POLAR SETS ON METRIC SPACES

Juha Kinnunen and Nageswari Shanmugalingam

Abstract. We show that if X is a proper metric measure space equipped with a doubling measure supporting a Poincaré inequality, then subsets of X with zero pcapacity are precisely the p-polar sets; that is, a relatively compact subset of a domain in X is of zero p-capacity if and only if there exists a p-superharmonic function whose set of singularities contains the given set. In addition, we prove that if X is a phyperbolic metric space, then the p-superharmonic function can be required to be p-superharmonic on the entire space X. We also study the the following question: If a set of zero p-capacity, does there exist a p-superharmonic function whose set of singularities is precisely the given set?

0. Introduction

Sets of zero capacity in potential theory play the same role played by sets of zero measure in the study of L^{p}-spaces. Given a metric measure space X, functions in the equivalence classes in the Sobolev space N^{1,p}(X) agree up to sets of zero p-capacity. The purpose of this note is to characterize sets of zero p-capacity in terms of the singular sets of p-superharmonic functions.

A polar set of a metric space is a set E ⊂ X so that there exists a p-superharmonic function which takes on the value of infinity at every point in E. When X is an Euclidean space and E is a subset of a domain in this Euclidean space with zero p-capacity, then there exists a p-superharmonic function on the domain taking on the value of infinity at each point of E. On the other hand, it is known that polar sets have zero p-capacity; hence, the collection of all sets of zero p-capacity in X and the collection of all polar sets in X are one and the same collection (see [HKM] and references therein). We extend this result to the setting of metric measure spaces that are equipped with a doubling measure and supporting a (1, q)-Poincaré inequality for some q with 1 ≤ q < p. For recent results on analysis and potential theory on metric spaces we refer to [B], [BMS], [C], [KaS], [KKM], [KinM1-2], [KSh], [KoM], [KST], [Sh1-3]. A very nice general overview of the theory can be found in [He].

It should be noted here that unlike in the setting of Riemannian manifolds, in general metric measure spaces p-superharmonic functions need not be associated in any way with a corresponding differential equation. Hence the technique of sweeping of measures developed in [Kil1] and [Kil2] is not applicable in the generality considered in this paper. However, we do follow the general idea of the proof given

Typeset by AMS-TEX

Key words and phrases. Minimizers, variational integrals, polar sets, zero capacity sets.

2000 Mathematics Subject Classification 31C45, 49N60.

in [Kil1] in proving the main result of this paper, Proposition 3.2. Proposition 3.2, which states that zero p-capacity sets are p-polar, together with Proposition 2.2, which states that p-polar sets have zero p-capacity, give a characterization of zero p-capacity sets in terms of p-polar sets. Observe that the p-superharmonic functions which identify a closed p-polar set E also act as a barrier function for the points in E, considered as boundary points of the open set Ω \ E whenever Ω is a domain containing E. Thus the study of zero p-capacity sets in terms of p-polar sets is of interest in potential theory.

This note is organized as follows. The first section gathers together the necessary background such as the definition of Newtonian (Sobolev) spaces, capacity, and superharmonic functions. In the second section we prove that p-polar sets are of zero p-capacity. The third section is focused toward proving that zero p-capacity sets are p-polar by using the tools developed in [KinM2] and the general idea of the proof from [Kil1]. It is also interesting to know when is it true that given a set of zero p-capacity there is a function that is p-superharmonic on the whole space X so that the given set is a subset of the singularity set of a global superharmonic function. In the fourth section we give a partial answer to this question by demonstrating that whenever X is a p-hyperbolic metric space such global p-superharmonic functions can be constructed. It is also interesting to know whether a zero p-capacity set is precisely the set of all singularities of a given p-superharmonic function. Sections 5 through 7 are devoted to answering this question under a modification of what is meant by a p-harmonic function using the approach of [C]. The method is based and potential estimates originally developed in [KilM1] and [KilM2] (see also [Mi]). On metric spaces potential estimates have been studied in [BMS].

1. Preliminaries

We assume that (X, d, μ) is a metric measure space endowed with a metric d and a non-trivial Borel-regular measure μ. We assume in addition that X is connected and proper, that is, closed and bounded subsets of X are compact. While the results in this note remain valid if we assume that X is locally compact rather than proper, we assume properness for the sake of simplifying notation. The proof of the results in this note can be extended to the locally compact case with some mild modifications.

In this section we recall basic definitions and describe the general setup of our study.

1.1. Upper gradients. Let u be a real-valued function on X. A non-negative Borel measurable function g: X → [0, ∞] is said to be an upper gradient of u if for all non-constant compact rectifiable paths γ joining points x and y in X we have

(1.2)  |u(x) − u(y)| ≤ ∫_{γ} g ds.

See [C] and [Sh2] for a discussion on upper gradients.

Let 1 ≤ p < ∞. A property is said to hold for p-almost all paths, if the set of non-constant paths for which the property fails is of zero p-modulus (see, for example, [Sh2] for the definition of the p-modulus). If (1.2) holds for p-almost all paths γ, then g is said to be a p-weak upper gradient of u.

1.3. Newtonian spaces. Let 1 ≤ p < ∞. We define the space Ñ^{1,p}(X) to be the collection of all functions u ∈ L^{p}(X) that have a p-weak upper gradient g ∈ L^{p}(X). This space is equipped with a seminorm

∥u∥_{Ñ^{1,p}(X)} = ∥u∥_{L^{p}(X)} + inf ∥g∥_{L^{p}(X)},

where the infimum is taken over all p-weak upper gradients of u. When p > 1, by the uniform convexity of L^{p}(X) we have that whenever u ∈ N^{1,p}(X) there is a function g_{u} in L^{p}(X)-convex hull formed by the set of all p-weak upper gradients of u, called the minimal p-weak upper gradient of u (or the minimal p-integrable p-weak upper gradient), so that g_{u} is a p-weak upper gradient of u and

∥g_{u}∥_{L^{p}(X)} = inf ∥g∥_{L^{p}(X)}

where the infimum is over all p-weak upper gradients g of u; see [C] or [Sh1]. Indeed, in this case, if g is another p-integrable p-weak upper gradient of u, then g_{u} ≤ g almost everywhere in X.

We define an equivalence relation in Ñ^{1,p}(X) by saying that u ∼ v if

∥u − v∥_{Ñ^{1,p}(X)} = 0.

The Newtonian space N^{1,p}(X) is defined to be the space Ñ^{1,p}(X)/ ∼ with the norm

∥u∥_{N^{1,p}(X)} = ∥u∥_{Ñ^{1,p}(X)}.

For basic properties of the Newtonian spaces we refer to [Sh2]. We recall here some facts for the future reference. It can be shown that N^{1,p}(X) is a Banach space. The functions in N^{1,p}(X) are absolutely continuous on p-almost every path, which means that u ∘ γ is absolutely continuous on [0, length(γ)] for p-almost every rectifiable arc-length parametrized path γ in X.

The p-capacity of a set E ⊂ X is defined by

cap_{p}(E) = inf_{u} ∥u∥_{N^{1,p}(X)}^{p},

where the infimum is taken over all functions u ∈ N^{1,p}(X), whose restriction to E is bounded below by 1. A property is said to hold p-quasieverywhere, if it holds except on a set of p-capacity zero. Capacity is the natural measure for exceptional sets of Sobolev functions. It is easy to see that sets of zero capacity are also of measure zero, but the converse is not true in general. See [KinM1] for more properties of the capacity.

Let E be an arbitrary subset of X. Following [KKM] we define N_{0}^{1,p}(E) to be the set of functions u ∈ N^{1,p}(X) for which

cap_{p}({x ∈ X \ E: u(x) ≠ 0}) = 0.

The space N_{0}^{1,p}(E) equipped with the norm

∥u∥_{N_{0}^{1,p}(E)} = ∥u∥_{N^{1,p}(X)},

is the Newtonian space with zero boundary values. The norm is unambiguously defined by [Sh1] and the space N_{0}^{1,p}(E) with this norm is a Banach space.

1.4. Poincaré inequalities. A metric measure space X is said to be doubling if there is a constant c_{d} ≥ 1 so that

(1.5)  μ(B(z, 2r)) ≤ c_{d}μ(B(z, r))

for every open ball B(z, r) in X. The constant c_{d} in (1.5) is called the doubling constant of μ. Note that by the doubling property, if B(y, R) is a ball in X, z ∈ B(y, R) and 0 < r ≤ R < ∞, then

\frac{μ(B(z, r))}{μ(B(y, R))} ≥ c(\frac{r}{R})^{Q}

for some c and Q depending only on the doubling constant. Let 1 ≤ q < ∞. In the following discussion, whenever A ⊂ X is a measurable set, u_{A} denotes the quantity

\frac{1}{μ(A)} ∫_{A} u dμ =\overline{∫} _{A} u dμ.

The space X is said to support a weak (1, q)-Poincaré inequality if there are constants c_{0} > 0 and τ ≥ 1 such that

(1.6)  \overline{∫} _{B(z,r)} |u − u_{B(z,r)}| dμ ≤ c_{0}r(\overline{∫} _{B(z,τr)} g^{q} dμ)^{1/q}

for all balls B(z, r) in X, for all integrable functions u in B(z, r) and for all upper gradients g of u. If τ = 1, then X is said to support a (strong) (1, q)-Poincaré inequality.

A result of [HaK] shows that in a doubling measure space a weak (1, q)-Poincaré inequality implies a weak (t, q)-Poincaré inequality for some t > q possibly with possibly different constants τ and c_{0}. More precisely, there exist constants c > 0 and τ′ ≥ 1 such that

(1.7)  (\overline{∫} _{B(z,r)} |u − u_{B(z,r)}|^{t} dμ)^{1/t} ≤ cr(\overline{∫} _{B(z,τ′r)} g^{q} dμ)^{1/q},

where 1 ≤ t ≤ Qq/(Q−q) if q < Q and t ≥ 1 if q ≥ Q, for all balls B(z, r) in X and for all integrable functions u in B(z, r) and all upper gradients g of u. Moreover, in (1.6) and (1.7) we can replace the upper gradient g with any p-weak upper gradient in L_{loc}^{p}(X), because of the result in [KoM] which states that every p-weak upper gradient in L_{loc}^{p}(X) can be approximated in L^{p}(X) by an upper gradient in L_{loc}^{p}(X). Indeed, given any p-weak upper gradient g_{1} ∈ L_{loc}^{p}(X) of u and any ϵ > 0 we can find an upper gradient g_{2} so that

∥g_{2} − g_{1}∥_{L^{p}(X)} < ϵ.

1.8. Sobolev inequalities. Next we prove a Sobolev type inequality for functions which vanish on a large set. The paper [B] has a better capacitary version of this inequality, but for our purposes it suffices to consider the more easily proved version below (see [KSh, Lemma 2.1])

1.9. Lemma. Let X be a doubling metric measure space supporting a weak (1, q)- Poincaré inequality for some 1 < q < p. Suppose that u ∈ N^{1,p}(X), and denote the minimal p-weak upper gradient g_{u}. Let

A = {x ∈ B(z, r) : |u(x)| > 0}.

If μ(A) ≤ γμ(B(z, r)) for some γ with 0 < γ < 1, then there is a constant c > 0 so that

(\overline{∫} _{B(z,r)} |u|^{t} dμ)^{1/t} ≤ cR(\overline{∫} _{B(z,τ′r)} g_{u}^{q} dμ)^{1/q},

where t and τ′ are as in (1.7). The constant c depends only on γ and the constants c and τ′ of (1.7).

We observe that Lemma 1.9 gives a Sobolev inequality for Sobolev functions with zero boundary values. To be more precise, there exists c > 0 so that for every ball B(z, r) with 0 < r ≤ diam(X)/3 and every u ∈ N_{0}^{1,p}(B(z, r)) we have

(1.10)  (\overline{∫} _{B(z,r)} |u|^{t} dμ)^{1/t} ≤ cR(\overline{∫} _{B(z,r)} g_{u}^{q} dμ)^{1/q}.

This follows easily from Lemma 1.9 after noting that g_{u} = 0 almost everywhere on X \ B(z, r) and by observing that there must be a point on the sphere ∂B(z, 2r). If there is no such point, then it is easy to construct a function which violates the Poincaré inequality.

1.11. Minimizers. Now we are ready to formulate the minimization problem for the p-Dirichlet integral in a metric measure space. By N_{loc}^{1,p}(Ω) we mean the space of all functions u ∈ L_{loc}^{p}(Ω) that have an upper gradient in L_{loc}^{p}(Ω), where L_{loc}^{p}(Ω) is the space of all measurable functions that are p-integrable on compact subsets of Ω.

Suppose that Ω ⊂ X is open and let 1 < p < ∞. A function u ∈ N_{loc}^{1,p}(Ω) is said to be p-harmonic on Ω if for every relatively compact open subset Ω′ of Ω with \overline{Ω′} ⊂ Ω and for every v ∈ N^{1,p}(Ω′) with u − v ∈ N_{0}^{1,p}(Ω′) we have

∫_{Ω′} g_{u}^{p} dμ ≤ ∫_{Ω′} g_{v}^{p} dμ,

where g_{u} and g_{v} are the minimal weak upper gradients of u and v respectively. p-harmonic functions on metric measure spaces have been studied in [C], [Sh1], and [KSh].

Let ϑ ∈ N^{1,p}(Ω) and ψ: Ω → [−∞, ∞] be an arbitrary function. Following [KinM2], we define

K_{ψ,ϑ}(Ω) = {v ∈ N^{1,p}(Ω): v − ϑ ∈ N_{0}^{1,p}(Ω), v ≥ ψ μ-a.e. in Ω}.

A function u ∈ K_{ψ,ϑ}(Ω) is a solution of the K_{ψ,ϑ}(Ω)-obstacle problem with the obstacle ψ and the boundary values ϑ, if

∫_{Ω} g_{u}^{p} dμ ≤ ∫_{Ω} g_{v}^{p} dμ

for every v ∈ K_{ψ,ϑ}(Ω). Here g_{u} and g_{v} are the minimal p-weak upper gradients of u and v in Ω, respectively. In other words, the solution of the obstacle problem minimizes the p-Dirichlet energy among all functions which lie above the obstacle function ψ and have the boundary values ϑ. If ψ = −∞, then the obstacle does not hinder and a solution of the K_{−∞,ϑ}(Ω)-obstacle problem is said to be a minimizer with boundary values ϑ in Ω.

We are mostly interested in local properties of solutions of the obstacle problem and especially in defining solutions locally without any requirements on the boundary values. A function u ∈ N_{loc}^{1,p}(Ω) is a p-minimizer in Ω, if it is a solution of the K_{−∞,u}(Ω′)-obstacle problem for every open set Ω′ ⊂⊂ Ω. On the other hand, a function u ∈ N_{loc}^{1,p}(Ω) is a p-superminimizer in Ω, if it is a solution of the K_{u,u}(Ω′)- obstacle problem for every open set Ω′ ⊂⊂ Ω. In the Euclidean case minimizers correspond to solutions and superminimizers correspond to supersolutions of the p-Laplace equation.

It is an easy observation that whenever ϑ is in N^{1,p}(X), the solution to the K_{ψ,ϑ} obstacle problem is also in N^{1,p}(X).

1.12. Definition. A function u : Ω → (−∞, ∞] is said to be p-superharmonic in Ω if

(i) u is lower semicontinuous and is not identically infinite on any component of Ω, and

(ii) for all relatively compact open subsets V ⊂⊂ Ω and for all functions v ∈ N^{1,p}(X) that are p-minimizers in V, whenever v ≤ u p-q.e. in Ω \ V we have v ≤ u p-q.e. in V.

Remark. The second condition above can be replaced with the condition that for all relatively compact subsets V ⊂⊂ Ω and for all functions v ∈ N^{1,p}(X) ∩ Lip(X) that are p-minimizers in V, whenever v ≤ u p-q.e. in Ω \ V we have v ≤ u pq.e. in V, because of the fact that Lipschitz functions are dense in N^{1,p}(X) (see [Sh1]) together with [Sh3]. If we can approximate general open sets by regular open sets for the Dirichlet problem, this definition coincides with the definition of p-superharmonic functions given in [HKM, Chapter 7].

Definition 1.1 is slightly different from the definition of p-superharmonicity given in [KinM2], the difference being in the requirement of comparison with harmonic functions. Let Ω be an open subset of X. Suppose that V ⊂⊂ Ω is open and v ∈ N^{1,p}(V). Let h_{V} (v) = h(v) be the unique continuous p-minimizer in V with v − h(v) ∈ N_{0}^{1,p}(V). Observe that every p-minimizer is locally Hölder continuous after redefinition on a set of measure zero by [KSh]. The definition given in [KinM2] replaces the requirement (ii) of Definition 1.1 with the condition

(ii’) for every relatively compact open set V ⊂⊂ Ω and for every function v ∈ C(\overline{V}) ∩ N^{1,p}(V), whenever v ≤ u on ∂V we have that h(v) ≤ u on V.

The object of the next lemma is to prove that the two definitions are indeed equivalent.

1.13. Lemma. A lower semicontinuous function on Ω that is not identically infinite on any component of Ω is p-superharmonic if and only if it satisfies the condition (ii’) above.

Proof. First suppose that u is a p-superharmonic function, let V ⊂⊂ Ω and let v ∈ C(\overline{V}) ∩ N^{1,p}(V) such that v ≤ u on ∂V. For each ϵ > 0 there is a compactly supported Lipschitz function φ_{ϵ} on X so that |v − φ_{ϵ}| ≤ ϵ on ∂V. Note that φ_{ϵ} ∈ N^{1,p}(X). Since

min{φ_{ϵ} + ϵ − v, 0} ∈ N_{0}^{1,p}(V),

and consequently

min{h(φ_{ϵ}) + ϵ − h(v), 0} ∈ N_{0}^{1,p}(V),

we have by the comparison theorem (see Lemma 3.4 in [KinM2]) that

h(φ_{ϵ}) − ϵ ≤ h(v) ≤ h(φ_{ϵ}) + ϵ

on V. We observe that the function

h = \{ \begin{matrix}h(φ_{ϵ}) & on V, \\ φ_{ϵ} & on X \ V, \end{matrix}

belongs to N^{1,p}(X). Moreover, since u ≥ v ≥ φ_{ϵ} − ϵ on ∂V and u is p-super-harmonic, we have u ≥ h(φ_{ϵ}) − ϵ on V. Hence u ≥ h(v) − 2ϵ on V. Letting ϵ → 0 we see that u ≥ h(v) on V.

Now suppose that u satisfies the condition (ii’). Then u is a p-superharmonic function in the sense of [KinM2], and hence by [KinM2, Corollary 7.8] the functions

u_{i} = min{u, i}, i = 1, 2, …,

are superminimizers on Ω. Let V ⊂⊂ Ω and v ∈ N^{1,p}(X) so that v is a p-minimizer on V and v ≤ u p-q.e. on Ω \ V. By [Sh1, Lemma 4.9] the sequence v_{i} = min{v, i} is in N^{1,p}(X) and v_{i} → v in N^{1,p}(X). Thus by [Sh3, Theorem 1.1], h(v_{i}) converges locally uniformly to v = h(v) in V. Since v ≤ u p-q.e. on ∂V, we have v_{i} ≤ u_{i} p-q.e. on ∂V. From this it follows that h(v_{i}) ≤ u_{i} ≤ u on V, see the discussion following the proof of Corollary 4.8 in [Sh3]. Letting i → ∞, we see that h(v) ≤ u on V, and hence u is a p-superharmonic function on Ω.

1.14. General assumptions. From now on we assume without further notice that the metric measure space X is connected, proper, and is equipped with a doubling Borel regular measure for which the measure of every open set is positive and that the measure of every bounded set is finite. Furthermore we assume that the space supports a weak (1, q)-Poincaré inequality for some q with 1 < q < p. We assume throughout this paper that Ω is a bounded open set in X with the property that cap_{p}(X \ Ω) > 0.

2. Polar sets are zero capacity sets.

Let Ω ⊂ X be open.

2.1. Definition. A set E ⊂ Ω is said to be p-polar if there is a p-superharmonic function in Ω so that u = ∞ on E.

2.2. Proposition. Let u be a p-superharmonic function on Ω, and let

E = {x ∈ Ω: u(x) = ∞}.

Then E has zero p-capacity.

Proof. Let F be any compact subset of E. It suffices to show that the p-capacity of F is zero.

Suppose that the p-capacity of F is positive. Let B be a relatively compact subdomain of Ω that contains F, and let v be the p-potential of the condenser (F; B); that is, v = 1 on F, v = 0 on X \ B, 0 ≤ v ≤ 1, v is p-minimizing on B \ F, and v is in N^{1,p}(X). Since p-superharmonic functions are lower semicontinuous and hence are locally bounded from below, we may assume that u is non-negative on B. As B \ F is a relatively compact open subset of Ω and kv ≤ u on Ω \ (B \ F) for every k = 1, 2, …, and u is a p-superharmonic function, we have kv ≤ u on B \ F for every k = 1, 2, … We can choose B so that there is a point y ∈ B with u(y) finite. Then y is in B \ F. Hence

0 ≤ kv(y) ≤ u(y) < ∞

for every positive k. Hence v(y) = 0.

As the p-capacity of F is positive, u is a non-constant p-minimizing function on B \ F, and hence by the strong comparison theorem, v(y) > 0, thus providing the contradiction.

2.3. Definition. A function f: Ω → [−∞, ∞] is p-quasicontinuous if for every ϵ > 0 there exists an open set E such that cap_{p}(E) < ϵ and the restriction of f to Ω \ E is continuous.

It should be noted that functions in N^{1,p}(X) and N_{loc}^{1,p}(X) are p-quasicontinuous; see Remark 4.4 of [Sh2].

2.4. Corollary. Let u be a p-superharmonic function on Ω. Then u is p-quasi-continuous.

Proof. By [KinM2, Corollary 7.8] the functions

u_{i} = min{u, i}, i = 1, 2, …,

are p-superminimizers in Ω, and in particular, they belong to N_{loc}^{1,p}(Ω). Hence the functions u_{i} are p-quasicontinuous and the result follows from Proposition 2.2 and the fact that the p-capacity is a Choquet capacity. Indeed, the sets

U_{i} = {x ∈ Ω : u(x) > i}

are open and cap_{p}(U_{i}) → 0 as i → ∞.

3. Zero capacity sets are polar sets.

3.1. Lemma. Let E ⊂⊂ Ω such that cap_{p}(E) = 0. Then there exists a sequence of functions φ_{i} ∈ N^{1,p}(X) so that φ_{i} = 0 on X \ Ω, 0 ≤ φ_{i} ≤ 1, φ_{i} = 1 on E, φ_{i} is a p-superharmonic p-superminimizer on Ω, and φ_{i} → 0 in N^{1,p}(X) and φ_{i} monotonically decreases to zero p-q.e. in Ω.

Proof. Since cap_{p}(E) = 0, there exists a sequence of open sets U_{i} ⊂⊂ Ω with E ⊂ U_{i} ⊂ U_{i−1} so that cap_{p}(U_{i}) ≤ 2^{−i}. Let φ̃_{i} be the p-potential of the condenser (U_{i}; Ω), and let φ_{i} be the lower semicontinuous regularization of φ̃_{i} given by [KinM2, Proposition 7.4]. The functions φ_{i} satisfy the conditions of the lemma.

3.2. Proposition. Let Ω ⊂ X be a bounded open set, and let ψ_{i} ∈ N^{1,p}(X) be a p-q.e. monotonic decreasing sequence of non-negative functions so that ψ_{i} → ψ in N^{1,p}(X). Let u_{i} be the solution to the K_{ψ_{i},ψ_{i}}(Ω)-obstacle problem. Then there exists a function u ∈ N^{1,p}(X) so that u_{i} monotonically decreases p-q.e. in Ω to u and u is the solution to the K_{ψ,ψ}(Ω)-obstacle problem.

The above proposition is a complementary result to [KinM2, Theorem 6.1]. To prove this proposition we need the following lemma. This lemma is from [KinM2, Lemma 3.5].

3.3. Lemma. Let ψ ∈ N^{1,p}(X) be a non-negative function and let v be a solution to the K_{ψ,ψ}(Ω)-obstacle problem. If u ∈ N^{1,p}(X) is a p-superminimizer on Ω with min{u, v} ∈ K_{ψ,ψ}(Ω), then u ≥ v p-q.e. on Ω. Moreover, if ũ and ṽ are the lower semicontinuous regularizations of u and v respectively, we have ũ ≥ ṽ everywhere on Ω.

Proof. For the proof of the first part, see [KinM2, Lemma 3.5]. While their proof involves comparing u and v outside of a relatively compact subdomain of Ω, as u and v are in N^{1,p}(X) the relative compact open subsets are unnecessary. Moreover, by [KinM2, Theorem 7.14] we have

ṽ(x) = ess lim inf_{y→x} ṽ(y) ≤ ess lim inf_{y→x} ũ(y) = ũ(x)

for every x ∈ Ω.

Now we are ready to prove Proposition 3.2.

Proof of Proposition 3.2. Since u_{i} ≥ ψ_{i} ≥ ψ_{i+1}, therefore u_{i} ∈ K_{ψ_{i+1},ψ_{i}}(Ω). As u_{i} is a p-superminimizer on Ω and u_{i+1} is the solution to the obstacle problem with obstacle ψ_{i+1} and boundary data ψ_{i+1} ≤ ψ_{i}, by Lemma 3.3 above we have u_{i} ≥ u_{i+1}. Without loss of generality we may assume that u_{i} is lower semicontinuous in Ω; see [KinM2, Theorem 5.1]. Hence (u_{i})_{i} is a monotone decreasing sequence. Let

u = lim_{i→∞} u_{n}.

As u_{i} ∈ L^{p}(Ω) and u ≥ 0, we have u ∈ L^{p}(Ω). Also, as ψ_{i} ∈ K_{ψ_{i},ψ_{i}}(Ω), we have

∫_{Ω} g_{u_{i}}^{p} dμ ≤ ∫_{Ω} g_{ψ_{i}}^{p} dμ.

Hence by [KaS, Lemma 3.1] we see that u ∈ N^{1,p}(X) with

∫_{Ω} g_{u}^{p} dμ ≤ lim inf_{i→∞} ∫_{Ω} g_{u_{i}}^{p} dμ ≤ lim inf_{i→∞} ∫_{Ω} g_{ψ_{i}}^{p} dμ ≤ ∫_{Ω} g_{ψ}^{p} dμ.

Since u_{i} ≥ ψ_{i} ≥ ψ, we conclude that u ≥ ψ and hence u ∈ K_{ψ,ψ}(Ω) because u_{i} − ψ_{i} ∈ N_{0}^{1,p}(Ω) and ψ_{i} converges to ψ in N^{1,p}(X). Let v be the solution to the K_{ψ,ψ}(Ω)-obstacle problem. Then

∫_{Ω} g_{v}^{p} dμ ≤ ∫_{Ω} g_{u}^{p} dμ.

Also by Lemma 3.3 again, u_{i} ≥ v, and hence u ≥ v. Let φ_{i} = max{v, ψ_{i}}. Then as ψ ≤ ψ_{i}, φ_{i} ∈ K_{ψ_{i},ψ_{i}}(Ω), and hence

∫_{Ω} g_{u_{i}}^{p} dμ ≤ ∫_{Ω} g_{φ_{i}}^{p} dμ.

As ψ_{i} monotonically decreases to ψ p-q.e. in Ω and v ≥ ψ, therefore φ_{i} → v pointwise p-q.e. in Ω. Moreover, as ψ_{i} → ψ in N^{1,p}(X), we see that φ_{i} → v in N^{1,p}(X). Therefore,

lim inf_{i→∞} ∫_{Ω} g_{u_{i}}^{p} dμ ≤ ∫_{Ω} g_{v}^{p} dμ,

and hence,

∫_{Ω} g_{u}^{p} dμ ≤ ∫_{Ω} g_{v}^{p} dμ.

Thus we see that

∫_{Ω} g_{u}^{p} dμ = ∫_{Ω} g_{v}^{p} dμ.

By the uniqueness of solutions to obstacle problems (see for example [KinM2, Theorem 3.2] or [HKM, Theorem 3.21]) we have u = v; that is, u is the solution to the K_{ψ,ψ}(Ω)-obstacle problem.

Now we state and prove the main theorem of this section.

3.4. Theorem. Let x_{0} ∈ Ω and E be a relatively compact subset of Ω \ {x_{0}} such that cap_{p}(E) = 0. Then there exists a non-negative p-superharmonic function u on Ω so that u(x_{0}) < ∞, u = 0 on X \ Ω, and

E ⊂ {x ∈ Ω: u(x) = ∞}.

This theorem was proved in the setting of X = R^{n} by Kilpeläinen; see [Kil1, Theorem 4.1]. The proof of this theorem follows the lead of [Kil1]; however, we avoid their use of Perron-Wiener-Brelot solutions by using Proposition 3.2 instead.

Proof. Let φ_{i} be the functions given by Lemma 3.1. There exists an index i_{1} so that φ_{i_{1}} ≤ \frac{1}{2} in a neighbourhood of x_{0} and φ_{i_{1}} = 1 on a neighbourhood of E. Let ṽ_{1} be the lower semicontinuous regularization of the p-superminimizer φ_{i_{1}}; see [KinM2, Theorem 5.1]. Note that as φ_{i_{1}} is already a p-superharmonic function as well, we have ṽ_{1} = φ_{i_{1}}. We inductively construct functions ṽ_{n} as follows.

Suppose ṽ_{1}, …, ṽ_{n−1} have already been chosen so that they are p-superharmonic p-superminimizers in Ω, ṽ_{i} = 0 on X \ Ω, ṽ_{i} = i on a neighbourhood of E, and

ṽ_{i}(x_{0}) ≤ ∑_{k=1}^{i} 2^{−k},

and ṽ_{i} ∈ N^{1,p}(X).

Let v_{n,j} be the solution to the obstacle problem on Ω with obstacle ṽ_{n−1} + φ_{j} and boundary data 0. Since φ_{j} → 0 in N^{1,p}(X) and φ_{j} monotonically decreases to 0 p-q.e., we see that ṽ_{n−1} + φ_{j} monotonically decreases to ṽ_{n−1} p-q.e. and in N^{1,p}(X). Now by Proposition 3.2, v_{n,j} monotonically decreases to ṽ_{n−1} p-q.e., and hence its lower semicontinuous regularization monotonically decreases to ṽ_{n−1} everywhere in Ω; see [KinM2, Theorem 5.1]. Hence there is an index i_{n} so that the lower semicontinuous regularization of v_{n,i_{n}} is bounded by ∑_{k=1}^{n} 2^{−k} at x_{0}, and as v_{n,i_{n}} = n on a neighbourhood of E, the lower semicontinuous regularization of v_{n,i_{n}} also takes on the value of n in a neighbourhood of E. Set ṽ_{n} to be this lower semicontinuous regularization of v_{n,i_{n}}.

The sequence (ṽ_{n}) has the following properties:

• ṽ_{n} ≥ 0 and is p-superharmonic in Ω,

• ṽ_{n} = 0 on X \ Ω,

• ṽ_{n} ∈ N^{1,p}(X),

• ṽ_{n} = n on E,

• ṽ_{n+1} ≥ ṽ_{n} pointwise on Ω, and

• ṽ_{n}(x_{0}) ≤ ∑_{k=1}^{n} 2^{−k}. Let u = lim_{n→∞} ṽ_{n}. By [KinM2, Lemma 7.1] and by Lemma 1.13 above, since

u(x_{0}) ≤ ∑_{k=1}^{∞} 2^{−k}

is finite, u is p-superharmonic on Ω. Moreover, it is easy to see that u = ∞ on E.

Remark. In the Euclidean setting, in the proof of Lemma 3.1 we can approximate Ω by regular open sets Ω_{j} ⊂ Ω and the open sets U_{j} by regular open sets Ũ_{j}; this technique has been used extensively in [HKM] to study the Perron-Wiener-Brelot solutions. In this case, we can ensure that the functions φ_{i} are all continuous, and then it follows from [KinM2, Theorem 5.5] that the functions ṽ_{i} constructed in the above proof are all continuous. Unfortunately, in the general setting of metric spaces considered in this note, such approximations are not known to be possible.

4. The existence of global p-superharmonic maps defining p-polar sets in a p-hyperbolic metric space.

Following [Ho], we say that a metric space is p-hyperbolic if there exists a compact set C ⊂ X so that the relative capacity Cap_{p}(C: X) is positive. Here

Cap_{p}(C: X) = lim_{i→∞} Cap_{p}(C: Ω_{i}),

where {Ω_{i}}_{i} is an increasing sequence of bounded domains in X such that X = ∪_{i}Ω_{i}. It is easily seen that a metric space is p-hyperbolic if and only if it is not possible to find a sequence of compactly supported functions f_{i} ∈ N^{1,p}(X) so that f_{i} → 1 in N_{loc}^{1,p}(X) while their energies

∫_{X} g_{f_{i}}^{p} dμ

remain bounded.

4.1. Theorem. If X is a p-hyperbolic metric measure space equipped with a doubling measure supporting a (1, q)-Poincaré inequality for some 1 ≤ q < p, and if E ⊂ X is a compact set so that cap_{p}(E) = 0, then there exists a non-negative function u on X so that u is p-superharmonic in X and

E ⊂ {x ∈ X: u(x) = ∞}.

The proof of this theorem follows along the same lines as the proof of Theorem 3.4, once an analog of Lemma 3.1 and Proposition 3.2 are proven in the setting Ω = X.

4.2. Definition. In what follows, the family N_{loc,0}^{1,p}(X) denotes the collection of all functions f on X for which there exists a sequence of compactly supported functions f_{i} ∈ N^{1,p}(X) so that f_{i} → f in N_{loc}^{1,p}(X) and the corresponding sequence of energies

∫_{X} g_{f_{i}}^{p} dμ

is bounded independent of i. Weak limits of the functions g_{f_{i}} can be easily seen to be p-weak upper gradients of f in X. From the fact that f_{i} → f in N_{loc}^{1,p}(X) we can infer that g_{f_{i}} → g_{f} in L^{p}(X) and that g_{f} ∈ L^{p}(X).

It is easy to see that if λ > 0 and f, h ∈ N_{loc,0}^{1,p}(X) then the functions min{λ, f} and max{f, h} are in N_{loc,0}^{1,p}(X). Furthermore, N_{loc,0}^{1,p}(X) is a vector space endowed with a family of seminorms inherited from N_{loc}^{1,p}(X).

4.3. Definition. Let f ∈ N_{loc}^{1,p}(X), and let K_{f} denote the collection of all functions u ∈ N_{loc}^{1,p}(X) such that u − f ∈ N_{loc,0}^{1,p}(X) and u ≥ f p-q.e. in X. By a solution to the obstacle problem K_{f} we mean a function u ∈ K_{f} so that

∫_{X} g_{u}^{p} dμ ≤ ∫_{X} g_{v}^{p} dμ

whenever v ∈ K_{f}. Observe that

∫_{X} g_{u}^{p} dμ ≤ ∫_{X} g_{f}^{p} dμ.

4.4. Lemma. Let X be a p-hyperbolic metric space. Given E ⊂ X so that cap_{p}(E) = 0 there exists a sequence of functions φ_{i} ∈ N_{loc,0}^{1,p}(X) so that φ_{i} = 1 on a neighbourhood of E, φ_{i} is a p-superharmonic p-superminimizer on X,

∫_{X} g_{φ_{i}}^{p} dμ → 0

as i → ∞, and φ_{i} monotonically decreases to 0 pointwise p-q.e. in X.

The proof of Lemma 4.4 depends on the following lemma.

4.5. Lemma. Let X be a p-hyperbolic space equipped with a doubling measure supporting a (1, p)-Poincaré inequality. Given a bounded function f ∈ N_{loc}^{1,p}(X) with finite energy there is one, and only one, solution to the obstacle problem K_{f} in the class N_{loc,0}^{1,p}(X). Such a solution is p-superharmonic and is a p-superminimizer in X.

Proof. We denote the bound of f by M. Let Ω_{i} be an increasing sequence of bounded domains in X so that

X = ⋃_{i=1}^{∞} Ω_{i}.

As \overline{Ω}_{i} is a compact set and N_{loc,0}^{1,p}(X) ⊂ N_{loc}^{1,p}(X), we see that the solution u_{i} to the obstacle problem K_{f,f}(Ω_{i}) exists and lies in N_{loc,0}^{1,p}(X). By Lemma 3.3 we see that f ≤ u_{i} ≤ u_{i+1} ≤ M p-q.e. on X.

On the other hand, by the energy minimization property of u_{i}, we have

∫_{X} g_{u_{i}}^{p} dμ = ∫_{Ω_{i}} g_{u_{i}}^{p} dμ + ∫_{X\Ω_{i}} g_{f}^{p} dμ ≤ ∫_{X} g_{f}^{p} dμ < ∞,

where we know that the last integral in the above chain of inequalities is finite by assumption. Hence the sequence u_{i} has the property that

∫_{X} g_{u_{i}}^{p} dμ

is bounded independent of i. Thus the limit function u = lim_{i→∞} u_{i} lies in the collection N_{loc}^{1,p}(X). Observe that u_{i} ∈ N_{loc}^{1,p}(X) and that ∥u_{i}∥_{N^{1,p}(Ω_{k})} are bounded above by a constant independent of i. Therefore, as

u − f = lim_{i→∞}(u_{i} − f)

with u_{i} − f of compact support, we see that u ∈ K_{f}.

Suppose v ∈ K_{f}. Then as v ≥ f, there exists a sequence of non-negative functions w_{i} ∈ N^{1,p}(X) with compact support so that

v = lim_{i→∞}(w_{i} + f)

and (because of Mazur’s lemma) so that

∫_{X} g_{v}^{p} dμ = lim_{i→∞} ∫_{X} g_{w_{i}+f}^{p} dμ.

Fix ϵ > 0 and choose w_{i} so that

∫_{X} g_{v}^{p} dμ ≥ ∫_{X} g_{w_{i}+f}^{p} dμ − ϵ.

Since

X = ⋃_{j=1}^{∞} Ω_{j},

we can find Ω_{j} so that the compact support of w_{i} is contained in Ω_{j}; thus w_{i} + f ∈ K_{f,f}(Ω_{j}) for sufficiently large j. Now by the minimal energy property of u_{j}, we see that

∫_{X} g_{u_{j}}^{p} dμ ≤ ∫_{X} g_{w_{i}+f}^{p} dμ.

Hence for sufficiently large j we have that

∫_{X} g_{u_{j}}^{p} dμ ≤ ∫_{X} g_{v}^{p} dμ + ϵ.

Letting j → ∞ and then ϵ → 0, we see that

∫_{X} g_{u}^{p} dμ ≤ ∫_{X} g_{v}^{p} dμ,

that is, u is a solution to the obstacle problem K_{f}.

Since the monotone increasing sequence of functions (u_{i})_{i≥j} are p-superharmonic and are p-superminimizers in Ω_{j}, by [KinM2] we see that u is itself such a function on Ω_{j} for every j; that is, u is p-superharmonic and is a p-superminimizer in X.

It now remains to show that no other solution to the obstacle problem exists. Suppose v is another solution to the obstacle problem K_{f}. Then as v ≥ f p-q.e. on X, we see by an application of Lemma 3.3 to the domains Ω_{j} that v ≥ u_{j} for each j. It follows that v ≥ u p-q.e. on X. As v ≢ u and X supports a (1, p)-Poincaré inequality (and hence is an MEC_{p}-space, see [Sh2]), we have g_{v−u} ≢ 0 or else v − u is a non-zero constant function. By the results in [Sh2], the support of g_{v−u} is contained in the support of v − u itself. As u ≢ v, we see that

μ ({x ∈ X: u(x) < v(x)}) > 0.

Note that the set {x ∈ X: u(x) < v(x)} is a countable union of sets of the form

{x ∈ X: u(x) < a < v(x)}

where a ∈ R is a positive rational number. Let a be a positive rational number such that

μ ({x ∈ X: u(x) < a < v(x)}) > 0,

and define

h(x) = \{ \begin{matrix}v(x), & if v(x) ≤ a, \\ a, & if u(x) < a < v(x), \\ u(x), & if u(x) ≥ a. \end{matrix}

It is easily seen that h ∈ K_{f}, and hence by the minimality of the energy of the solutions,

∫_{X} g_{u}^{p} dμ = ∫_{X} g_{v}^{p} dμ ≤ ∫_{X} g_{h}^{p} dμ = ∫_{{v≤a}} g_{v}^{p} dμ + ∫_{{u≥a}} g_{u}^{p} dμ.

By the above inequality,

∫_{{v≤a}} g_{v}^{p} dμ = ∫_{{v≤a}} g_{u}^{p} dμ and ∫_{{u≥a}} g_{v}^{p} dμ = ∫_{{u≥a}} g_{u}^{p} dμ.

Hence

∫_{X} g_{h}^{p} dμ = ∫_{X} g_{u}^{p} dμ − ∫_{{u<a<v}} g_{u}^{p} dμ ≥ ∫_{X} g_{u}^{p} dμ,

which leads us to the conclusion that g_{u} = 0 almost everywhere in the set

{x ∈ X: u(x) < a < v(x)},

and a similar argument shows that g_{v} = 0 almost everywhere on this set. Thus g_{v−u} ≤ g_{v} + g_{u} = 0 almost everywhere on the set

{x ∈ X: u(x) < a < v(x)}.

Thus g_{v−u} = 0 almost everywhere on the set

{x ∈ X: u(x) < v(x)}.

Hence it follows that u + c ≡ v for some constant c > 0. On the other hand, as u − f, v − f ∈ N_{loc,0}^{1,p}(X) we have v − u ∈ N_{loc,0}^{1,p}(X). Hence as X is p-hyperbolic, we have c = 0, yielding the uniqueness result for the solution to the obstacle problem K_{f}.

Proof of Lemma 4.4. By Lemma 3.1 we obtain a sequence of compactly supported functions ϕ_{i} ∈ N^{1,p}(X) so that ϕ_{i} = 1 on E,

∫_{X} g_{ϕ_{i}}^{p} dμ ≤ 2^{−ip},

and ϕ_{i} monotonically decreases to zero p-q.e. in X. Let φ_{i} be the solution to the K_{ϕ_{i}}-obstacle problem constructed in the proof of Lemma 4.5. As ϕ_{i} is compactly supported, φ_{i} ∈ N_{loc,0}^{1,p}(X) with

∫_{X} g_{φ_{i}}^{p} dμ ≤ 2^{−ip}.

Moreover, by Lemma 4.5 we see that φ_{i} are p-superharmonic p-superminimizers in X.

As ϕ_{i} is monotonic decreasing, by construction and by Lemma 3.3 we see that (φ_{i})_{i} is a monotonic decreasing sequence in X. Let φ := lim_{i} φ_{i}. Since

∫_{X} g_{φ}^{p} dμ ≤ ∫_{X} g_{φ_{i}}^{p} dμ ≤ 2^{−ip},

we see that g_{φ} = 0 almost everywhere in X. As X supports a (1, p)-Poincaré inequality (and hence is an MEC_{p}-space; see [Sh2]), we see that φ ≥ 0 is constant p-a.e. in X. To see that this constant is zero, it suffices to know that φ ∈ N_{loc,0}^{1,p}(X).

Indeed, as ϕ_{i} has compact support, we see that φ_{i} ∈ N_{loc,0}^{1,p}(X). As φ_{i} are bounded by 1, we see that the sequence ϕ_{i,j}, obtained as the solution to the obstacle problem K_{ϕ_{i},ϕ_{i}}(Ω_{j}), is a bounded sequence in N_{loc}^{1,p}(X). Fix one such relatively compact domain Ω_{k}. Hence by Mazur’s lemma we can choose a subsequence of convex combination (ϕ̂_{i,j})_{j} of the sequence (ϕ_{i,j})_{j} so that ϕ̂_{i,j} → φ_{i} in N^{1,p}(Ω_{k}) for every relatively compact domain Ω_{k} in X. Fix ϵ > 0. Then for every positive integer i we can find a cutoff integer j_{i} so that whenever j ≥ j_{i} we have

∥φ_{i} − ϕ̂_{i,j}∥_{N^{1,p}(Ω_{k})} < ϵ/2.

Similarly, as φ_{i} → φ in a monotonic decreasing fashion and is a bounded sequence in N^{1,p}(Ω_{k}), we can obtain a convex combination subsequence

ϕ̃_{n} = ∑_{i=n}^{N_{n}} λ_{i,n}φ_{i}

that converges in N^{1,p}(Ω_{k}) to φ. Thus a corresponding convex combination of the sequence ϕ̂_{i,j} converges in N^{1,p}(Ω_{k}) to φ. More specifically, with ϵ > 0 fixed as above, we can find a positive integer i_{0} so that

∥φ̃_{i_{0}} − φ∥_{N^{1,p}(Ω_{k})} < ϵ/2.

Choosing j_{0} = max{j_{i_{0}}, j_{i_{0}+1}, …, j_{N_{i_{0}}} } and letting

ϕ̃_{i_{0}} := ∑_{i=i_{0}}^{N_{i_{0}}} λ_{i,i_{0}}ϕ_{i},

it is easy to see that

∥ϕ̃_{i_{0}} − φ̃_{i_{0}}∥_{N^{1,p}(Ω_{k})} < ϵ/2.

Thus we obtain an approximation ϕ̃_{i_{0}} of φ so that this approximation is compactly supported, in N^{1,p}(X), and in addition,

∥ϕ̃_{i_{0}} − φ∥_{N^{1,p}(Ω_{k})} < ϵ.

We can now conclude that φ ∈ N_{loc,0}^{1,p}(X), and hence cannot be a non-zero constant. Thus φ = 0, and the proof is complete.

Remark. The above proof also shows that if u_{i} ∈ N_{loc,0}^{1,p}(X) is a sequence that converges to a function u ∈ N_{loc}^{1,p}(X), with the convergence occuring in N_{loc}^{1,p}(X), and furthermore, the energies

∫_{X} g_{u_{i}}^{p} dμ

are bounded independent of i, then u is also in N_{loc,0}^{1,p}(X). Hence N_{loc,0}^{1,p}(X) is a complete space.

4.6. Proposition. Let ψ_{i} ∈ N_{loc,0}^{1,p}(X) be a sequence of p-q.e. monotone decreasing sequence of non-negative functions so that ψ_{i} → ψ ∈ N_{loc,0}^{1,p}(X), the convergence occuring also in N^{1,p}(X). Let u_{i} be the solution to the obstacle problem K_{ψ_{i}}. Then u_{i} → u both monotonically p-q.e. in X and weakly in N_{loc}^{1,p}(X), where u is the solution to the obstacle problem K_{ψ}.

The proof of Proposition 4.6 is along the same lines as that of Proposition 3.2.

Proof. Lemma 3.3 together with the proof of Lemma 4.5 shows that u_{i} ≥ u_{i+1} p-q.e. since ψ_{i} ≥ ψ_{i+1}. Let

u = lim_{i→∞} u_{i}.

As u_{i} ≥ ψ_{i} ≥ ψ, we see that u ≥ ψ. Furthermore, as

∫_{X} g_{ψ_{i}}^{p} dμ

is bounded independent of i, we see that so is the sequence of energies

∫_{X} g_{u_{i}}^{p} dμ.

Hence by the remark following the proof of Lemma 4.5, u ∈ N_{loc,0}^{1,p}(X) and thus we can conclude that u ∈ K_{ψ}. Moreover, as

∫_{X} g_{u_{i}}^{p} dμ ≤ ∫_{X} g_{ψ_{i}}^{p} dμ → ∫_{X} g_{ψ}^{p} dμ,

we see that

∫_{X} g_{u}^{p} dμ ≤ ∫_{X} g_{ψ}^{p} dμ.

Let v denote the solution to the obstacle problem K_{ψ} (Lemma 4.5 guarantees such a solution). From the fact that u ∈ K_{ψ} we see that

∫_{X} g_{u}^{p} dμ ≥ ∫_{X} g_{v}^{p} dμ.

By the uniqueness of solutions to obstacle problems (Lemma 4.5), it therefore suffices to show that

∫_{X} g_{u}^{p} dμ ≤ ∫_{X} g_{v}^{p} dμ.

To this end, as in the proof of Proposition 3.2, let φ_{i} = max{v, ψ_{i}}. Then φ_{i} ∈ N_{loc,0}^{1,p}(X) and φ_{i} ≥ ψ_{i}. Therefore, φ_{i} ∈ K_{ψ_{i}} and

∫_{X} g_{u_{i}}^{p} dμ ≤ ∫_{X} g_{φ_{i}}^{p} dμ.

On the other hand, as ψ_{i} → ψ in N^{1,p}(X) and v ≥ ψ, we see that φ_{i} → v in N^{1,p}(X); that is,

∫_{X} g_{φ_{i}}^{p} dμ → ∫_{X} g_{v}^{p} dμ.

Therefore

∫_{X} g_{u}^{p} dμ ≤ lim inf_{i→∞} ∫_{X} g_{u_{i}}^{p} dμ ≤ ∫_{X} g_{v}^{p} dμ,

completing the proof of the proposition.

Proof of Theorem 4.1. The proof of Theorem 4.1 follows the exact lines as the proof of Theorem 3.4, with Proposition 4.6 playing the role of Proposition 3.2 and Lemma 3.1, Lemma 4.5, and Lemma 4.4 together playing the role of Lemma 3.1. We leave the details up to the reader.

5. Whitney decomposition and sweeping of measures.

The aim of the next section is to generalize the result of [Kil2, Theorem 3.1] to the setting of metric spaces considered in this note. The proof in [Kil2] used the classical Whitney decomposition of the complement of a closed polar set. One of the primary characteristics of a Whitney decomposition is that the corresponding covering family of cubes is pairwise disjoint. In our setting we are unable to obtain this characteristic, but the decomposition we use in this section has a bounded overlap property that compensates for this shortcoming.

5.1. Definition. Let K be a compact subset of X. A countable cover F of balls in X \K is said to be a Whitney decomposition of X \K if there is a constant C > 0 so that the following three conditions hold:

(1) ∑_{B∈F} χ_{2B} ≤ C;

(2) \frac{1}{2}rad(B) ≤ dist(B, K) ≤ 8 rad(B) for every B ∈ F;

(3) 2B ⊂ X \ K and ⋃_{B∈F} B = X \ K.

The following proposition demonstrates the existance of such a decomposition.

5.2. Proposition. Given a compact subset K of X, there exists a Whitney decomposition F of X \ K.

We refer to the proof of [CW, Theorem 1.3] in page 70 for a proof of this proposition.

We define the following Wolff potential for every compactly supported Radon measure ν on X.

5.5. Definition. Given a Radon measure ν on X, the Wolff potential of ν is the function

W_{p}^{ν}(x, r) := ∫_{0}^{r} (t^{p} \frac{ν(B(x, t))}{μ(B(x, t))})^{1/(p−1)} \frac{dt}{t}

for every x ∈ X and r > 0.

Following [Kil2, Theorem 2.15], we prove the existence of a “swept” measure of ν supported on K for every finite Radon measure ν on X and any compact set K ⊂ X.

5.6. Lemma. Let K ⊂ X be a compact set and ν be a finite Radon measure on X (that is, ν(X) is finite). Then there exists a finite Radon measure ν̃, supported on K, so that ν(X)/C ≤ ν̃(X) ≤ Cν(X) and for every point x ∈ K and r > 0,

W ν̃ ; p(x, 21r) ≥ \frac{1}{C} W_{p}^{ν}(x, r)

where C > 0 is independent of x and r.

Proof. Let F be the Whitney decomposition of X \ K given by Proposition 5.2. For each ball B ∈ F, choose a point x_{B} ∈ K so that d(x_{B}, B) = dist(K, B) (this is possible because K is compact). By the definition of Whitney decompositions, we see that d(x_{B}, B) ≤ 8 rad(B). We now construct one possible measure ν̃ as follows: for sets E ⊂ X, let

ν̃(E) := ν(E ∩ K) + ∑_{B∈F} ν(B)δ_{x_{B}}(E),

where δ_{x_{B}} is the Dirac mass supported at the point x_{B}. Clearly ν̃ is a Radon measure supported inside K, and by the bounded overlap property of F and by the covering property of F, it is easily seen that

ν(X)/C ≤ ν̃(X) ≤ Cν(X)

where C is the constant associated with the bounded overlap property of F.

Let x ∈ K and r > 0, and consider the family F(x, r) to be the collection of all balls B ∈ F which intersect B(x, r). If B ∈ F(x, r) and y ∈ B ∩ B(x, r), then

d(x_{B}, y) ≤ d(x_{B}, B) + 2 rad(B) ≤ 10 rad(B)
≤ 20 dist(B, K) ≤ 20 d(x, y) ≤ 20r.

Therefore, d(x_{B}, x) ≤ 21r, and hence

ν(B(x, r)) ≤ ν ((B(x, r) ∩ K) ⋃_{B∈F(x,r)} B)

≤ ν(B(x, r) ∩ K) + ∑_{B∈F(x,r)} ν(B)

≤ ν(B(x, r) ∩ K) + ∑_{x_{B}∈B(x,21r)} ν(B) = ν̃(B(x, 21r)).

Now from the definition of Wolff potential and by the doubling property of the measure μ,

W ν̃ ; p(x, 21r) = ∫_{0}^{21r} (t^{p} \frac{ν̃(B(x, t))}{μ(B(x, t))})^{1/(p−1)} \frac{dt}{t}

≥ ∫_{0}^{21r} (t^{p} \frac{ν(B(x, t/21))}{μ(B(x, t))})^{1/(p−1)} \frac{dt}{t}

≥ \frac{1}{C} ∫_{0}^{r} (t^{p} \frac{ν(B(x, t))}{μ(B(x, t))})^{1/(p−1)} \frac{dt}{t} = W_{p}^{ν}(x, r),

where the constant C depends on the doubling constant of μ but is indeed independent of x and r.

6. Finite Radon measures and p-superharmonic functions.

The p-superharmonic functions, defined using p-minimizers, are influenced by what is meant by p-minimizers. If, as in [KinM2], [KSh], [Sh1], and in the first four sections of this note, we define p-minimizers v to be functions that locally minimize the integral

∫ g_{w}^{p} dμ

amongst the class of all functions w ∈ N_{0}^{1,p}(V) + v, where g_{v} is the minimal p-weak upper gradient of v, then the corresponding p-minimizers may or may not satisfy an Euler-Lagrange equation. This leads to a disadvantage in that we cannot explicitly identify the space of all finite Radon measures on Ω with the corresponding space of non-negative p-superharmonic functions on Ω as in [HKM], [KilM1-2] or [Mi]. To address this shortcoming, in this section we modify the definition of p-minimizers by using the results of [C] and [BMS].

In [C] Cheeger gives another definition of a Sobolev type space, but for indices p > 1 Cheeger’s construction yields the same space as N^{1,p}(X); see [Sh2, Theorem 4.10]. It is a deep theorem of Cheeger that if a metric measure space supports a doubling measure and a (1, p)-Poincaré inequality, then the corresponding Sobolevtype space is reflexive; see [C, Theorem 4.48]. The results of [C] can easily be extended to the case of spaces supporting a locally Q-regular measure and a local (1, p)-Poincaré inequality.

In [C, Theorem 4.38] it was shown that if the measure on X is doubling and if X supports a (1, p)-Poincaré inequality for some p, then there is a “differentiable” structure on X compatible with the notion of upper gradient. More precisely, the following theorem of Cheeger holds.

6.1. Theorem. Let X be a metric measure space equipped with a locally doubling Borel regular measure μ. Assume that X admits a weak (1, p)-Poincaré inequality for some 1 < p < ∞. Then there exist measurable sets U_{α} with μ(X \ ⋃_{α} U_{α}) = 0 and there exist Lipschitz “coordinate” functions

\overline{X}^{α} = (X_{1}^{α}, …, X_{k(α)}^{α}): X → R^{k(α)}

so that

(i) for each α, the functions X_{1}^{α}, …, X_{k(α)}^{α} are all linearly independent on U_{α} and 1 ≤ k(α) ≤ N, where N is a constant depending only on the doubling constant of μ and the constants from the Poincaré inequality,

(ii) if f : X → R is Lipschitz, then there exist unique bounded vector-valued functions d^{α}f : U_{α} → R^{k(α)} such that

lim_{r→0+} sup_{x∈B(x_{0},r)} \frac{|f(x) − f(x_{0}) − d^{α}f(x_{0}) · (\overline{X}^{α}(x) − \overline{X}^{α}(x_{0}))|}{r} = 0

for μ-a.e. x_{0} ∈ U_{α}.

In other words, there exists a finite dimensional L^{∞}-vector bundle of Banach spaces T^{∗}X = {F_{x}}_{x∈X}, the generalized cotangent bundle, and a derivation operator D on the algebra of locally Lipschitz functions on X taking values in the sections of T^{∗}X. For x in the “coordinate chart” U_{α}, the fiber F_{x} is the dual space to the space R^{k(α)} mentioned in the theorem. Given a Newtonian function f, a norm on the space F_{x} is given by

||λ||_{α,x} = g_{⟨λ,\overline{X}^{α}⟩}(x),

where, for λ = (λ_{1}, …, λ_{k(α)}) in R^{k(α)} we set ⟨λ, \overline{X}^{α}⟩ to be the Lipschitz function given by

∑_{i=1}^{k(α)} λ_{i} X_{i}^{α}.

By linear independence of X_{1}^{α}, …, X_{k(α)}^{α} we mean that whenever g_{⟨λ,\overline{X}^{α}⟩}(x) = 0 for some x ∈ U_{α} we have λ = 0. Finally, in a given chart U_{α}, Df is given by d^{α}f.

Note that the generalized cotangent bundle is only a measurable vector bundle, that is, the charts U_{α} are measurable subsets of X and the change of coordinate functions \overline{X}^{β} ∘ (\overline{X}^{α})^{−1} are bimeasurable invertible transformations.

By [Sh2, Theorem 4.10] and [C, Theorem 4.47], the Newtonian space N^{1,p}(X) is equal to the closure in the N^{1,p}(X)-norm of the collection of locally Lipschitz functions on X, and by [FHK, Theorem 10], the derivation operator D can be extended to all of N^{1,p}(X) so that

|Df(x)| ≈ g_{f}(x)

for all f ∈ N^{1,p}(X) and μ-a.e. x ∈ X. Here the norm | · | is obtained from a measurable inner product on T^{∗}X, chosen to validate the above equivalence with g_{f}; see the discussion in [C].

One easily verifies (see also [C, p. 458]) that the derivation operator Du satisfies the product and chain rules: if v is a bounded Lipschitz function on X, u ∈ N^{1,p}(X), and Φ : R → R is continuously differentiable with bounded derivative, then uv and Φ(u) belong to N^{1,p}(X) and

D(uv) = u Dv + v Du,
D(Φ ∘ u) = Φ′(u) Du.

We modify the notion of p-harmonicity of [KinM2], [Sh2], and [KSh] by minimizing |Du| rather than g_{u}.

6.2. Definition. Let Ω ⊂ X be an open set. A function u : X → [−∞, ∞] is said to be p-harmonic on Ω if u ∈ N_{loc}^{1,p}(Ω) and for all relatively compact subsets U of Ω and for all functions φ ∈ N_{0}^{1,p}(U),

∫_{U} |Du|^{p} dμ ≤ ∫_{U} |D(u + φ)|^{p} dμ.

It must be noted that by the results in [C], the p-harmonic functions as defined above are p-quasiminimizers in the sense of [KSh]. Moreover, p-harmonic functions can be characterized in terms of a weak formulation of the appropriate Euler-Lagrange equation: f is p-harmonic if and only if

∫_{U} |Du|^{p−2}Du · Dϕ dμ = 0

for all U and φ as in Definition 6.2. Here we make use of the fact that the norms |·| on the fibers of T^{∗}X can be chosen to be inner product norms; see the discussion in [C, p. 460].

6.3. Definition. A function u ∈ N_{loc}^{1,p}(Ω) is said to be a p-supersolution if for all relatively compact open subset V of Ω and for all functions φ ∈ N_{0}^{1,p}(V) so that φ ≥ 0,

∫_{V} |Du|^{p−2}Du · Dφ dμ ≥ 0,

or equivalently,

∫_{V} |Du|^{p} dμ ≤ ∫_{V} |D(u + φ)|^{p} dμ.

Remarks.

(i) A function is p-harmonic if and only if it is both a p-supersolution and a p-subsolution (a function u is a p-subsolution if −u is a p-supersolution). For the remainder of this section we consider the p-superharmonic functions obtained from Definition 1.12 by using p-harmonic functions instead of p-minimizers in the comparison tests. We also define solutions to obstacle problems by minimizing the integral ∫ |Du|^{p} dμ rather than ∫ g_{u}^{p} dμ.

(ii) Since p-harmonic functions and p-supersolutions exhibit the minimizing property of p-minimizers and p-superminimizers respectively, the results of the first three sections of this note as well as the results of [Sh1], [KinM2], and [KSh] apply to the modified versions of p-superharmonic functions.

(iii) The above definition of p-harmonic functions and p-supersolutions has been studied in [BMS] using the techniques developed in [KilM1-2] and [Mi], and it has been shown in [BMS] that the set of all non-negative p-supersolutions u ∈ N_{0}^{1,p}(Ω) and the Radon measures ν in the positive cone in the dual space N_{0}^{1,p}(Ω)^{∗} are in one-to-one correspondence given by

(6.4)  ∫_{Ω} |Du|^{p−2}Du · Dφ dμ = ∫_{Ω} φ dν

whenever φ ∈ N_{0}^{1,p}(Ω).

We extend the results of [BMS] and construct a correspondence between compactly supported finite Radon measures on Ω and non-negative p-superharmonic functions with vanishing boundary values, thus extending the results of [KilM1-2] and [Mi] to the metric space setting. We will use this correspondence together with the sweeping technique of the previous section to construct a p-superharmonic function on Ω whose infinity set is precisely the given zero capacity set.

We first show that given a p-superharmonic function there is an associated finite Radon measure ν supported on Ω, related to u via equation (6.4) above. It was shown in [KinM2, Corollary 7.8] that if k is a positive integer, then

u_{k} = min{u, k}

is in N_{loc}^{1,p}(Ω), and hence Du_{k} is well-defined a.e., and moreover, if j > k, then Du_{j} = Du_{k} on the set

{x ∈ Ω: u(x) ≤ k}.

6.5. Definition. Let u be a non-negative p-superharmonic function on Ω. Then

Du := lim_{k→∞} Du_{k}

is well-defined a.e. on the set {x ∈ Ω: u(x) < ∞}.

The arguments in [HKM, Theorem 7.46] and [KilM1, Theorem 1.15] show that in the Euclidean setting with Du = ∇u, |Du| ∈ L^{1}(Ω). Their argument is quite general, and is easily adapted to our situation; see below.

6.6. Lemma. Let Ω be a bounded domain with cap_{p}(X \ Ω) > 0 and u be a non-negative p-superharmonic function on Ω. Then |Du| ∈ L_{loc}^{p−1}(Ω).

Proof. By [KinM2, Corollary 7.8] the function

u_{k} = min{u, k}

is a p-supersolution and u_{k} ∈ N_{loc}^{1,p}(Ω). By the Caccioppoli type inequality [BMS, Lemma 4.4], if B is a ball in X so that 3B ⊂ Ω and η ≥ 0 is a Lipschitz function with support in 2B so that η = 1 on B, then

∫_{B} |Du_{k}|^{p}u_{k}^{−(1+ϵ)} dμ ≤ ∫_{2B} |Du_{k}|^{p}η^{p}u_{k}^{−(1+ϵ)} dμ
≤ C ∫_{2B} u_{k}^{p−(1+ϵ)}|Dη|^{p} dμ ≤ C_{B} ∫_{2B} u_{k}^{p−(1+ϵ)} dμ.

Let B be a ball in X so that 5B ⊂ Ω. By [BMS, Lemma 4.5] and by the fact that X supports a (p_{0}, p)-Poincaré inequality for some p_{0} > p (see [HaK, Section 5]), we have that

(6.7)  (\overline{∫} _{B} u^{γ} dμ)^{1/γ} ≤ C_{γ} inf_{2B} u

whenever γ < p_{0}(p − 1)/p. Here the constant C_{γ} is independent of u and B, but may depend on γ. Just as in the alternative proof of [HKM, Theorem 7.46], if ϵ > 0, then

∫_{B} |Du_{k}|^{p−1} dμ = ∫_{B} |Du_{k}|^{p−1}u_{k}^{−(1+ϵ)(p−1)/p} u_{k}^{(1+ϵ)(p−1)/p} dμ

≤ (∫_{B} |Du_{k}|^{p}u_{k}^{−(1+ϵ)} dμ)^{(p−1)/p} (∫_{B} u_{k}^{(1+ϵ)(p−1)} dμ)^{1/p}

≤ (∫_{B} |Du_{k}|^{p}u_{k}^{−(1+ϵ)} dμ)^{(p−1)/p} (∫_{B} u^{(1+ϵ)(p−1)} dμ)^{1/p}

≤ C_{B} (∫_{2B} u^{p−(1+ϵ)} dμ)^{(p−1)/p} (∫_{B} u^{(1+ϵ)(p−1)} dμ)^{1/p} .

Since p_{0}/p > 1, we can find ϵ > 0 so that 1 + ϵ < p_{0}/p, that is, (1 + ϵ)(p − 1) < p_{0}(p − 1)/p. Thus by Proposition 2.2 and inequality (6.7) above,

∫_{B} |Du_{k}|^{p−1} dμ ≤ C_{B,p,p_{0}}(inf_{B} u)^{p−1} < ∞.

Therefore the sequence |Du_{k}| is a bounded sequence in L_{loc}^{p−1}(Ω), and hence by the reflexivity of L_{loc}^{p−1}(Ω) we conclude that |Du| ∈ L_{loc}^{p−1}(Ω).

The next proposition is a metric space analog of [KilM1, Theorem 1.15], and associates to each non-negative p-superharmonic function a Radon measure ν.

6.8. Proposition. If u is a non-negative p-superharmonic function on domain Ω, then there is a Radon measure ν, supported on \overline{Ω}, so that for every φ ∈ Lip_{c}(Ω),

∫_{Ω} |Du|^{p−2}Du · Dφ dμ = ∫_{Ω} φ dν.

Observe that the equation given in the above proposition is precisely equation (6.4).

Proof. Let u_{k} = min{u, k}, and φ ∈ Lip_{c}(Ω). Since u_{k} ∈ N_{loc}^{1,p}(Ω) and φ has compact support in Ω, by [BMS, Proposition 3.5] there is a Radon measure ν_{k} supported in \overline{Ω} so that

∫_{Ω} |Du_{k}|^{p−2}Du_{k} · Dφ dμ = ∫_{Ω} φ dν_{k}.

By Lemma 6.6 above, |Du_{k}|^{p−2}Du_{k} · Dφ converges to |Du|^{p−2}Du · Dφ in L^{1}(Ω). Therefore, (ν_{k})_{k} is a bounded sequence of Radon measures on compact subsets of Ω, and hence weakly converge to a Radon measure ν; that is,

∫_{Ω} φ dν_{k} → ∫_{Ω} φ dν

for every φ ∈ Lip_{c}(Ω). Thus by the convergence of |Du_{k}|^{p−2}Du_{k}·Dφ to |Du|^{p−2}Du· Dφ again, we see that

∫_{Ω} |Du|^{p−2}Du · Dφ dμ = ∫_{Ω} φ dν.

Remark. Unlike in [BMS, Proposition 3.5], we cannot replace the condition of φ ∈ Lip_{c}(Ω) with φ ∈ N_{0}^{1,p}(Ω) because unlike the Radon measures associated with psupersolutions, the Radon measures associated to p-superharmonic functions as in the above proposition may charge sets of zero p-capacity.

The aim of the next section is to construct a p-superharmonic function whose infinity set is precisely the closed zero p-capacity set E under study; this would be a refinement of Theorem 3.4. In order to construct such a p-superharmonic function we follow the lead of [Kil2]. We sweep the Radon measure associated with the p-superharmonic function constructed in Theorem 3.4 onto the set E using the technique developed in Section 5, and use the swept measure to construct the desired p-superharmonic function. The rest of this section is devoted to showing that for any finite Radon measure there is a corresponding p-harmonic function related to it as in (6.4). The argument closely follows the proofs of [KilM1, Theorem 2.4] and [Mi, Theorem 2.21] and uses the machinery set up in [BMS].

6.9. Proposition. Given a finite Radon measure ν on the domain Ω, there exists a non-negative p-superharmonic function u on Ω so that for all φ ∈ Lip_{c}(Ω),

∫_{Ω} |Du|^{p−2}Du · Dφ dμ = ∫_{Ω} φ dν.

Proof. By [BMS, Lemma 3.12] there exists a sequence (ν_{k})_{k} of Radon measures on Ω so that ν_{k} → ν weakly and ν_{k}(Ω) ≤ ν(Ω), and by [BMS, Proposition 3.9] there is an associated sequence of p-supersolutions u_{k} ∈ N_{0}^{1,p}(Ω) so that for every k,

∫_{Ω} |Du_{k}|^{p−2}Du_{k} · Dφ dμ = ∫_{Ω} φ dν_{k}.

Note that

∫_{Ω} φ dν_{k} → ∫_{Ω} φ dν

as k → ∞. For positive integers j set

u_{k,j} = min{u_{k}, j}.

Then as |Du_{k,j}| = 0 on the set where u_{k} ≥ j and u_{k,j} ∈ N_{0}^{1,p}(Ω), we see that

∫_{Ω} |Du_{k,j}|^{p} dμ = ∫_{Ω} |Du_{kj}|^{p−2}Du_{k,j} · Du_{k,j} dμ
= ∫_{Ω} |Du_{k}|^{p−2}Du_{k} · Du_{k,j} dμ
= ∫_{Ω} u_{k,j} dν_{k} ≤ j ν_{k}(Ω) ≤ j ν(Ω).

Therefore, by the Sobolev inequality [KSh, Lemma 2.10], we have

∫_{Ω} u_{k,j}^{p} dμ ≤ C ∫_{Ω} |Du_{k,j}|^{p} dμ ≤ Cν(Ω) j.

Now,

k^{p} μ ({x ∈ Ω : j ≤ u_{k}(x) < 2j}) ≤ ∫_{Ω} u_{k,2j}^{p} dμ ≤ 2Cj.

Thus, by Hölder’s inequality, when 0 < q < 1,

∫_{{j≤u_{k}<2j}} |Du_{k,2j}|^{q(p−1)} dμ

≤ μ ({j ≤ u_{k} < 2j})^{1−q(p−1)/p} (∫_{Ω} |Du_{k,2j}|^{p} dμ)^{q(p−1)/p} ≤ Cj^{−(1−q)(p−1)}.

Note that 1 − q > 0. Hence, when 0 < i < 2^{m+1},

∫_{Ω} |Du_{k,i}|^{q(p−1)} dμ ≤ ∑_{j=0}^{m} ∫_{{2^{j}<u_{k}≤2^{j+1}}} |Du_{k,2^{j}}|^{q(p−1)} dμ

≤ C ∑_{j=0}^{m} 2^{−j(1−q)(p−1)} ≤ C(p, q).

Therefore, as u_{k,i} → u_{k} in N_{0}^{1,p}(Ω), we have

∫_{Ω} |Du_{k}|^{q(p−1)} dμ ≤ C(p, q),

the bound being independent of k. Hence without loss of generality we can assume that the sequence |Du_{k}|^{p−2}Du_{k} is a weakly convergent sequence in L^{1}(Ω; R^{N}).

For every positive integer j the sequence (u_{k,j})_{k} is bounded in N_{0}^{1,p}(Ω). By the reflexivity of N_{0}^{1,p}(Ω) we can assume that for every positive integer j the sequence (u_{k,j})_{k} converges weakly to a function v_{j} in N_{0}^{1,p}(Ω) and

|Du_{k,j}|^{p−2}Du_{k,j} → |Dv_{j}|^{p−2}Dv_{j}

weakly in L^{1}(Ω; R^{N}).

Moreover, we claim that v_{j} ≤ v_{j+1}. To see this, we use a careful diagonalization argument. Since (u_{k,j})_{k} converges weakly to v_{j}, by Mazur’s lemma we can choose a convex combination of this sequence converging strongly to v_{j} in N_{0}^{1,p}(Ω). Consider the corresponding convex combination sequence obtained from (u_{k,j+1})_{j}; it converges weakly to v_{j+1} as well; now obtain a corresponding convex combination of this new sequence converging strongly to v_{j+1}. Since u_{k,j} ≤ u_{k,j+1}, the convex combination of the convex combination that gave v_{j+1} has every term larger than the corresponding convex combination of convex combination of (u_{k,j})_{k} as well, which again converges strongly to v_{j}. Hence v_{j} ≤ v_{j+1}.

Since

v_{j} = lim_{i→∞} inf_{k≥i} u_{k,j} = lim_{i→∞} lim_{l→∞} inf_{i≤k≤l} u_{k,j}

is p-superharmonic on Ω and v_{j} ≤ j, by [KinM2, Corollary 7.9] we conclude that v_{j} is a p-supersolution on Ω. Hence by [KinM2, Lemma 7.1] we see that the lower semicontinuous regularization of the function v = lim_{j→∞} v_{j} is either a p-superharmonic function on Ω or is identically infinite on Ω.

Suppose the lower semicontinuous regularization of v is identically infinite. Then v is infinite a.e.; that is, v_{j}(x) → ∞ for a.e. x ∈ Ω. Since

v_{j} = lim inf_{k→∞} u_{k,j},

this implies that v_{j} = j a.e. on Ω. This contradicts the fact that v_{j} ∈ N_{0}^{1,p}(Ω) because of the (1, p)-Poincaré inequality. Thus v is p-superharmonic on Ω, and hence by Proposition 2.2 v is finite p-q.e. in Ω. Consequently,

v = lim inf_{k→∞} u_{k}

on the set where v is finite. Moreover, as min{v, j} = v_{j}, we see by the definition of Dv and by the proof of Lemma 6.6 that

|Dv_{j}|^{p−2}Dv_{j} → |Dv|^{p−2}Dv

weakly in L^{1}(Ω; R^{N}). The weak convergence of Du_{k,j} to Dv_{j} now indicates that

∫_{Ω} |Du_{k,j}|^{p−2}Du_{k,j} · Dφ dμ → ∫_{Ω} |Dv|^{p−2}Dv · Dφ dμ

as k and then j tend to infinity. Since Du_{k,j} = Du_{k} wherever u_{k} ≤ j (almost everywhere), and Du_{k,j} = 0 wherever u_{k} > j, we see by the weak convergence of |Du_{k}|^{p−2}Du_{k} that

∫_{Ω} |Du_{k}|^{p−2}Du_{k} · Dφ dμ → ∫_{Ω} |Dv|^{p−2}Dv · Dφ dμ,

and hence

∫_{Ω} |Dv|^{p−2}Dv · Dφ dμ = ∫_{Ω} φ dν.

The lower semicontinuous regularization u of v now satisfies the conditions of the proposition.

7. Compact zero p-capacity sets and ∞-sets of modified p-superharmonic functions.

In the previous section we studied the relations of the p-superharmonic functions to the Radon measures. The next lemma is an adaptation of [KilM2, Theorem 1.6] (see also [Mi, Theorem 3.1]) to the metric setting.

7.1. Lemma. Let u be a non-negative p-superharmonic function on Ω in the sense of section 6 and let ν be the measure associated with u as in Propositions 6.8 and 6.9. Then there is a constant C > 0 so that for every ball B = B(x, r) with 4B ⊂ Ω,

\frac{1}{C} W_{p}^{ν}(x, r) ≤ u(x) ≤ C (inf_{B} u + W_{p}^{ν}(x, 2r)) .

The proof of this lemma follows the same lines as the proofs of [BMS, Proposition 4.9] and [BMS, Proposition 4.10] once [BMS, Lemma 4.8] is extended to p-superharmonic functions; this can be done just as in [KilM1, Lemma 3.5] and [Mi, Lemma 3.5], and is left to the reader to verify.

Now we are ready to state and prove the main result of this section.

7.2. Theorem. Let E be a compact subset of Ω so that cap_{p}(E) = 0. Then there exists a non-negative p-superharmonic function u on Ω so that

E = {x ∈ Ω: u(x) = ∞}.

Proof. By Theorem 3.4 there exists a non-negative p-superharmonic function v on Ω so that E ⊂ {x ∈ Ω : v(x) = ∞}. Let ν be the Radon measure associated with v as in Proposition 6.8. Let ν̃ be the swept measure corresponding to ν constructed via Lemma 5.6, and let u be the non-negative p-superharmonic function on Ω associated with ν̃ as in Proposition 6.9. We show that u satisfies the above theorem.

Since v = ∞ on E, by Lemma 7.1 we see that W_{p}^{ν}(x, r) = ∞ for all x ∈ E and for all r > 0 so that B(x, 4r) ⊂ Ω. Hence by Lemma 5.6, W ν̃ ; p(x, r) = ∞ whenever x ∈ E and B(x, 84r) ⊂ Ω. Again by Lemma 7.1, we have u(x) = ∞ when x ∈ E.

On the other hand, since ν̃ is supported on E and E is compact, if x ∈ Ω \ E there is a radius r > 0 so that W ν̃ ; p(x, 2r) = 0, and hence by Lemma 7.1 and Proposition 2.2 we see that u(x) is finite. Thus

E = {x ∈ Ω: u(x) = ∞}.

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J.K., Department of Mathematical Sciences, P.O. Box 3000, FIN-90014 University of Oulu, Finland

   E-mail address: juha.kinnunen@oulu.fi

N. S., Department of Mathematical Sciences, P.O. Box 210025, University of Cincinnati, Cincinnati, OH 45221-0025, U.S.A.

   E-mail address: nages@math.uc.edu