The Kellogg property and boundary regularity for p-harmonic functions with respect to the Mazurkiewicz boundary and other compactifications

In this paper boundary regularity for p-harmonic functions is studied with respect to the Mazurkiewicz boundary and other compactifications. In particular, the Kellogg property (which says that the set of irregular boundary points has capacity zero) is obtained for a large class of compactifications, but also two examples when it fails are given. This study is done for complete metric spaces equipped with doubling measures supporting a p-Poincar\'e inequality, but the results are new also in unweighted Euclidean spaces.


Introduction
In this paper we study boundary regularity for p-harmonic functions on metric spaces, including R n as an important case. Such studies have earlier been done with respect to the given metric boundary, but the novelty here is that we consider boundary regularity with respect to the Mazurkiewicz boundary and other compactifications.
This builds on the earlier work for the Dirichlet problem with respect to the Mazurkiewicz boundary, by Björn-Björn-Shanmugalingam [15], and more recently with respect to arbitrary compactifications, by Björn-Björn-Sjödin [17]. Boundary regularity was however not considered therein.
To be more precise, let 1 < p < ∞ and let X be a complete metric space equipped with a doubling measure µ supporting a p-Poincaré inequality. X can e.g. be unweighted R n , in which case a function u is p-harmonic if it is a continuous weak solution of the p-Laplace equation div(|∇u| p−2 ∇u) = 0.
The definition of p-harmonic functions on metric spaces is more involved, see Section 4.
Let Ω be a bounded domain. (If X is bounded we also require that C p (X \ Ω) > 0.) The Mazurkiewicz distance d M on Ω is defined by where the infimum is taken over all connected sets E ⊂ Ω containing x, y ∈ Ω. (The Mazurkiewicz distance was first used by Mazurkiewicz [38] in 1916, but goes under different names in the literature, see Remark 4.2 in [15].) The Mazurkiewicz boundary ∂ M Ω is the boundary of Ω in the completion of (Ω, d M ). For instance, in the slit disc B(0, 1) \ [0, 1] ⊂ R 2 this gives two boundary points corresponding to each point in the slit (but for the tip), while for smooth domains ∂ M Ω = ∂Ω.
Assume that the completion of (Ω, d M ) is compact, which happens if and only if Ω is finitely connected at the boundary, see Section 3. Then, to each point in the given metric boundary ∂Ω there corresponds one or more (at most countably many) points in the Mazurkiewicz boundary ∂ M Ω, while conversely to every point in ∂ M Ω there corresponds a unique point in ∂Ω, see Björn-Björn-Shanmugalingam [16]. There is therefore a natural projection Φ : ∂ M Ω → ∂Ω between these boundaries.
The following is the first new result, and the key tool to obtaining the Kellogg property. This leads directly to the following consequence. Moreover, u equals the Perron solution P Ω M f .
We are also able to show that boundary regularity is a local property for the Mazurkiewicz boundary in the following sense. Theorem 1.4. Assume that Ω is a bounded domain which is finitely connected at the boundary. Letx ∈ ∂ M Ω and let G be an Ω M -neighbourhood ofx.
Thenx is regular with respect to Ω M if and only if it is regular with respect to G Ω M , where G Ω M is G equipped with the boundary inherited from Ω M .
Throughout the paper we also study to what extent such results are true for other compactifications of Ω. The details describing the different results and cases are quite involved.
Boundary regularity for p-harmonic functions with respect to the given metric boundary has been studied for a long period, especially on R n . The first significant result was Maz ′ ya's [37] sufficiency part of the Wiener criterion in 1970. Later on the full Wiener criterion was obtained in various situations including weighted R n and for Cheeger p-harmonic functions on metric spaces, see [25], [32], [35], [39] and [18]. The full Wiener criterion remains open (for the given metric boundary) in the generality considered here, but the sufficiency has been obtained, see [21] and [19], and a weaker necessity condition, see [20].
The Wiener criterion characterizes the regularity of a boundary point using the complement of the domain (beyond the boundary). Many other results, such as the Kellogg property, do not directly involve the complement, but the proofs of most boundary regularity results do use the complement (beyond the boundary) in significant ways.
In our situation we have a boundary of the domain, but no complement beyond that. Thus most of the techniques used to study boundary regularity with respect to the given metric boundary are not available to us. Instead we will mainly depend on comparing boundary regularity between different boundaries. In particular, most of our stronger results are for boundaries larger than the given metric boundary.
We also give several counterexamples showing that the specific assumptions in our results are at least to some extent essential. These include two examples where the Kellogg property fails.

Notation and preliminaries
We will need quite a bit of notation, which we will introduce in this and the next two sections. We will be brief, see Björn-Björn-Shanmugalingam [15] and Björn-Björn-Sjödin [17] for more details. Proofs of the results in this section can be found in the monographs Björn-Björn [10] and Heinonen-Koskela-Shanmugalingam-Tyson [27].
We assume throughout the paper that 1 < p < ∞ and that X = (X, d, µ) is a metric space equipped with a metric d and a positive complete Borel measure µ such that 0 < µ(B) < ∞ for all balls B ⊂ X.
We will only consider curves which are nonconstant, compact and rectifiable (i.e. have finite length), and thus each curve can be parameterized by its arc length ds. A property is said to hold for p-almost every curve if it fails only for a curve family Γ with zero p-modulus, i.e. there exists 0 ≤ ρ ∈ L p (X) such that γ ρ ds = ∞ for every curve γ ∈ Γ.
where the left-hand side is considered to be ∞ whenever at least one of the terms therein is infinite.
If f has a p-weak upper gradient in L p loc (X), then it has an a.e. unique minimal p-weak upper gradient g f ∈ L p loc (X) in the sense that for every p-weak upper gradient g ∈ L p loc (X) of f we have g f ≤ g a.e., see Shanmugalingam [42]. Following Shanmugalingam [41], we define a version of Sobolev spaces on the metric space X.
where the infimum is taken over all p-weak upper gradients g of f . The Newtonian space on X is [41]. We also define D p (X) = {f : f is measurable and has a p-weak upper gradient in L p (X)}.
In this paper we assume that functions in N 1,p (X) and D p (X) are defined everywhere (with values in R), not just up to an equivalence class in the corresponding function space. For a measurable set E ⊂ X, the Newtonian space N 1,p (E) is defined by considering (E, d| E , µ| E ) as a metric space in its own right. We say that where the infimum is taken over all u ∈ N 1,p (X) such that u ≥ 1 on E.
The measure µ is doubling if there exists a doubling constant C > 0 such that Definition 2.4. X supports a p-Poincaré inequality if there exist constants C > 0 and λ ≥ 1 such that for all balls B ⊂ X, all integrable functions f on X and all p-weak upper gradients g of f , In this paper neighbourhoods are always open and continuous functions are real-valued, whereas semicontinuous functions may take the values ±∞. Let Ω be a locally compact noncompact metric space. A couple (∂ ′ Ω, τ ) is said to compactify Ω if ∂ ′ Ω is a set with ∂ ′ Ω ∩ Ω = ∅ and τ is a Hausdorff topology on Ω ′ := Ω ∪ ∂ ′ Ω such that (i) Ω ′ is compact with respect to τ ;

Compactifications and the capacity C p
(ii) Ω is dense in Ω ′ with respect to τ ; (iii) the topology induced on Ω by τ coincides with the given topology on Ω.
The space Ω ′ with the topology τ is a compactification of Ω.
Since Ω ′ is a compact Hausdorff space it is normal.
We assume from now on that X is a complete metric space supporting a p-Poincaré inequality, that µ is doubling, and that 1 < p < ∞. We also assume that Ω is a nonempty bounded open set such that C p (X \Ω) > 0, and that Ω j = Ω j ∪∂ j Ω, j = 1, 2, are compactifications of Ω, where Ω j = Ω with the intended boundary ∂ j Ω and where the topologies on Ω j are denoted by τ j . Furthermore, we reserve ∂Ω and Ω for the given metric boundary and closure induced by X on Ω.
As µ is doubling and X is complete, it follows that X is proper (i.e. all closed bounded sets are compact).
We define ∂ 1 Ω ≺ ∂ 2 Ω to mean that there is a continuous mapping, which is called projection, An example of a compactification is the Mazurkiewicz completion discussed in the introduction, which is a compactification of Ω if and only if Ω is a domain which is finitely connected at the boundary (in the following sense), by Theorem 1.1 in Björn-Björn-Shanmugalingam [16] or Theorem 1.3.8 in Karmazin [28].
Definition 3.2. A bounded domain Ω ⊂ X is finitely connected at the boundary if for every x ∈ ∂Ω and r > 0 there is an open set G (in X) such that x ∈ G ⊂ B(x, r) and G ∩ Ω has only finitely many components.
In addition to the Sobolev capacity mentioned above, we will also need the following capacity. It was introduced for Ω and Ω M by Björn-Björn-Shanmugalingam [15], and generalized to arbitrary compactifications as here in Björn-Björn-Sjödin [17, Definition 4.1]. A similar capacity was considered in Kilpeläinen-Malý [31].
When proving the Kellogg property we will need the following lemma.
Lemma 3.4. Assume that ∂ 1 Ω ≺ ∂ 2 Ω and let Φ : Ω 2 → Ω 1 be the projection. Let We will use nets to study convergence in our compactifications, see e.g. Pedersen [40] for the key results on nets.
Conversely, assume that u ∈ A Φ −1 (E) . Let x ∈ E ∩ ∂ 1 Ω. Assume that there is a net y λ ∈ Ω such that y λ τ 1 −→x and lim inf λ u(y λ ) < 1. By taking a subnet we may assume that lim λ u(y λ ) exists and is less than 1. Then there is a further subnet y µ which converges to some pointx ∈ Ω 2 . Hence lim µ u(y µ ) < 1. As Φ is continuous we see that y µ τ 1 −→Φ(x), and thus Φ(x) = x. But together with lim µ u(y µ ) < 1, this contradicts the assumption u ∈ A Φ −1 (E) . Hence Therefore the infima defining the two capacities are taken over the same set and the two capacities agree.
As a consequence we can obtain the following result, which shows that the Kellogg property is not seeking something trivial.
Proof. Let Ω 2 be the one-point compactification of Ω. We will first show that j=1 is a Cauchy sequence in N 1,p (X) and thus, by Corollary 1.72 in [10], it has a subsequence which converges q.e. to v ∈ N 1,p (X). We get directly that v ≡ 0 in X \ Ω. Moreover, if 0 < δ < 1 2 , then which tends to 0 as j → ∞. It follows that v = 1 a.e. in Ω.
We are now ready to introduce the Perron solutions with respect to Ω 1 . We follow Björn-Björn-Sjödin [17], although therein Perron solutions were only defined in domains.
for all x ∈ ∂ 1 Ω. The upper Perron solution of f is then defined to be while the lower Perron solution of f is defined by If P Ω 1 f = P Ω 1 f and it is real-valued, then we let P Ω 1 f := P Ω 1 f and f is said to be resolutive with respect to Ω 1 . Furthermore, let DU f (Ω 1 ) = U f (Ω 1 ) ∩ D p (Ω), and define the Sobolev-Perron solutions of f by In every component of Ω the upper/lower (Sobolev)-Perron solutions are pharmonic or identically ±∞, see Theorem 4.1 in Björn-Björn-Shanmugalingam [14] (or Theorem 10.10 in [10]). The Sobolev-Perron solutions were introduced in Björn-Björn-Sjödin [17]; we will only use them in Corollary 7.4.
The given metric boundary ∂Ω is resolutive by Theorem 6.1 in [14] (or Theorem 10.22 in [10]). It is also Sobolev-resolutive by Theorem 6.4 and Proposition 7.3 in [17]. If Ω is finitely connected at the boundary, then ∂ M Ω is resolutive by Theorem 8.2 in Björn-Björn-Shanmugalingam [15]. Also ∂ M Ω is Sobolev-resolutive (if Ω is finitely connected at the boundary), which again follows from Theorem 6.4 and Proposition 7.3 in [17] since continuous functions on ∂ M Ω can be uniformly approximated by Lipschitz functions on Ω M .
The following results from [17] will be important for us.
One consequence of Theorem 4.3 is that (almost always) there are plenty of resolutive functions.

Boundary regularity
Resolutivity will play an important role in several of our boundary regularity results. One possibility would have been to restrict our attention to resolutive boundaries. Here we have instead chosen a more general approach introducing both regular and resolutive-regular boundary points. The idea of studying resolutive-regularity is due to Sjödin (private communication).
We also say that ∂ 1 Ω is (resolutive)-regular if all its boundary points are (resolutive)-regular.
Note that if all continuous functions are resolutive, then regularity and resolutiveregularity of course coincide. Example 10.5 shows that this is not true in general.
The following result shows that we can equivalently replace lim by lim sup and = by ≤ in Definition 5.1.
The following result shows that the boundary regularity classification (into regular and irregular boundary points) can be useful also for noncontinuous boundary data. As regularity and resolutive-regularity are different, by Example 10.5, the latter cannot be characterized in a similar fashion. Proposition 5.3. Let x 0 ∈ ∂ 1 Ω. Then the following are equivalent : for all bounded f : for all functions f : Together with (5.2) this gives the desired conclusion.
Proof. Since x 0 is (resolutive)-irregular there is, due to Lemma 5.2, a (resolutive) function f ∈ C(∂ 1 Ω) such that We may assume that M = 2 and f ( As h(x) = f (x 0 ) = 0 and h is continuous (and resolutive if f is), this shows thatx is (resolutive)-irregular with respect to Ω 2 .
6. The proof of Theorem 1.1 We are now ready to consider our generalization of Theorem 1.1. For this we need an additional assumption, which we now define.
We say that On the way to proving Theorem 6.2 we first obtain the following result, which generalizes both (c) ⇒ (b) and (d) ⇒ (b) in Theorem 6.2. It shows in particular that the niceness assumption can be dropped for the implication (d) ⇒ (b). Theorem 6.3. Assume that ∂Ω ≺ ∂ 2 Ω, where ∂Ω is the given metric boundary. Let Φ : Ω 2 → Ω be the projection. If there are finitely many, and at least one, resolutive-irregular boundary points in Φ −1 (x 0 ) with respect to Ω 2 , or more general there is a resolutive-irregular boundary pointx ∈ Φ −1 (x 0 ), with respect to Ω 2 , which has arbitrarily small τ 2 -neighbourhoods U such that x ′ is resolutive-irregular with respect to Ω 2 } = ∅, (6.1) then x 0 is irregular with respect to Ω.
Note thatf need not be continuous, but since (6.1) holds,f is continuous at x 0 .
If u ∈ U f (Ω 2 ), then u ∈ Uf (G), where G is equipped with the given metric topology of Ω. Hence, P Gf ≤ P Ω 2 f in G. It thus follows that which, together with Proposition 5.3, shows that x 0 is irregular with respect to G. Since ∂Ω is the given metric boundary, Corollary 4.4 in Björn-Björn [8] (or Corollary 11.3 in [10]) shows that x 0 is irregular with respect to Ω. (c) ⇒ (d) Letx 1 ,x 2 ∈ Φ −1 (x 0 ) be resolutive-irregular with respect to Ω 2 . Assume thatx 1 =x 2 . We can proceed as in the proof of Theorem 6.3 finding functions f j ,f j and sets U j , G j corresponding tox j , j = 1, 2. We may require that U 1 ∩U 2 = ∅. As in the proof of Theorem 6.3, we see that x 0 is irregular with respect to G 1 and also with respect to G 2 . Since G 1 and G 2 are disjoint, this contradicts Lemma 7.4 in Björn [6] (or Lemma 11.32 in [10]). We also do not know if "resolutive-regular" can be replaced by "regular" in

The Kellogg property and uniqueness results
Our aim in this section is to establish Theorems 1.2 and 1.3 and suitable generalizations of them. As an application of Theorem 6.2, we can obtain the following so-called Kellogg property under the assumption that ∂Ω ≺ ∂ 2 Ω.
If in addition ∂ 2 Ω is resolutive, then we obtain the Kellogg property for ∂ 2 Ω, i.e. C p (Irr 2 ; Ω 2 ) = 0, where Irr 2 is the set of irregular boundary points with respect to Ω 2 . Example 10.2 shows that the Kellogg property does not hold for arbitrary resolutive compactifications, while Example 10.5 shows that it does not hold for Irr 2 with respect to arbitrary compactifications ∂ 2 Ω ≻ ∂Ω. Note also that by Proposition 3.5 the full boundary of any compactification always has positive capacity, and hence the Kellogg property is never trivial.
To establish Theorem 1.3, and its generalization Theorem 7.3 below, we need the following two conditions: One may think that we also need to require that ∂ 1 Ω is resolutive, but in fact this is a consequence of the two assumptions above, as we show in Proposition 7.2 below. Note also that by Proposition 3.5 the weak Kellogg property is never trivial. The equality in (7.1) can equivalently be replaced by the inequality in Lemma 5.2, see the proof of that lemma. We do not know if all boundaries are q.e.-invariant.
Example 10.5 shows that the weak Kellogg assumption cannot be dropped, nor can it be replaced by the resolutive Kellogg property.
Proof. Let f ∈ C(∂ 1 Ω) and h = ∞χ E −f , where E −f comes from the weak Kellogg property for −f . Then P Ω 1 f ∈ U f −h , and hence by the q.e.-invariance and Proposition 4.2, P from which it follows that f is resolutive. Moreover, u = P Ω 1 f .
Examples 10.2 and 10.5 show that the weak Kellogg assumption cannot be dropped, and the latter example also shows that it cannot be replaced by the resolutive Kellogg property.
Proof. By Proposition 7.2, f is resolutive and, by the weak Kellogg property, u = P Ω 1 f satisfies (7.2), which establishes the existence.
As for the uniqueness, let u be a bounded p-harmonic function and E ⊂ ∂ 1 Ω be such that C p (E; Ω 1 ) = 0 and Let h = ∞χ E . Then u ∈ U f −h and thus u ≥ P Ω 1 (f − h) = P Ω 1 f , by the q.e.invariance. By applying this to −u and −f we also see that u ≤ P Ω 1 f . Hence u = P Ω 1 f .
The weak Kellogg property of course follows from the usual Kellogg property (but not from the resolutive Kellogg property, see Example 10.5). However, if Ω 1 is metrizable then the weak Kellogg property is equivalent to the usual Kellogg property. Observe that we do not assume that Ω 1 is resolutive. In the proof below we use that it follows from the metrizability that C(∂ 1 Ω) is separable, and instead we could have used this assumption. However, by Theorem 2.10 in Björn-Björn-Sjödin [17] these two assumptions are equivalent. The name "weak Kellogg property" was coined in Björn [2] where it was obtained for quasiminimizers with respect to the given metric boundary. For quasiminimizers, it is not known if the Kellogg property holds or not. Similarly, when C(∂ 1 Ω) is not separable we do not know if the weak Kellogg property for p-harmonic functions implies the usual Kellogg property.
Proof. Assume that the weak Kellogg property holds. By Theorem 2.10 in [17], where E f comes from the weak Kellogg property for f . As the capacity is countably subadditive, we see that C p (E; Ω 1 ) = 0.
If f ∈ C(∂ 1 Ω), then we can find f j ∈ A such that f j → f uniformly. Then also P Ω 1 f j → P Ω 1 f uniformly and it follows that Hence Irr 1 ⊂ E and the Kellogg property follows. The converse implication is trivial.

Further results when
The results in Section 5-7 can be strengthened when Φ −1 (x 0 ) is finite. The following are the two main results in this section, which improve upon Proposition 5.4 and Theorem 6.2 under more restrictive assumptions.
Let x 0 ∈ ∂ 1 Ω and assume that Φ −1 (x 0 ) consists of just one pointx. Then x 0 is regular with respect to Ω 1 if and only ifx is regular with respect to Ω 2 .
Example 10.3 shows that Proposition 8.1 cannot be generalized to the case when Φ −1 (x 0 ) is finite. But if we assume that ∂ 1 Ω = ∂Ω is the given metric boundary, we do obtain the following characterization. The implication (c ′ ) ⇒ (b ′ ) is of course trivial and the implication (a ′ ) ⇒ (b ′ ) follows from Proposition 5.4. One may ask how much of this result remains true without assuming that the smaller boundary is the given metric boundary ∂Ω. If we instead would assume that the larger boundary is ∂Ω, then in fact no other than the two implications mentioned above would be true, see Examples 10.1 and 10.3.
As a consequence of Theorem 8.2 we can obtain the non-resolutive Kellogg property under some conditions, but without requiring resolutivity of the boundary. Theorem 8.3. (The Kellogg property) Assume that ∂Ω ≺ ∂ 2 Ω, where ∂Ω is the given metric boundary, and that Φ −1 (x 0 ) is finite for C p ( · ; Ω)-q.e. x 0 ∈ ∂Ω.
Proof. The proof is almost identical to the proof of Theorem 7.1, but using Theorem 8.2 instead of Theorem 6.2.
To prove Proposition 8.1 we will use the following characterization, which may be of independent interest. P Ω 1 f K (y) = 0 for all nonempty compact If there is a nonnegative function h ∈ C(∂ 1 Ω) which is zero only at x 0 , then we can use that function alone, i.e. we may let f K = h for all K. This is however possible if and only if x 0 has a countable base of neighbourhoods (which in particular holds if Ω 1 is first countable).
Proof. If ∂ 1 Ω = {x 0 } is the one-point compactification of Ω, then x 0 is regular as it is the only boundary point, and the equivalence is trivial. So assume that Assume first that (8.1) holds. Let f ∈ C(∂ 1 Ω). We may assume that f (x 0 ) = 0. Let ε > 0. Then there is a τ 1 -neighbourhood G Letting ε → 0 shows that lim sup As f was arbitrary, Lemma 5.2 yields that x 0 is regular. The converse implication is trivial.
Proof of Proposition 8.1. One direction follows from Proposition 5.3 but we will nevertheless show the full equivalence directly.
For each nonempty compact K ⊂ ∂ 1 Ω \ {x 0 }, let f K ∈ C(∂ 1 Ω) be nonnegative and such that f (x 0 ) = 0 < inf K f K (which exists by Tietze's extension theorem). When proving Theorem 8.2 we also need the following restriction result.
Boundary regularity with respect to the given metric is a local property by Theorem 6.1 in Björn-Björn [8] (or Theorem 11.11 in [10]). Proposition 8.5 shows that one direction of this equivalence holds in full generality, while Example 10.3 shows that the other does not. We will discuss this further in Section 9.
Proof of Theorem 8.2. Let x 1 , ... , x m be the points in Φ −1 (x 0 ). As Ω 2 is normal we can for each x j find a τ 2 -neighbourhood G j of x j whose closure avoids the other points. After having chosen all G j we can make each one smaller (if necessary, and still denoting it G j ) to make sure that its closure does not intersect the other closures either. Let next U j = G j ∩ Ω. We will now consider U j both with its given metric closure U j and with the τ 2 -closure U ) we see that x 0 is regular with respect to U j . Hence, by Proposition 8.1, x j is regular with respect to U 2 j . Thus, x j is regular with respect to Ω 2 , by Proposition 8.5. ( Assume first that there are (at least) two irregular boundary points in Φ −1 (x 0 ) with respect to Ω 2 , which we may assume to be x 1 and x 2 . It then follows from Proposition 8.5 that x j is also irregular with respect to U 2 j , j = 1, 2. Hence, by Proposition 8.1, x 0 is irregular with respect to U j , j = 1, 2. Since U 1 and U 2 are disjoint, this contradicts Lemma 7.4 in Björn [6] (or Lemma 11.32 in [10]). We thus conclude that if (c ′ ) fails, then there is no irregular boundary point in Φ −1 (x 0 ) (with respect to Ω 2 ), and thus (b ′ ) also fails.
The last part now follows from Theorem 6.2 (together with the obvious fact that a regular point is resolutive-regular).

Regularity as a local property
Boundary regularity with respect to the given metric is a local property by Theorem 6.1 in Björn-Björn [8] (or Theorem 11.11 in [10]). The following result extends this to a large class of boundaries greater than the given metric boundary, and also deduces a "restriction result" for the same boundaries (quite different from the restriction result in Proposition 8.5).
Example 10.3 shows that neither of these two facts hold in general. We do not know if they may hold for arbitrary boundaries larger than the given metric boundary. As already noted, one direction in the equivalence does hold for arbitrary compactifications, by Proposition 8.5.
Proof. We assume first that Φ −1 (x 0 ) is finite. In order to prove the first part we need to consider two cases.
Case 1. x 0 is regular with respect to Ω (with the given metric boundary). In this case it follows from Corollary 4.4 in Björn-Björn [8] (or Corollary 11.3 in [10]), that x 0 is regular with respect to U . It then follows from Theorem 8.2 (applied to U ) thatx is regular with respect to U 2 . Case 1. x 0 is irregular with respect to Ω. By Theorem 8.2 there is x ′ ∈ Φ −1 (x 0 ) which is irregular with respect to Ω 2 . Sincex is regular with respect to Ω 2 , we must have x ′ =x. As Ω 2 is a normal space there are τ 2 -neighbourhoods G and G ′ ofx and x ′ , respectively, with disjoint τ 2 -closures.
. By Proposition 8.5, x ′ is irregular with respect to V 2 . Thus, by Theorem 8.2, (b ′ ) ⇒ (c ′ ), applied to V ,x must be regular with respect to V 2 . As V and V ′ are disjoint the Perron solution P V 2 f within V only depends on the boundary values on ∂ 2 V . Sincex / ∈ V ′ 2 , it follows thatx is regular also with respect to V 2 . As V ⊂ U , it follows from Proposition 8.5 thatx is regular with respect to U 2 .
One direction of the second part follows directly from the first part, while the other one is a direct consequence of Proposition 8.5.
The proof in case (ii) is similar, but using Theorem 6.2 instead of Theorem 8.2.
The rather complicated condition (ii) above is essential for our proof (as Theorem 6.2 is applied to V ) and it may seem hard to know when it is satisfied. However, the main way of showing that the boundary ∂ 2 Ω is resolutive (and almost the only available way) is to show that continuous functions on ∂ 2 Ω can be uniformly approximated (on ∂ 2 Ω) by functions in from which the resolutivity (and even Sobolev-resolutivity) of ∂ 2 Ω follows by Theorem 6.4 and Proposition 7.3 in Björn-Björn-Sjödin [17]. If one instead require that continuous functions on Ω 2 can be uniformly approximated (on Ω 2 ) by functions in A, then not only ∂ 2 Ω is resolutive, but also ∂ 2 Ω ′ for any open Ω ′ ⊂ Ω. To see this one just need to take restrictions to Ω ′ 2 , and apply the same resolutivity results. Note however that it is not trivial that the restriction of a C p ( · ; Ω 2 )quasicontinuous function is C p ( · ; (Ω ′ ) 2 )-quasicontinuous, since the conditions (3.1) and (3.2) are different, but this follows from the fact that C p ( · ; Ω 2 ) is an outer capacity, by Proposition 4.2 in [17].
In particular, this is true for the Mazurkiewicz boundary ∂ M Ω, if Ω is finitely connected at the boundary, since Lipschitz functions on Ω M belong to N 1,p (Ω).
Proof of Theorem 1.4. It follows from Theorem 11.2 in Björn-Björn-Shanmugalingam [15] and the discussion after Definition 6.1 that condition (ii) in Theorem 9.1 is satisfied, and thus the result follows from Theorem 9.1.

Counterexamples
In this section we have collected a number of counterexamples demonstrating the sharpness of our results (to the extent known to us). These examples are all in R 2 .
To simplify notation we will consider R to be embedded into R 2 in the usual way.
showing that0 is irregular. Hence both the weak and the usual Kellogg properties fail for Ω 1 , and also the conclusion in Theorem 7.3 fails even though ∂ 1 Ω is q.e.invariant (as the empty set is the only boundary set with zero capacity). Let G = (Ω \ B(4, 2)) ∪ {0}, which is a τ 1 -neighbourhood of0, U = {x ∈ G ∩ Ω : |x| < 4} and f = χ ∂ 1 U\{0} ∈ C(∂ 1 G). Since 0 is irregular with respect to G, we see that P G 1 f = χ U and thus0 is irregular with respect to (G ∩ Ω) 1 . Hence the converse implication to the one in Proposition 8.5 does not hold in general, and regularity is not a local property in this situation. In particular, neither of the two parts in Theorem 9.1 hold in this case.
Example 10.5. Assume now that Ω and Ω 2 are as in Example 10.4, but this time with p > 2. In this case C p ({0}) > 0 and thus 0 is regular with respect to Ω. Let again f ∈ C(∂ 2 Ω) and let f 1 and f 2 be given by (10.2). If a < sup I f then, because of (10.1), any function u ∈ U f (Ω 2 ) is necessarily greater than a on concentric circles which are arbitrarily close to 0, and hence it is greater than a in a neighbourhood of 0, by the minimum principle for superharmonic functions, see Heinonen-Kilpeläinen-Martio [25, Theorem 7.12] (or [10, Theorem 9.13]). As this holds for all a < sup I f , we see that u ∈ U f2 (Ω). Conversely, any function in U f2 (Ω) necessarily belongs to U f (Ω 2 ). We therefore conclude that Similarly, P Ω 2 f = P Ω f 1 . As P Ω f 1 ≡ P Ω f 2 if and only if f is constant on I, only such f are resolutive with respect to Ω 2 , and thus ∂ 2 Ω is not resolutive. From this we can easily conclude that all the points in I are irregular, but resolutive-regular. As C p (I; Ω 2 ) = C p ({0}, Ω) > 0, by Lemma 3.4, we see that neither the weak nor the usual Kellogg property hold with respect to Ω 2 . On the other hand the resolutive Kellogg property does hold, as there are no resolutiveirregular boundary points. Moreover, this shows that (c) ⇒ (b) in Theorem 6.2 can fail when x 0 does not split nicely, even if ∂ 2 Ω is assumed to be resolutive.
In fact, a similar argument (using concentric circles) shows that Hence ∂ 2 Ω is q.e.-invariant (as the only set E ⊂ ∂ 2 Ω with zero capacity is the empty set). Since any bounded p-harmonic function on Ω has a limit as x → 0 (see below), we also conclude from (10.3) that if f ∈ C(∂ 2 Ω) is nonconstant on I, then the (existence) conclusion in Theorem 7.3 fails for f . We also see that the weak Kellogg property in Proposition 7.2 and Theorem 7.3 neither can be dropped nor replaced by the resolutive Kellogg property. It remains to show that for any bounded p-harmonic function u ≥ 0 on Ω the limit lim x→0 u(x) exists. To this end, let m(r) = inf |x|=r u(x) and M (r) = sup |x|=r u(x), 0 < r < 1, which are both continuous functions that, by the strong maximum principle (see [25,Theorem 7.12] or [10, Theorem 9.13]), can have at most one local extreme point each. Hence the limits m := lim r→0 m(r) and M := lim r→0 M (r) exist. By Harnack's inequality, there is a constant A such that M 1 2 ≤ Am 1 2 , and by scaling invariance we can apply it also in smaller punctured balls. Let ε > 0. Then there is ρ > 0 such that for 0 < r < ρ we have u > m − ε in B(0, 2r). Applying the Harnack inequality to u − (m − ε) shows that M (r) − m + ε ≤ A(m(r) − m + ε).
Letting first r → 0 and then ε → 0 shows that M = m.