 3.6. In the event that the domain Ω is all of X, using Semmes’ idea [S1] and [Sh, Theorem 4.1], globally Lipschitz approximations can be obtained in Proposition 1.2 instead of merely locally Lipschitz approximations. Hence a better result is obtained: Cont-Cap_{p}(E, F, Ω) = Lip − Cap_{p}(E, F, Ω) = Mod_{p}(E, F, Ω).

References

[AO] Aikawa, H., and M. Ohtsuka: Extremal length of vector measures. - Ann. Acad. Sci. Fenn. Math. 24, 1999, 61–88.

[F] Fuglede, B.: Extremal length and functional completion. - Acta Math. 98, 1957, 171–218.

[Ha] Hajłasz, P.: Geometric approach to Sobolev spaces and badly degenerated elliptic equations. - Nonlinear Analysis and Applications, Warsaw 1994, 141–168.

[HeK1] Heinonen, J., and P. Koskela: Quasiconformal maps in metric space with controlled geometry. - Acta Math. 181, 1998, 1–61.

[HeK2] Heinonen, J., and P. Koskela: A note on Lipschitz functions, upper gradients, and the Poincaré inequality. - New Zealand J. Math. 28, 1999, 37–42.

[H] Hesse, J.: A p-extremal length and p-capacity equality. - Ark. Mat. 13, 1975, 131–144.

[KM] Koskela, P., and P. MacManus: Quasiconformal mappings and Sobolev spaces. -Studia Math. 131, 1998, 1–17.

[KST] Koskela, P., N. Shanmugalingam, and H. Tuominen: Removable sets for the Poincaré inequality on metric spaces. - Indiana Univ. Math. J. 49, 2000, 333–352.

[O] Ohtsuka, M.: Extremal lengths and p-precise functions in 3-space. - Manuscript.

[S1] Semmes, S.: Personal communication.

[S2] Semmes, S.: Finding curves on general spaces through quantitative topology with applications to Sobolev and Poincaré inequalities. - Selecta Math. 2, 1996, 155–295.

[Sh] Shanmugalingam, N.: Newtonian spaces: a generalization of Sobolev spaces. - Revista Math. (to appear).

[Shl1] Shlyk, V.A.: On the equality between p-capacity and p-modulus. - Siberian Math. J. 34, 1993, 216–221.

[Shl2] Shlyk, V.A.: Weighted capacities, moduli of condensers and Fuglede exceptional sets. - Dokl. Akad. Nauk 332, 1993, 428–431.

[Y] Yosida, K.: Functional Analysis. - Springer-Verlag, Berlin, 1980.

[Z1] Ziemer, W.: Extremal length and p-capacity. - Michigan Math. J. 16, 1969, 43–51.

[Z2] Ziemer, W.: Extremal length as a capacity. - Michigan Math. J. 17, 1970, 117–128.

Received 17 March 2000