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

TL;DR

This paper investigates boundary regularity for p-harmonic functions in metric spaces, establishing the Kellogg property for various compactifications and providing examples where it fails, extending known Euclidean results.

Contribution

It extends the Kellogg property to the Mazurkiewicz boundary and other compactifications in metric spaces, including new results in Euclidean spaces.

Findings

Kellogg property holds for many compactifications in metric spaces

Examples are provided where the Kellogg property fails

Results are new even in unweighted Euclidean spaces

Abstract

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.

Equations

$$\mathrm{div}(|\nabla u{|}^{p-2}\nabla u)=0.$$

$$d_M(x,y)=\underset{E}{\inf }\mathrm{diam}E,$$

$$\lim_{\Omega\ni y\overset{d_{M}}{\longrightarrow}x}u(y)=f(x)\quad\text{for ${{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}$\hss}{C}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}$\hss}{C}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}$\hss}{C}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}$\hss}{C}}}_{p}}(\,\cdot\,;{\Omega^{M}})$-q.e. }x\in{\partial^{M}}\Omega.$$

$$|f(\gamma (0))-f(\gamma (l_{\gamma }))|\le {\int }_{\gamma }gds,$$

$$\|f\|_{N^{1,p}(X)}=\biggl{(}\int_{X}|f|^{p}\,d\mu+\inf_{g}\int_{X}g^{p}\,d\mu\biggr{)}^{1/p},$$

$$N^{1,p}(X)=\{f:\Vert f{\Vert }_{N^{1,p}(X)}<\infty \}.$$

$$D^p(X)=\{f:f\text{ is measurable and has a pmspace1.0mu-weak upper gradient in }{L}^p(X)\}.$$

$$C_p(E)=\underset{u}{\inf }\Vert u{\Vert }_{N^{1,p}(X)}^p,$$

$$0<\mu (2B)\le C\mu (B)<\infty$$

$$\mathop{\mathchoice{\kern 1.28333pt\vbox{\hrule width=3.50002pt}\kern-4.78336pt}{\kern 1.16664pt\vbox{\hrule width=3.50002pt}\kern-4.66666pt}{\kern 0.91664pt\vbox{\hrule width=2.75pt}\kern-3.66664pt}{\kern 0.74997pt\vbox{\hrule width=2.25pt}\kern-2.99998pt}}\mathopen{}\int_{B}|f-f_{B}|\,d\mu\leq C\operatorname{diam}(B)\biggl{(}\mathop{\mathchoice{\kern 1.28333pt\vbox{\hrule width=3.50002pt}\kern-4.78336pt}{\kern 1.16664pt\vbox{\hrule width=3.50002pt}\kern-4.66666pt}{\kern 0.91664pt\vbox{\hrule width=2.75pt}\kern-3.66664pt}{\kern 0.74997pt\vbox{\hrule width=2.25pt}\kern-2.99998pt}}\mathopen{}\int_{\lambda B}g^{p}\,d\mu\biggr{)}^{1/p},$$

$$\Phi :{\overline{\Omega }}^2⟶{\overline{\Omega }}^1\text{with }\Phi {|}_{\Omega }=\mathrm{id}.$$

$${{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}$\hss}{C}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}$\hss}{C}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}$\hss}{C}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}$\hss}{C}}}_{p}}(E;{\Omega^{1}})=\inf_{u\in\mathcal{A}_{E}({\Omega^{1}})}\|u\|_{N^{1,p}(\Omega)}^{p},$$

$$u\ge 1\text{on }E\cap \Omega$$

$$\underset{\Omega \ni y\stackrel{{\tau }^1}{⟶}x}{\mathrm{liminf}}u(y)\ge 1\text{for all }x\in E\cap {\partial }^1\Omega .$$

$$\underset{y\stackrel{{\tau }^1}{⟶}x}{\mathrm{liminf}}u(y)\ge 1$$

$$v_j=\{\begin{array}{ll}(1-2u_j{)}_{\text{vbox}+}& \text{in }\Omega ,\\ 0& \text{in }X\setminus \Omega ,\end{array}\text{and}v(x)=\underset{j\to \infty }{\mathrm{limsup}}{v}_j(x),x\in X.$$

$$\mu (\{x\in \Omega :v_j(x)\le 1-2\delta \}=\mu (\{x\in \Omega :u_j(x)\ge \delta \}\le \frac{1}{{\delta }^p}{\int }_{\Omega }{u}_j^{p}d\mu \le \frac{1}{(j\delta {)}^p},$$

$$0<\mathop{\mathchoice{\kern 1.28333pt\vbox{\hrule width=3.50002pt}\kern-4.78336pt}{\kern 1.16664pt\vbox{\hrule width=3.50002pt}\kern-4.66666pt}{\kern 0.91664pt\vbox{\hrule width=2.75pt}\kern-3.66664pt}{\kern 0.74997pt\vbox{\hrule width=2.25pt}\kern-2.99998pt}}\mathopen{}\int_{B}|v-v_{B}|\,d\mu\leq C\operatorname{diam}(B)\biggl{(}\mathop{\mathchoice{\kern 1.28333pt\vbox{\hrule width=3.50002pt}\kern-4.78336pt}{\kern 1.16664pt\vbox{\hrule width=3.50002pt}\kern-4.66666pt}{\kern 0.91664pt\vbox{\hrule width=2.75pt}\kern-3.66664pt}{\kern 0.74997pt\vbox{\hrule width=2.25pt}\kern-2.99998pt}}\mathopen{}\int_{\lambda B}g_{v}^{p}\,d\mu\biggr{)}^{1/p}=0,$$

$${{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}$\hss}{C}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}$\hss}{C}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}$\hss}{C}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}$\hss}{C}}}_{p}}({\partial^{1}}\Omega;{\Omega^{1}})={{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}$\hss}{C}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}$\hss}{C}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}$\hss}{C}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}$\hss}{C}}}_{p}}({\partial^{2}}\Omega;{\Omega^{2}})>0.\qed$$

$${\int }_{\phi \ne 0}{g}_u^{p}d\mu \le {\int }_{\phi \ne 0}{g}_{u+\phi }^{p}d\mu \text{for all (nonnegative) }\phi \in N_0^{1,p}(\Omega ).$$

$$v(x):=\underset{y\to x}{\mathrm{essliminf}}v(y):=\underset{r\to 0}{\lim }\underset{B(x,r)}{\mathrm{essinf}}v.$$

$$\underset{\Omega \ni y\stackrel{{\tau }^1}{⟶}x}{\mathrm{liminf}}u(y)\ge f(x)$$

$${\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle P}\kern 0.0pt}$\hss}{P}}}_{{\Omega^{1}}}f(x)=\inf_{u\in\mathcal{U}_{f}({\Omega^{1}})}u(x),\quad x\in\Omega,$$

$${\mathchoice{\hbox to0.0pt{$\underline{\phantom{\displaystyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\underline{\phantom{\textstyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\underline{\phantom{\scriptstyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\underline{\phantom{\scriptscriptstyle P}\kern 0.0pt}$\hss}{P}}}_{{\Omega^{1}}}f=-{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle P}\kern 0.0pt}$\hss}{P}}}_{{\Omega^{1}}}(-f).$$

$${\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle S}\kern 0.0pt}$\hss}{S}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle S}\kern 0.0pt}$\hss}{S}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle S}\kern 0.0pt}$\hss}{S}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle S}\kern 0.0pt}$\hss}{S}}}_{{\Omega^{1}}}f(x)=\inf_{u\in\mathcal{DU}_{f}({\Omega^{1}})}u(x),\ x\in\Omega,\quad\text{and}\quad{\mathchoice{\hbox to0.0pt{$\underline{\phantom{\displaystyle S}\kern 0.0pt}$\hss}{S}}{\hbox to0.0pt{$\underline{\phantom{\textstyle S}\kern 0.0pt}$\hss}{S}}{\hbox to0.0pt{$\underline{\phantom{\scriptstyle S}\kern 0.0pt}$\hss}{S}}{\hbox to0.0pt{$\underline{\phantom{\scriptscriptstyle S}\kern 0.0pt}$\hss}{S}}}_{{\Omega^{1}}}f=-{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle S}\kern 0.0pt}$\hss}{S}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle S}\kern 0.0pt}$\hss}{S}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle S}\kern 0.0pt}$\hss}{S}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle S}\kern 0.0pt}$\hss}{S}}}_{{\Omega^{1}}}(-f).$$

$${\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle P}\kern 0.0pt}$\hss}{P}}}_{\Omega^{1}}f={\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle P}\kern 0.0pt}$\hss}{P}}}_{\Omega^{2}}({f\circ\Phi})\quad\text{and}\quad{\mathchoice{\hbox to0.0pt{$\underline{\phantom{\displaystyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\underline{\phantom{\textstyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\underline{\phantom{\scriptstyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\underline{\phantom{\scriptscriptstyle P}\kern 0.0pt}$\hss}{P}}}_{\Omega^{1}}f={\mathchoice{\hbox to0.0pt{$\underline{\phantom{\displaystyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\underline{\phantom{\textstyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\underline{\phantom{\scriptstyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\underline{\phantom{\scriptscriptstyle P}\kern 0.0pt}$\hss}{P}}}_{\Omega^{2}}({f\circ\Phi}).$$

$$\lim_{\Omega\ni y{\overset{\tau^{1}}{\longrightarrow}}x_{0}}{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle P}\kern 0.0pt}$\hss}{P}}}_{{\Omega^{1}}}f(y)=f(x_{0})$$

$$\limsup_{\Omega\ni y{\overset{\tau^{1}}{\longrightarrow}}x_{0}}{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle P}\kern 0.0pt}$\hss}{P}}}_{{\Omega^{1}}}f(y)\leq f(x_{0})$$

$$f(x_{0})\leq\liminf_{\Omega\ni y{\overset{\tau^{1}}{\longrightarrow}}x_{0}}{\mathchoice{\hbox to0.0pt{$\underline{\phantom{\displaystyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\underline{\phantom{\textstyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\underline{\phantom{\scriptstyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\underline{\phantom{\scriptscriptstyle P}\kern 0.0pt}$\hss}{P}}}_{{\Omega^{1}}}f(y)\leq\liminf_{\Omega\ni y{\overset{\tau^{1}}{\longrightarrow}}x_{0}}{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle P}\kern 0.0pt}$\hss}{P}}}_{{\Omega^{1}}}f(y)\leq\limsup_{\Omega\ni y{\overset{\tau^{1}}{\longrightarrow}}x_{0}}{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle P}\kern 0.0pt}$\hss}{P}}}_{{\Omega^{1}}}f(y)\leq f(x_{0}),$$

$$\lim_{\Omega\ni y{\overset{\tau^{1}}{\longrightarrow}}x_{0}}{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle P}\kern 0.0pt}$\hss}{P}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle P}\kern 0.0pt}$\hss}{P}}}_{{\Omega^{1}}}f(y)=f(x_{0}).$$

Full text

The Kellogg property for $p\mspace{1.0mu}$-harmonic functions

with respect to the Mazurkiewicz boundary

Anders Björn

  

Anders Björn

*Department of Mathematics, Linköping University,

SE-581 83 Linköping, Sweden; anders.bjorn@liu.se *

The Kellogg property and boundary regularity for $p\mspace{1.0mu}$-harmonic functions

with respect to the Mazurkiewicz boundary and other compactifications

Anders Björn

  

Anders Björn

*Department of Mathematics, Linköping University,

SE-581 83 Linköping, Sweden; anders.bjorn@liu.se *

**Abstract. In this paper boundary regularity for $p\mspace{1.0mu}$-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 $p\mspace{1.0mu}$-Poincaré inequality, but the results are new also in unweighted Euclidean spaces. **

Key words and phrases: boundary regularity, capacity, Dirichlet problem, doubling measure, Kellogg property, metric space, $p\mspace{1.0mu}$-harmonic function, Poincaré inequality, regular point, resolutive-regular point, weak Kellogg property.

Mathematics Subject Classification (2010): Primary: 31C45; Secondary: 31E05, 35J66, 35J92, 49Q20.

1 Introduction

In this paper we study boundary regularity for $p\mspace{1.0mu}$-harmonic functions on metric spaces, including $\mathbf{R}^{n}$ 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 $1<p<\infty$ and let $X$ be a complete metric space equipped with a doubling measure $\mu$ supporting a $p\mspace{1.0mu}$-Poincaré inequality. $X$ can e.g. be unweighted $\mathbf{R}^{n}$, in which case a function $u$ is $p\mspace{1.0mu}$-harmonic if it is a continuous weak solution of the $p\mspace{1.0mu}$-Laplace equation

[TABLE]

The definition of $p\mspace{1.0mu}$-harmonic functions on metric spaces is more involved, see Section 4.

Let $\Omega$ be a bounded domain. (If $X$ is bounded we also require that ${C_{p}}(X\setminus\Omega)>0$.) The Mazurkiewicz distance $d_{M}$ on $\Omega$ is defined by

[TABLE]

where the infimum is taken over all connected sets $E\subset\Omega$ containing $x,y\in\Omega$. (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 ${\partial^{M}}\Omega$ is the boundary of $\Omega$ in the completion of $(\Omega,d_{M})$. For instance, in the slit disc $B(0,1)\setminus[0,1]\subset\mathbf{R}^{2}$ this gives two boundary points corresponding to each point in the slit (but for the tip), while for smooth domains ${\partial^{M}}\Omega=\partial\Omega$.

Assume that the completion of $(\Omega,d_{M})$ is compact, which happens if and only if $\Omega$ is finitely connected at the boundary, see Section 3. Then, to each point in the given metric boundary $\partial\Omega$ there corresponds one or more (at most countably many) points in the Mazurkiewicz boundary ${\partial^{M}}\Omega$, while conversely to every point in ${\partial^{M}}\Omega$ there corresponds a unique point in $\partial\Omega$, see Björn–Björn–Shanmugalingam [16]. There is therefore a natural projection $\Phi:{\partial^{M}}\Omega\to\partial\Omega$ between these boundaries.

The following is the first new result, and the key tool to obtaining the Kellogg property.

Theorem 1.1**.**

Assume that $\Omega$ is a bounded domain which is finitely connected at the boundary. Let $x_{0}\in\partial\Omega$. Then the following are equivalent:**

The point $x_{0}$ is an irregular boundary point with respect to $\Omega$. 2. 2.

There is at least one irregular boundary point in $\Phi^{-1}(x_{0})$ with respect to $\Omega^{M}$. 3. 3.

There is exactly one irregular boundary point in $\Phi^{-1}(x_{0})$ with respect to ${\Omega^{M}}$.

This leads directly to the following consequence.

Theorem 1.2**.**

(The Kellogg property)* Assume that $\Omega$ is a bounded domain which is finitely connected at the boundary. Let $\operatorname{Irr}^{M}$ be the set of irregular boundary points with respect to ${\Omega^{M}}$. Then the capacity {{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}\hss}{C}}}_{p}}(\operatorname{Irr}^{M},{\Omega^{M}})=0.*

Note that by Proposition 3.5, {{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}\hss}{C}}}_{p}}({\partial^{M}}\Omega,{\Omega^{M}})>0, so the Kellogg property is never trivial. The following uniqueness result is also new.

Theorem 1.3**.**

Assume that $\Omega$ is a bounded domain which is finitely connected at the boundary. Let $f\in C({\partial^{M}}\Omega)$. Then there exists a unique bounded $p\mspace{1.0mu}$-harmonic function $u$ on $\Omega$ such that

[TABLE]

Moreover, $u$ equals the Perron solution $P_{{\Omega^{M}}}f$.

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 $\Omega$ is a bounded domain which is finitely connected at the boundary. Let $\hat{x}\in{\partial^{M}}\Omega$ and let $G$ be an ${\overline{\Omega}\mspace{1.0mu}}^{M}$-neighbourhood of $\hat{x}$.

Then $\hat{x}$ is regular with respect to ${\Omega^{M}}$ if and only if it is regular with respect to $G^{{\Omega^{M}}}$, where $G^{{\Omega^{M}}}$ is $G$ equipped with the boundary inherited from ${\overline{\Omega}\mspace{1.0mu}}^{M}$.

Throughout the paper we also study to what extent such results are true for other compactifications of $\Omega$. The details describing the different results and cases are quite involved.

Boundary regularity for $p\mspace{1.0mu}$-harmonic functions with respect to the given metric boundary has been studied for a long period, especially on $\mathbf{R}^{n}$. 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 $\mathbf{R}^{n}$ and for Cheeger $p\mspace{1.0mu}$-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].

In the nonlinear potential theory, the Kellogg property was first obtained by Hedberg [23] and Hedberg–Wolff [24] on $\mathbf{R}^{n}$ (see also Kilpeläinen [29]), who also obtained the more general fine Kellogg property and the even more general Choquet property. The Kellogg property was extended to homogeneous spaces by Vodop*′yanov [43] (who also obtained the fine Kellogg property), to weighted $\mathbf{R}^{n}$ by Heinonen–Kilpeläinen–Martio [25], to subelliptic equations by Markina–Vodop′*yanov [36], and to metric spaces by Björn–Björn–Shanmugalingam [13]. The fine Kellogg property, as well as the Choquet property, was deduced on metric spaces by Björn–Björn–Latvala [11]. Other aspects of boundary regularity and boundary behaviour for $p\mspace{1.0mu}$-harmonic functions have been studied in [3]–[9], [22], [25], [29], [30] and [36].

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.

Acknowledgement. The author was supported by the Swedish Research Council, grants 621-2007-6187, 621-2011-3139 and 2016-03424. The idea to study resolutive-regularity is due to Tomas Sjödin (private communication).

2 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 $1<p<\infty$ and that $X=(X,d,\mu)$ is a metric space equipped with a metric $d$ and a positive complete Borel measure $\mu$ such that $0<\mu(B)<\infty$ for all balls $B\subset X$.

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 $p\mspace{1.0mu}$-almost every curve if it fails only for a curve family $\Gamma$ with zero $p\mspace{1.0mu}$-modulus, i.e. there exists $0\leq\rho\in L^{p}(X)$ such that $\int_{\gamma}\rho\,ds=\infty$ for every curve $\gamma\in\Gamma$.

Following Koskela–MacManus [34] (see also Heinonen–Koskela [26]) we introduce weak upper gradients as follows.

Definition 2.1**.**

A measurable function $g:X\to[0,\infty]$ is a $p\mspace{1.0mu}$-weak upper gradient of a function $f:X\to{\overline{\mathbf{R}}}:=[-\infty,\infty]$ if for $p\mspace{1.0mu}$-almost all curves $\gamma:[0,l_{\gamma}]\to X$,

[TABLE]

where the left-hand side is considered to be $\infty$ whenever at least one of the terms therein is infinite.

If $f$ has a $p\mspace{1.0mu}$-weak upper gradient in $L^{p}_{\rm loc}(X)$, then it has an a.e. unique minimal $p\mspace{1.0mu}$-weak upper gradient $g_{f}\in L^{p}_{\rm loc}(X)$ in the sense that for every $p\mspace{1.0mu}$-weak upper gradient $g\in L^{p}_{\rm loc}(X)$ of $f$ we have $g_{f}\leq g$ a.e., see Shanmugalingam [42]. Following Shanmugalingam [41], we define a version of Sobolev spaces on the metric space $X$.

Definition 2.2**.**

For a measurable function $f:X\to{\overline{\mathbf{R}}}$, let

[TABLE]

where the infimum is taken over all $p\mspace{1.0mu}$-weak upper gradients $g$ of $f$. The Newtonian space on $X$ is

[TABLE]

The space $N^{1,p}(X)/{\sim}$, where $f\sim h$ if and only if $\|f-h\|_{N^{1,p}(X)}=0$, is a Banach space, see [41]. We also define

[TABLE]

In this paper we assume that functions in $N^{1,p}(X)$ and $D^{p}(X)$ are defined everywhere (with values in ${\overline{\mathbf{R}}}$), not just up to an equivalence class in the corresponding function space. For a measurable set $E\subset X$, the Newtonian space $N^{1,p}(E)$ is defined by considering $(E,d|_{E},\mu|_{E})$ as a metric space in its own right. We say that $f\in N^{1,p}_{\rm loc}(E)$ if for every $x\in E$ there exists a ball $B_{x}\ni x$ such that $f\in N^{1,p}(B_{x}\cap E)$.

Definition 2.3**.**

The (Sobolev) capacity of an arbitrary set $E\subset X$ is

[TABLE]

where the infimum is taken over all $u\in N^{1,p}(X)$ such that $u\geq 1$ on $E$.

The measure $\mu$ is doubling if there exists a doubling constant $C>0$ such that

[TABLE]

for all balls $B=B(x_{0},r):=\{x\in X:d(x,x_{0})<r\}$ in $X$, where $\lambda B=B(x_{0},\lambda r)$.

Definition 2.4**.**

$X$ supports a $p\mspace{1.0mu}$-Poincaré inequality if there exist constants $C>0$ and $\lambda\geq 1$ such that for all balls $B\subset X$, all integrable functions $f$ on $X$ and all $p\mspace{1.0mu}$-weak upper gradients $g$ of $f$,

[TABLE]

where $f_{B}:=\mathop{\mathchoice{\kern 1.28333pt\vbox{\hrule width=3.50002pt}\kern-4.78336pt}{\kern 1.16664pt\vbox{\hrule width=3.50002pt}\kern-4.66666pt}{\kern 0.91664pt\vbox{\hrule width=2.75pt}\kern-3.66664pt}{\kern 0.74997pt\vbox{\hrule width=2.25pt}\kern-2.99998pt}}\mathopen{}\int_{B}f\,d\mu:=\int_{B}f\,d\mu/\mu(B)$.

In this paper neighbourhoods are always open and continuous functions are real-valued, whereas semicontinuous functions may take the values $\pm\infty$.

3 Compactifications and the capacity

{{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}\hss}{C}}}_{p}}

Definition 3.1**.**

Let $\Omega$ be a locally compact noncompact metric space. A couple $(\partial^{\prime}\Omega,\tau)$ is said to compactify $\Omega$ if $\partial^{\prime}\Omega$ is a set with $\partial^{\prime}\Omega\cap\Omega=\varnothing$ and $\tau$ is a Hausdorff topology on ${\overline{\Omega}\mspace{1.0mu}}^{\prime}:=\Omega\cup\partial^{\prime}\Omega$ such that

1. (i).

${\overline{\Omega}\mspace{1.0mu}}^{\prime}$ is compact with respect to $\tau$; 2. (ii).

$\Omega$ is dense in ${\overline{\Omega}\mspace{1.0mu}}^{\prime}$ with respect to $\tau$; 3. (iii).

the topology induced on $\Omega$ by $\tau$ coincides with the given topology on $\Omega$.

The space ${\overline{\Omega}\mspace{1.0mu}}^{\prime}$ with the topology $\tau$ is a compactification of $\Omega$.

Since ${\overline{\Omega}\mspace{1.0mu}}^{\prime}$ is a compact Hausdorff space it is normal.

We assume from now on that $X$ is a complete metric space supporting a $p\mspace{1.0mu}$-Poincaré inequality, that $\mu$ is doubling, and that $1<p<\infty$. We also assume that $\Omega$ is a nonempty bounded open set such that ${C_{p}}(X\setminus\Omega)>0$, and that ${\overline{\Omega}^{j}}=\Omega^{j}\cup{\partial^{j}}\Omega$, $j=1,2$, are compactifications of $\Omega$, where $\Omega^{j}=\Omega$ with the intended boundary ${\partial^{j}}\Omega$ and where the topologies on ${\overline{\Omega}^{j}}$ are denoted by ${\tau^{j}}$. Furthermore, we reserve $\partial\Omega$ and ${\overline{\Omega}}$ for the given metric boundary and closure induced by $X$ on $\Omega$.

As $\mu$ is doubling and $X$ is complete, it follows that $X$ is proper (i.e. all closed bounded sets are compact).

We define $\partial^{1}\Omega\prec\partial^{2}\Omega$ to mean that there is a continuous mapping, which is called projection,

[TABLE]

An example of a compactification is the Mazurkiewicz completion discussed in the introduction, which is a compactification of $\Omega$ if and only if $\Omega$ 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 $\Omega\subset X$ is finitely connected at the boundary if for every $x\in\partial\Omega$ and $r>0$ there is an open set $G$ (in $X$) such that $x\in G\subset B(x,r)$ and $G\cap\Omega$ has only finitely many components.

In addition to the Sobolev capacity mentioned above, we will also need the following capacity. It was introduced for ${\overline{\Omega}}$ and ${\overline{\Omega}\mspace{1.0mu}}^{M}$ 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].

Definition 3.3**.**

For $E\subset{\overline{\Omega}^{1}}$ let

[TABLE]

where $u\in\mathcal{A}_{E}({\Omega^{1}})$ if $u\in N^{1,p}(\Omega)$ is such that

[TABLE]

and

[TABLE]

When proving the Kellogg property we will need the following lemma.

Lemma 3.4**.**

Assume that ${\partial^{1}}\Omega\prec{\partial^{2}}\Omega$ and let $\Phi:{\overline{\Omega}^{2}}\to{\overline{\Omega}^{1}}$ be the projection. Let $E\subset{\partial^{1}}\Omega$. Then {{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}\hss}{C}}}_{p}}(E;{\Omega^{1}})={{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}\hss}{C}}}_{p}}(\Phi^{-1}(E),{\Omega^{2}}).

We will use nets to study convergence in our compactifications, see e.g. Pedersen [40] for the key results on nets.

• Proof.

We first show that $\mathcal{A}_{E}\subset\mathcal{A}_{\Phi^{-1}(E)}$. Take $u\in\mathcal{A}_{E}$. Let $\hat{x}\in\Phi^{-1}(E)\cap{\partial^{2}}\Omega$ and take a net $y_{\lambda}\in\Omega$ such that $y_{\lambda}{\overset{\tau^{2}}{\longrightarrow}}\hat{x}$. Since $\Phi$ is continuous it follows that $y_{\lambda}{\overset{\tau^{1}}{\longrightarrow}}\Phi(\hat{x})$. Hence, because $\Phi(\hat{x})\in E\cap{\partial^{1}}\Omega$, we see that $\liminf_{\lambda}u(y_{\lambda})\geq 1$, and thus $u\in\mathcal{A}_{\Phi^{-1}(E)}$.

Conversely, assume that $u\in\mathcal{A}_{\Phi^{-1}(E)}$. Let $x\in E\cap{\partial^{1}}\Omega$. Assume that there is a net $y_{\lambda}\in\Omega$ such that $y_{\lambda}{\overset{\tau^{1}}{\longrightarrow}}x$ and $\liminf_{\lambda}u(y_{\lambda})<1$. By taking a subnet we may assume that $\lim_{\lambda}u(y_{\lambda})$ exists and is less than $1$. Then there is a further subnet $y_{\mu}$ which converges to some point $\hat{x}\in{\Omega^{2}}$. Hence $\lim_{\mu}u(y_{\mu})<1$. As $\Phi$ is continuous we see that $y_{\mu}{\overset{\tau^{1}}{\longrightarrow}}\Phi(\hat{x})$, and thus $\Phi(\hat{x})=x$. But together with $\lim_{\mu}u(y_{\mu})<1$, this contradicts the assumption $u\in\mathcal{A}_{\Phi^{-1}(E)}$. Hence

[TABLE]

and $u\in A_{E}$.

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.

Proposition 3.5**.**

{{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}\hss}{C}}}_{p}}({\partial^{1}}\Omega;{\Omega^{1}})>0.

• Proof.

Let ${\overline{\Omega}^{2}}$ be the one-point compactification of $\Omega$. We will first show that {{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}\hss}{C}}}_{p}}(\partial{\Omega^{2}};{\Omega^{2}})>0. Assume on the contrary that {{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}\hss}{C}}}_{p}}(\partial{\Omega^{2}};{\Omega^{2}})=0. Then for each $j=1,2,\mathinner{.}\nonscript\!.\nonscript\!\mathinner{.}$, there is $u_{j}\in\mathcal{A}_{\partial{\Omega^{2}}}({\Omega^{2}})$ with $0\leq u_{j}\leq 1$ and $\|u_{j}\|_{N^{1,p}(\Omega)}<1/j$. Let

[TABLE]

By (3.2), the set (X\setminus\Omega)\cup\bigl{\{}x\in\Omega:u_{j}(x)>\tfrac{1}{2}\bigr{\}} contains a neighbourhood of $\partial\Omega$, and therein $v_{j}\equiv 0$. As $v_{j}\in N^{1,p}(\Omega)$ we conclude that $v_{j}\in N^{1,p}(X)$. Moreover, $\{v_{j}\}_{j=1}^{\infty}$ 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 $v\in N^{1,p}(X)$. We get directly that $v\equiv 0$ in $X\setminus\Omega$. Moreover, if $0<\delta<\tfrac{1}{2}$, then

[TABLE]

which tends to [math] as $j\to\infty$. It follows that $v=1$ a.e. in $\Omega$.

We next need to consider two cases separately. First, if $\mu(X\setminus\Omega)=0$, then $v$ equals $1$ a.e.  but not q.e. in $X$ (because ${C_{p}}(X\setminus\Omega)>0$), contradicting Proposition 1.59 in [10].

On the other hand, if $\mu(X\setminus\Omega)>0$, then we let $B$ be a large enough ball so that $\Omega\subset B$ and $\mu(B\setminus\Omega)>0$. By Corollary 2.21 in [10], we know that $g_{v}=0$ a.e. Hence, by the $p\mspace{1.0mu}$-Poincaré inequality,

[TABLE]

a contradiction.

Thus we must have {{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}\hss}{C}}}_{p}}(\partial{\Omega^{2}};{\Omega^{2}})>0, and by Lemma 3.4,

[TABLE]

4 $p\mspace{1.0mu}$-harmonic functions and Perron solutions

A function $u\in N^{1,p}_{\rm loc}(\Omega)$ is a (super)minimizer in $\Omega$ if

[TABLE]

A $p\mspace{1.0mu}$-harmonic function is a continuous minimizer.

A function $u:\Omega\to(-\infty,\infty]$ is superharmonic in $\Omega$ if

1. (i)

$u$ is not identically $\infty$ in any component of $\Omega$; 2. (ii)

$\min\{u,k\}$ is an lsc-regularized superminimizer for all $k\in\mathbf{R}$,

where $v$ is lsc-regularized if

[TABLE]

By Theorem 6.1 in Björn [1] (or [10, Theorems 9.24 and 14.10]), this definition of superharmonicity is equivalent to the ones used both in the Euclidean and metric space literature, e.g. in Heinonen–Kilpeläinen–Martio [25], Kinnunen–Martio [33] and Björn–Björn [10].

We are now ready to introduce the Perron solutions with respect to ${\Omega^{1}}$. We follow Björn–Björn–Sjödin [17], although therein Perron solutions were only defined in domains.

Definition 4.1**.**

Given $f:{\partial^{1}}\Omega\to{\overline{\mathbf{R}}}$, let $\mathcal{U}_{f}({\Omega^{1}})$ be the set of all superharmonic functions $u$ on $\Omega$, bounded from below, such that

[TABLE]

for all $x\in{\partial^{1}}\Omega$. The upper Perron solution of $f$ is then defined to be

[TABLE]

while the lower Perron solution of $f$ is defined by

[TABLE]

If {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle P}\kern 0.0pt}\hss}{P}}}_{{\Omega^{1}}}f={\mathchoice{\hbox to0.0pt{\underline{\phantom{\displaystyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\underline{\phantom{\textstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\underline{\phantom{\scriptstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\underline{\phantom{\scriptscriptstyle P}\kern 0.0pt}\hss}{P}}}_{{\Omega^{1}}}f and it is real-valued, then we let P_{{\Omega^{1}}}f:={\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle P}\kern 0.0pt}\hss}{P}}}_{{\Omega^{1}}}f and $f$ is said to be resolutive with respect to ${\Omega^{1}}$.

Furthermore, let $\mathcal{DU}_{f}({\Omega^{1}})=\mathcal{U}_{f}({\Omega^{1}})\cap D^{p}(\Omega)$, and define the Sobolev–Perron solutions of $f$ by

[TABLE]

If {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle S}\kern 0.0pt}\hss}{S}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle S}\kern 0.0pt}\hss}{S}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle S}\kern 0.0pt}\hss}{S}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle S}\kern 0.0pt}\hss}{S}}}_{{\Omega^{1}}}f={\mathchoice{\hbox to0.0pt{\underline{\phantom{\displaystyle S}\kern 0.0pt}\hss}{S}}{\hbox to0.0pt{\underline{\phantom{\textstyle S}\kern 0.0pt}\hss}{S}}{\hbox to0.0pt{\underline{\phantom{\scriptstyle S}\kern 0.0pt}\hss}{S}}{\hbox to0.0pt{\underline{\phantom{\scriptscriptstyle S}\kern 0.0pt}\hss}{S}}}_{{\Omega^{1}}}f and it is real-valued, then $f$ is said to be Sobolev-resolutive with respect to ${\Omega^{1}}$. The boundary ${\partial^{1}}\Omega$ is (Sobolev)-resolutive if all functions $f\in C({\partial^{1}}\Omega)$ are (Sobolev)-resolutive.

In every component of $\Omega$ the upper/lower (Sobolev)–Perron solutions are $p\mspace{1.0mu}$-harmonic or identically $\pm\infty$, 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 $\partial\Omega$ 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 $\Omega$ is finitely connected at the boundary, then ${\partial^{M}}\Omega$ is resolutive by Theorem 8.2 in Björn–Björn–Shanmugalingam [15]. Also ${\partial^{M}}\Omega$ is Sobolev-resolutive (if $\Omega$ is finitely connected at the boundary), which again follows from Theorem 6.4 and Proposition 7.3 in [17] since continuous functions on ${\partial^{M}}\Omega$ can be uniformly approximated by Lipschitz functions on ${\overline{\Omega}\mspace{1.0mu}}^{M}$.

The following results from [17] will be important for us.

Proposition 4.2**.**

([17, Corollary 6.3])* If $f:{\partial^{1}}\Omega\to{\overline{\mathbf{R}}}$, then {\mathchoice{\hbox to0.0pt{\underline{\phantom{\displaystyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\underline{\phantom{\textstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\underline{\phantom{\scriptstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\underline{\phantom{\scriptscriptstyle P}\kern 0.0pt}\hss}{P}}}_{{\Omega^{1}}}f\leq{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle P}\kern 0.0pt}\hss}{P}}}_{{\Omega^{1}}}f.*

Theorem 4.3**.**

([17, Theorem 6.7 and Proposition 8.2])* Assume that $\partial^{1}\Omega\prec\partial^{2}\Omega$ and let $\Phi:{\overline{\Omega}^{2}}\to{\overline{\Omega}^{1}}$ denote the projection. If $f:\partial^{1}\Omega\rightarrow{\overline{\mathbf{R}}}$, then*

[TABLE]

In particular, if $\partial^{2}\Omega$ is resolutive then so is $\partial^{1}\Omega$.

One consequence of Theorem 4.3 is that (almost always) there are plenty of resolutive functions.

5 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).

Definition 5.1**.**

A boundary point $x_{0}\in{\partial^{1}}\Omega$ is (resolutive)-regular if

[TABLE]

for all (resolutive) $f\in C({\partial^{1}}\Omega)$. Otherwise, it is (resolutive)-irregular.

We also say that $\partial^{1}\Omega$ is (resolutive)-regular if all its boundary points are (resolutive)-regular.

Note that if all continuous functions are resolutive, then regularity and resolutive-regularity 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 $\limsup$ and $=$ by $\leq$ in Definition 5.1.

Lemma 5.2**.**

A boundary point $x_{0}\in{\partial^{1}}\Omega$ is (resolutive)-regular if and only if

[TABLE]

for all (resolutive) $f\in C({\partial^{1}}\Omega)$.

• Proof.

Assume that (5.1) holds for all (resolutive) $f\in C({\partial^{1}}\Omega)$. Let $f:{\partial^{1}}\Omega\to\mathbf{R}$ be continuous (and resolutive). Then also $-f$ is continuous (and resolutive). Hence, by Proposition 4.2,

[TABLE]

and therefore

[TABLE]

Thus $x_{0}$ is (resolutive)-regular. The converse is trivial. ∎

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 $x_{0}\in{\partial^{1}}\Omega$. Then the following are equivalent:**

1. (i).

The point $x_{0}$ is a regular boundary point. 2. (ii).

It is true that

[TABLE]

for all bounded $f:{\partial^{1}}\Omega\to\mathbf{R}$ which are continuous at $x_{0}$. 3. (iii).

It is true that

[TABLE]

for all functions $f:{\partial^{1}}\Omega\to[-\infty,\infty)$ which are bounded from above on ${\partial^{1}}\Omega$ and upper semicontinuous at $x_{0}$.

• Proof.

(ii) $\Rightarrow$ (i) This is trivial.

(i) $\Rightarrow$ (iii) Let $A>f(x_{0})$ be real and M=\sup_{{\partial^{1}}\Omega}f_{\mathchoice{\vbox{\hbox{\scriptstyle+}}}{\vbox{\hbox{\scriptstyle+}}}{\vbox{\hbox{\scriptscriptstyle+}}}{\vbox{\hbox{\scriptscriptstyle+}}}}. Since $f$ is upper semicontinuous at $x_{0}$ we can find a ${\tau^{1}}$-neighbourhood $G\subset{\partial^{1}}\Omega$ containing $x_{0}$ such that $f<A$ in $G$. By Tietze’s extension theorem, we can also find $h\in C({\partial^{1}}\Omega)$ such that $h(x_{0})=A$, $h=M$ on ${\partial^{1}}\Omega\setminus G$ and $h\geq A$ everywhere. Then $f\leq h$ on ${\partial^{1}}\Omega$, and thus

[TABLE]

from which (5.2) follows after taking infiumum over all $A>f(x_{0})$.

(iii) $\Rightarrow$ (ii) Applying (iii) to $-f$ yields

[TABLE]

Together with (5.2) this gives the desired conclusion. ∎

Proposition 5.4**.**

Assume that ${\partial^{1}}\Omega\prec{\partial^{2}}\Omega$ and let $\Phi:{\overline{\Omega}^{2}}\to{\overline{\Omega}^{1}}$ be the projection. If $x_{0}\in{\partial^{1}}\Omega$ is (resolutive)-irregular with respect to ${\Omega^{1}}$, then there is at least one (resolutive)-irregular boundary point in $\Phi^{-1}(x_{0})$ with respect to $\Omega^{2}$.

• Proof.

Since $x_{0}$ is (resolutive)-irregular there is, due to Lemma 5.2, a (resolutive) function $f\in C(\partial^{1}\Omega)$ such that

[TABLE]

We may assume that $M=2$ and $f(x_{0})=0$. Let G=\{y\in\Omega:{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle P}\kern 0.0pt}\hss}{P}}}_{{\Omega^{1}}}f(y)>1\}, which is a nonempty open subset of $\Omega$ such that x_{0}\in{{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}\hss}{G}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern 0.0pt}\hss}{G}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}\hss}{G}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern 0.0pt}\hss}{G}}}^{1}}, where {{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}\hss}{G}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern 0.0pt}\hss}{G}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}\hss}{G}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern 0.0pt}\hss}{G}}}^{1}} is the closure of $G$ within ${\overline{\Omega}^{1}}$. In particular there is a net $\{x_{\lambda}\}_{\lambda}$ such that $G\ni x_{\lambda}{\overset{\tau^{1}}{\longrightarrow}}x_{0}$. As $G$ is not closed in ${\overline{\Omega}^{1}}$ it is not compact.

By compactness of ${\overline{\Omega}^{2}}$, there is a ${\tau^{2}}$-convergent subnet $\{x_{\mu}\}_{\mu}$, with ${\tau^{2}}$-limit $\hat{x}$. It follows that $\Phi(\hat{x})=x_{0}$. Set $h=f\circ\Phi$. By Theorem 4.3, {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle P}\kern 0.0pt}\hss}{P}}}_{\Omega^{2}}h={\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle P}\kern 0.0pt}\hss}{P}}}_{\Omega^{1}}f (and $h$ is resolutive if $f$ is). Thus

[TABLE]

As $h(\hat{x})=f(x_{0})=0$ and $h$ is continuous (and resolutive if $f$ is), this shows that $\hat{x}$ is (resolutive)-irregular with respect to ${\Omega^{2}}$. ∎

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.

Definition 6.1**.**

Assume that ${\partial^{1}}\Omega\prec{\partial^{2}}\Omega$ and let $\Phi:{\overline{\Omega}^{2}}\to{\overline{\Omega}^{1}}$ be the projection. We say that $x_{0}\in{\partial^{1}}\Omega$ splits nicely if every $\hat{x}\in\Phi^{-1}(x_{0})$ has arbitrarily small ${\tau^{2}}$-neighbourhoods $U$ such that ${\partial^{2}}U\cap\Phi^{-1}(x_{0})=\varnothing$.

Note that if $\Phi^{-1}(x_{0})$ is finite, or if ${\overline{\Omega}^{2}}$ is metrizable and $\Phi^{-1}(x_{0})$ is at most countable, then $x_{0}$ splits nicely. In particular, all points in $\partial\Omega$ split nicely with respect to the Mazurkiewicz boundary ${\partial^{M}}\Omega$. When ${\overline{\Omega}^{2}}$ is nonmetrizable, we do not know if $x_{0}$ always splits nicely whenever $\Phi^{-1}(x_{0})$ is countable.

Theorem 6.2**.**

Assume that $\partial\Omega\prec{\partial^{2}}\Omega$, where $\partial\Omega$ is the given metric boundary. Let $\Phi:{\overline{\Omega}^{2}}\to{\overline{\Omega}}$ be the projection. Assume that $x_{0}\in\partial\Omega$ splits nicely. Then the following are equivalent:**

The point $x_{0}$ is irregular with respect to $\Omega$. 2. 2.

The point $x_{0}$ is resolutive-irregular with respect to $\Omega$. 3. (c).

There is at least one resolutive-irregular boundary point in $\Phi^{-1}(x_{0})$ with respect to ${\Omega^{2}}$. 4. (d).

There is exactly one resolutive-irregular boundary point in $\Phi^{-1}(x_{0})$ with respect to ${\Omega^{2}}$.

On the way to proving Theorem 6.2 we first obtain the following result, which generalizes both (c) $\Rightarrow$ 2 and (d) $\Rightarrow$ 2 in Theorem 6.2. It shows in particular that the niceness assumption can be dropped for the implication (d) $\Rightarrow$ 2.

Theorem 6.3**.**

Assume that $\partial\Omega\prec{\partial^{2}}\Omega$, where $\partial\Omega$ is the given metric boundary. Let $\Phi:{\overline{\Omega}^{2}}\to{\overline{\Omega}}$ be the projection. If there are finitely many, and at least one, resolutive-irregular boundary points in $\Phi^{-1}(x_{0})$ with respect to ${\Omega^{2}}$, or more general there is a resolutive-irregular boundary point $\hat{x}\in\Phi^{-1}(x_{0})$, with respect to ${\Omega^{2}}$, which has arbitrarily small ${\tau^{2}}$-neighbourhoods $U$ such that

[TABLE]

then $x_{0}$ is irregular with respect to $\Omega$.

• Proof.

As $\hat{x}$ is resolutive-irregular, there is a resolutive $f\in C({\partial^{2}}\Omega)$ such that

[TABLE]

We may assume that $f\geq 0$ and that $m<1<f(\hat{x})$. Then there is a ${\tau^{2}}$-neighbourhood $U\subset{\overline{\Omega}^{2}}$ of $\hat{x}$ satisfying (6.1) and such that $f\geq 1$ on ${\overline{U}^{2}}\cap{\partial^{2}}\Omega$. Let $G=U\cap\Omega$ and

[TABLE]

Note that $\tilde{f}$ need not be continuous, but since (6.1) holds, $\tilde{f}$ is continuous at $x_{0}$.

If $u\in\mathcal{U}_{f}({\Omega^{2}})$, then $u\in\mathcal{U}_{\tilde{f}}(G)$, where $G$ is equipped with the given metric topology of ${\overline{\Omega}}$. Hence, {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle P}\kern 0.0pt}\hss}{P}}}_{G}\tilde{f}\leq P_{{\Omega^{2}}}f in $G$. It thus follows that

[TABLE]

which, together with Proposition 5.3, shows that $x_{0}$ is irregular with respect to $G$. Since $\partial\Omega$ 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 $\Omega$. ∎

• Proof of Theorem 6.2.

1 $\mathchoice{\quad\Longleftrightarrow\quad}{\Leftrightarrow}{\Leftrightarrow}{\Leftrightarrow}$ 2 This is a direct consequence of the fact that the given metric boundary is resolutive.

2 $\Rightarrow$ (c) This follows from Proposition 5.4.

(c) $\Rightarrow$ 1 This follows from Theorem 6.3.

(c) $\Rightarrow$ (d) Let $\hat{x}_{1},\hat{x}_{2}\in\Phi^{-1}(x_{0})$ be resolutive-irregular with respect to ${\Omega^{2}}$. Assume that $\hat{x}_{1}\neq\hat{x}_{2}$. We can proceed as in the proof of Theorem 6.3 finding functions $f_{j}$, $\tilde{f}_{j}$ and sets $U_{j}$, $G_{j}$ corresponding to $\hat{x}_{j}$, $j=1,2$. We may require that $U_{1}\cap U_{2}=\varnothing$. 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]).

(d) $\Rightarrow$ (c) This is trivial. ∎

• Proof of Theorem 1.1.

Since ${\partial^{M}}\Omega$ is resolutive (by Theorem 8.2 in Björn–Björn–Shanmugalingam [15]), and all points in $\partial\Omega$ split nicely with respect to the Mazurkiewicz boundary ${\partial^{M}}\Omega$, this follows directly from Theorem 6.2, ∎

A natural question is to which extent the assumptions in Theorem 6.2 are essential. For two arbitrary compactifications ${\partial^{1}}\Omega\prec{\partial^{2}}\Omega$, the implication (corresponding to) 2 $\Rightarrow$ (c) holds by Proposition 5.4, while (d) $\Rightarrow$ (c) is trivial. Also obviously 1 $\Rightarrow$ 2, while the converse implication fails by Example 10.5. (All the counterexamples are collected in Section 10.) What about the other four implications not containing 1?

Consider first the case when $x_{0}$ splits nicely. In this case Examples 10.1 and 10.3 show that no other implication holds. In both cases ${\overline{\Omega}^{2}}={\overline{\Omega}}$ and both boundaries are resolutive (so that regularity and resolutive-regularity coincide).

If we instead keep the assumption that ${\partial^{1}}\Omega=\partial\Omega$, but drop the assumption that $x_{0}$ splits nicely, then (d) $\Rightarrow$ 2 by Theorem 6.3. On the other hand, Example 10.4 shows that 2 $\not\Rightarrow$ (d) and (c) $\not\Rightarrow$ (d), even under the assumption that ${\partial^{2}}\Omega$ is resolutive. We do not know if (c) $\Rightarrow$ 2 holds without the niceness assumption.

We also do not know if “resolutive-regular” can be replaced by “regular” in Theorem 6.2. But if we also drop the niceness assumption, then 2 $\not\Rightarrow$ (d), (c) $\not\Rightarrow$ (d) and (c) $\not\Rightarrow$ 2, by Examples 10.4 and 10.5, while 2 $\Rightarrow$ (c) and (d) $\Rightarrow$ (c) remain true, and it is only the implication (d) $\Rightarrow$ 2 that we do not know if it holds in this case.

See Section 8 for some sharper results when $\Phi^{-1}(x_{0})$ is finite.

7 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 $\partial\Omega\prec{\partial^{2}}\Omega$.

Theorem 7.1**.**

(The resolutive Kellogg property)* Assume that $\partial\Omega\prec{\partial^{2}}\Omega$, where $\partial\Omega$ is the given metric boundary, and that {{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}\hss}{C}}}_{p}}(\,\cdot\,;\Omega)-q.e. boundary point in $\partial\Omega$ splits nicely.*

Then {{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}\hss}{C}}}_{p}}(\operatorname{Irr}_{\operatorname{res}}^{2};{\Omega^{2}})=0, where $\operatorname{Irr}_{\operatorname{res}}^{2}$ is the set of resolutive-irregular boundary points with respect to ${\Omega^{2}}$.

If in addition ${\partial^{2}}\Omega$ is resolutive, then we obtain the Kellogg property for ${\partial^{2}}\Omega$, i.e. {{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}\hss}{C}}}_{p}}(\operatorname{Irr}^{2};{\Omega^{2}})=0, where $\operatorname{Irr}^{2}$ is the set of irregular boundary points with respect to ${\Omega^{2}}$. 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 $\operatorname{Irr}^{2}$ with respect to arbitrary compactifications ${\partial^{2}}\Omega\succ\partial\Omega$. 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.

• Proof.

Let $\operatorname{Irr}\subset\partial\Omega$ be the set of irregular boundary points with respect to $\Omega$, and $E\subset\partial\Omega$ be the set of boundary points which do not split nicely. Then ${C_{p}}(\operatorname{Irr})=0$, by the Kellogg property in Theorem 3.9 in Björn–Björn–Shanmugalingam [13] and Theorem 6.1 in Björn–Björn–Shanmugalingam [14] (or Theorems 10.5 and 10.22 in [10]). By Theorem 6.2, $\operatorname{Irr}_{\operatorname{res}}^{2}\subset\Phi^{-1}(\operatorname{Irr}\cup E)$, where $\Phi:{\overline{\Omega}^{2}}\to{\overline{\Omega}}$ is the projection. Also, {{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}\hss}{C}}}_{p}}(\operatorname{Irr}\cup E;\Omega)\leq{C_{p}}(\operatorname{Irr}\cup E)=0, by Lemma 5.2 in Björn–Björn–Shanmugalingam [15]. Hence, by Lemma 3.4,

[TABLE]

• Proof of Theorem 1.2.

Since ${\partial^{M}}\Omega$ is resolutive (by Theorem 8.2 in Björn–Björn–Shanmugalingam [15]), and all points in $\partial\Omega$ split nicely with respect to the Mazurkiewicz boundary ${\partial^{M}}\Omega$, the Kellogg property for ${\partial^{M}}\Omega$ follows directly from Theorem 7.1. ∎

To establish Theorem 1.3, and its generalization Theorem 7.3 below, we need the following two conditions:

1. (i).

${\partial^{1}}\Omega$ is q.e.-invariant if whenever $f\in C({\partial^{1}}\Omega)$ and $h:{\partial^{1}}\Omega\to{\overline{\mathbf{R}}}$ satisfies {{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}\hss}{C}}}_{p}}(\{x\in{\partial^{1}}\Omega:h(x)\neq 0\};{\Omega^{1}})=0, then {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle P}\kern 0.0pt}\hss}{P}}}_{{\Omega^{1}}}f={\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle P}\kern 0.0pt}\hss}{P}}}_{{\Omega^{1}}}(f+h); 2. (ii).

the weak Kellogg property holds for ${\partial^{1}}\Omega$ if for every $f\in C({\partial^{1}}\Omega)$ there is a set $E_{f}$, with {{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}\hss}{C}}}_{p}}(E_{f};{\Omega^{1}})=0, such that

[TABLE]

One may think that we also need to require that ${\partial^{1}}\Omega$ 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.

Proposition 7.2**.**

If ${\partial^{1}}\Omega$ is q.e.-invariant and that the weak Kellogg property holds, then ${\partial^{1}}\Omega$ is resolutive.

Example 10.5 shows that the weak Kellogg assumption cannot be dropped, nor can it be replaced by the resolutive Kellogg property.

• Proof.

Let $f\in C({\partial^{1}}\Omega)$ and $h=\infty\chi_{E_{-f}}$, where $E_{-f}$ comes from the weak Kellogg property for $-f$. Then {\mathchoice{\hbox to0.0pt{\underline{\phantom{\displaystyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\underline{\phantom{\textstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\underline{\phantom{\scriptstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\underline{\phantom{\scriptscriptstyle P}\kern 0.0pt}\hss}{P}}}_{{\Omega^{1}}}f\in\mathcal{U}_{f-h}, and hence by the q.e.-invariance and Proposition 4.2,

[TABLE]

from which it follows that $f$ is resolutive. ∎

Theorem 7.3**.**

Assume that ${\partial^{1}}\Omega$ is q.e.-invariant and that the weak Kellogg property holds. Let $f\in C({\partial^{1}}\Omega)$. Then there exists a unique bounded $p\mspace{1.0mu}$-harmonic function $u$ on $\Omega$ such that

[TABLE]

Moreover, $u=P_{{\Omega^{1}}}f$.

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, $u=P_{{\Omega^{1}}}f$ satisfies (7.2), which establishes the existence.

As for the uniqueness, let $u$ be a bounded $p\mspace{1.0mu}$-harmonic function and $E\subset{\partial^{1}}\Omega$ be such that {{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}\hss}{C}}}_{p}}(E;{\Omega^{1}})=0 and

[TABLE]

Let $h=\infty\chi_{E}$. Then $u\in\mathcal{U}_{f-h}$ and thus u\geq{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle P}\kern 0.0pt}\hss}{P}}}_{{\Omega^{1}}}(f-h)=P_{{\Omega^{1}}}f, by the q.e.-invariance. By applying this to $-u$ and $-f$ we also see that $u\leq P_{{\Omega^{1}}}f$. Hence $u=P_{{\Omega^{1}}}f$. ∎

• Proof of Theorem 1.3.

By Theorem 8.2 in Björn–Björn–Shanmugalingam [15], ${\partial^{M}}\Omega$ is q.e.-invariant, and by Theorem 1.2 the Kellogg property holds for ${\partial^{M}}\Omega$. Hence, the conclusion follows directly from Theorem 7.3. ∎

Corollary 7.4**.**

Assume that $\partial\Omega\prec{\partial^{1}}\Omega$, that {{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}\hss}{C}}}_{p}}(\,\cdot\,;\Omega)-q.e. boundary point in $\partial\Omega$ splits nicely, and that ${\partial^{1}}\Omega$ is Sobolev-resolutive. Then the assumptions, and therefore the conclusion, in Theorem 7.3 hold for ${\partial^{1}}\Omega$.

• Proof.

As ${\partial^{1}}\Omega$ is Sobolev-resolutive, it is automatically resolutive. Moreover it is q.e.-invariant by Proposition 7.1 in Björn–Björn–Sjödin [17]. It follows from Theorem 7.1 that the Kellogg property holds. Hence the conclusion follows from Theorem 7.3. ∎

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 ${\overline{\Omega}^{1}}$ is metrizable then the weak Kellogg property is equivalent to the usual Kellogg property.

Theorem 7.5**.**

Assume that ${\overline{\Omega}^{1}}$ is metrizable. Then the weak Kellogg property holds if and only if the usual Kellogg property holds.

Observe that we do not assume that ${\overline{\Omega}^{1}}$ is resolutive. In the proof below we use that it follows from the metrizability that $C({\partial^{1}}\Omega)$ 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 $C({\partial^{1}}\Omega)$ is not separable we do not know if the weak Kellogg property for $p\mspace{1.0mu}$-harmonic functions implies the usual Kellogg property.

• Proof.

Assume that the weak Kellogg property holds. By Theorem 2.10 in [17], $C({\partial^{1}}\Omega)$ is separable, i.e. it contains a dense countable subset $A$. Let

[TABLE]

where $E_{f}$ comes from the weak Kellogg property for $f$. As the capacity is countably subadditive, we see that {{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}\hss}{C}}}_{p}}(E;{\Omega^{1}})=0.

If $f\in C({\partial^{1}}\Omega)$, then we can find $f_{j}\in A$ such that $f_{j}\to f$ uniformly. Then also {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle P}\kern 0.0pt}\hss}{P}}}_{{\Omega^{1}}}f_{j}\to{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle P}\kern 0.0pt}\hss}{P}}}_{{\Omega^{1}}}f uniformly and it follows that

[TABLE]

Hence $\operatorname{Irr}^{1}\subset E$ and the Kellogg property follows. The converse implication is trivial. ∎

8 Further results when $\Phi^{-1}(x_{0})$

is finite

The results in Section 5–7 can be strengthened when $\Phi^{-1}(x_{0})$ 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.

Proposition 8.1**.**

Assume that ${\partial^{1}}\Omega\prec{\partial^{2}}\Omega$, and let $\Phi:{\overline{\Omega}^{2}}\to{\overline{\Omega}^{1}}$ be the projection. Let $x_{0}\in{\partial^{1}}\Omega$ and assume that $\Phi^{-1}(x_{0})$ consists of just one point $\hat{x}$. Then $x_{0}$ is regular with respect to ${\Omega^{1}}$ if and only if $\hat{x}$ is regular with respect to $\Omega^{2}$.

Example 10.3 shows that Proposition 8.1 cannot be generalized to the case when $\Phi^{-1}(x_{0})$ is finite. But if we assume that ${\partial^{1}}\Omega=\partial\Omega$ is the given metric boundary, we do obtain the following characterization.

Theorem 8.2**.**

Assume that $\partial\Omega\prec{\partial^{2}}\Omega$, where $\partial\Omega$ is the given metric boundary. Let $\Phi:{\overline{\Omega}^{2}}\to{\overline{\Omega}}$ be the projection and $x_{0}\in\partial\Omega$. Assume that $\Phi^{-1}(x_{0})$ is finite. Then the following are equivalent:**

1. (a*′*).

The point $x_{0}$ is irregular with respect to $\Omega$. 2. (b*′*).

There is at least one irregular boundary point in $\Phi^{-1}(x_{0})$ with respect to ${\Omega^{2}}$. 3. (c*′*).

There is exactly one irregular boundary point in $\Phi^{-1}(x_{0})$ with respect to ${\Omega^{2}}$.

Moreover, if $\hat{x}\in\Phi^{-1}(x_{0})$, then $\hat{x}$ is regular if and only if it is resolutive-regular (with respect to ${\overline{\Omega}^{2}}$).

The implication (c*′*) $\Rightarrow$ (b*′*) is of course trivial and the implication (a*′*) $\Rightarrow$ (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 $\partial\Omega$. If we instead would assume that the larger boundary is $\partial\Omega$, 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 $\partial\Omega\prec{\partial^{2}}\Omega$, where $\partial\Omega$ is the given metric boundary, and that $\Phi^{-1}{(x_{0})}$ is finite for {{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}\hss}{C}}}_{p}}(\,\cdot\,;\Omega)-q.e. $x_{0}\in\partial\Omega$.*

Then {{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}\hss}{C}}}_{p}}(\operatorname{Irr}^{2};{\Omega^{2}})=0.

• 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.

Proposition 8.4**.**

Let $x_{0}\in{\partial^{1}}\Omega$. For each nonempty compact $K\subset{\partial^{1}}\Omega\setminus\{x_{0}\}$, let $f_{K}\in C({\partial^{1}}\Omega)$ be nonnegative and such that $f(x_{0})=0<\inf_{K}f_{K}$ (it exists by Tietze’s extension theorem).

Then $x_{0}$ is regular if and only if

[TABLE]

If there is a nonnegative function $h\in C({\partial^{1}}\Omega)$ 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 ${\overline{\Omega}^{1}}$ is first countable).

• Proof.

If ${\partial^{1}}\Omega=\{x_{0}\}$ is the one-point compactification of $\Omega$, then $x_{0}$ is regular as it is the only boundary point, and the equivalence is trivial. So assume that ${\partial^{1}}\Omega\neq\{x_{0}\}$.

Assume first that (8.1) holds. Let $f\in C({\partial^{1}}\Omega)$. We may assume that $f(x_{0})=0$. Let $\varepsilon>0$. Then there is a ${\tau^{1}}$-neighbourhood $G\varsubsetneq{\partial^{1}}\Omega$ of $x_{0}$ such that $f<\varepsilon$ in $G\cap{\partial^{1}}\Omega$. Let $K={\partial^{1}}\Omega\setminus G$ and $M=\sup_{{\partial^{1}}\Omega}f/{\inf_{K}f_{K}}$. Then $f\leq Mf_{K}+\varepsilon$ on ${\partial^{1}}\Omega$. Hence

[TABLE]

Letting $\varepsilon\to 0$ shows that

[TABLE]

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 $K\subset{\partial^{1}}\Omega\setminus\{x_{0}\}$, let $f_{K}\in C({\partial^{1}}\Omega)$ be nonnegative and such that $f(x_{0})=0<\inf_{K}f_{K}$ (which exists by Tietze’s extension theorem). Also let $\hat{f}_{\widehat{K}}=f_{\Phi(\widehat{K})}\circ\Phi$ for each compact $\widehat{K}\subset{\partial^{2}}\Omega\setminus\{\hat{x}\}$.

If a net $\{y_{\lambda}\}_{\lambda}$ in $\Omega$ converges to $x_{0}$ in ${\overline{\Omega}^{1}}$ then it must converge to $\hat{x}$ in ${\overline{\Omega}^{2}}$, and conversely. It thus follows from Theorem 4.3 that

[TABLE]

if and only if

[TABLE]

which together with Proposition 8.4 completes the proof. ∎

When proving Theorem 8.2 we also need the following restriction result.

Proposition 8.5**.**

Let $x_{0}\in{\partial^{1}}\Omega$, $G\subset{\overline{\Omega}^{1}}$ be a ${\tau^{1}}$-neighbourhood of $x_{0}$ and ${U^{1}}=G\cap{\Omega^{1}}$. If $x_{0}$ is regular with respect to ${U^{1}}$, then $x_{0}$ is regular with respect to ${\Omega^{1}}$.

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.

Let $f\in C({\partial^{1}}\Omega)$. By Tietze’s extension theorem we can consider $f$ to be defined on ${\overline{\Omega}^{1}}$. We can also assume that $|f|\leq 1$ and that $f(x_{0})=0$. Let $K={\partial^{1}}U\setminus G$. Then, by Tietze’s extension theorem, there is a nonnegative $h\in C({\partial^{1}}U)$ such that $h(x_{0})=0$ and $h=1$ on $K$. Let now \hat{f}=\min\{h+f_{\mathchoice{\vbox{\hbox{\scriptstyle+}}}{\vbox{\hbox{\scriptstyle+}}}{\vbox{\hbox{\scriptscriptstyle+}}}{\vbox{\hbox{\scriptscriptstyle+}}}},1\} on ${\partial^{1}}U$ and $\hat{u}\in\mathcal{U}_{\hat{f}}({U^{1}})$. Then

[TABLE]

is superharmonic in $\Omega$, by Lemma 3.13 in Björn–Björn–Mäkäläinen–Parviainen [12] (or [10, Lemma 10.27]). It follows that $u\in\mathcal{U}_{f}({\Omega^{1}})$ from which we conclude that {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle P}\kern 0.0pt}\hss}{P}}}_{{\Omega^{1}}}f\leq{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle P}\kern 0.0pt}\hss}{P}}}_{{U^{1}}}\hat{f} in $U$. Hence,

[TABLE]

As $f$ was arbitrary, Lemma 5.2 yields that $x_{0}$ is regular with respect to ${\Omega^{1}}$. ∎

• Proof of Theorem 8.2.

Let $x_{1},\mathinner{.}\nonscript\!.\nonscript\!\mathinner{.},x_{m}$ be the points in $\Phi^{-1}(x_{0})$. As ${\overline{\Omega}^{2}}$ is normal we can for each $x_{j}$ find a ${\tau^{2}}$-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 $U_{j}=G_{j}\cap\Omega$. We will now consider $U_{j}$ both with its given metric closure $\overline{U}_{j}$ and with the ${\tau^{2}}$-closure ${\overline{U}^{2}_{j}}$. Note that $(\Phi|_{\overline{U}^{2}_{j}})^{-1}(x_{0})=\{x_{j}\}$ and thus Proposition 8.1 is available.

(a*′*) $\Rightarrow$ (b*′*) This follows from Proposition 5.4.

$\neg$ (a*′*) $\Rightarrow$ $\neg$ (b*′*) Let $j\in\{1,\mathinner{.}\nonscript\!.\nonscript\!\mathinner{.},m\}$. By Corollary 4.4 in Björn–Björn [8] (or Corollary 11.3 in [10]) we see that $x_{0}$ is regular with respect to $U_{j}$. Hence, by Proposition 8.1, $x_{j}$ is regular with respect to $U_{j}^{2}$. Thus, $x_{j}$ is regular with respect to $\Omega^{2}$, by Proposition 8.5.

(c*′*) $\Rightarrow$ (b*′*) This is trivial.

$\neg$ (c*′*) $\Rightarrow$ $\neg$ (b*′*) Assume first that there are (at least) two irregular boundary points in $\Phi^{-1}(x_{0})$ with respect to ${\Omega^{2}}$, 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 $U_{j}^{2}$, $j=1,2$. 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 $\Phi^{-1}(x_{0})$ (with respect to ${\Omega^{2}}$), 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). ∎

9 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).

Theorem 9.1**.**

Assume that $\partial\Omega\prec{\partial^{2}}\Omega$, where $\partial\Omega$ is the given metric boundary. Let $\Phi:{\overline{\Omega}^{2}}\to{\overline{\Omega}}$ be the projection, $\hat{x}\in{\partial^{2}}\Omega$ and $x_{0}=\Phi(\hat{x})$. Assume that either

1. (i).

$\Phi^{-1}(x_{0})$* is finite*;* or* 2. (ii).

$x_{0}$* splits nicely and ${\partial^{2}}\Omega^{\prime}$ is resolutive for every open $\Omega^{\prime}\subset\Omega$.*

If $U\subset\Omega$ is an open set such that $\hat{x}\in{\overline{U}^{2}}$ and $\hat{x}$ is regular with respect to ${\Omega^{2}}$, then $\hat{x}$ is regular with respect to ${U^{2}}$.

Moreover, if $G^{2}\subset{\overline{\Omega}^{2}}$ is a ${\tau^{2}}$-neighbourhood of $\hat{x}$, then $\hat{x}$ is regular with respect to ${\Omega^{2}}$ if and only if it is regular with respect to $(G\cap\Omega)^{2}$.

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 $\Phi^{-1}(x_{0})$ is finite. In order to prove the first part we need to consider two cases.

Case 1. $x_{0}$ is regular with respect to $\Omega$ (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 $U$) that $\hat{x}$ is regular with respect to ${U^{2}}$.

Case 1. $x_{0}$ is irregular with respect to $\Omega$. By Theorem 8.2 there is $x^{\prime}\in\Phi^{-1}(x_{0})$ which is irregular with respect to ${\Omega^{2}}$. Since $\hat{x}$ is regular with respect to ${\Omega^{2}}$, we must have $x^{\prime}\neq\hat{x}$. As ${\overline{\Omega}^{2}}$ is a normal space there are ${\tau^{2}}$-neighbourhoods $\widehat{G}$ and $G^{\prime}$ of $\hat{x}$ and $x^{\prime}$, respectively, with disjoint ${\tau^{2}}$-closures.

Let $V=\widehat{V}\cup V^{\prime}:=(\widehat{G}\cap U)\cup(G^{\prime}\cap\Omega)$. By Proposition 8.5, $x^{\prime}$ is irregular with respect to ${V^{2}}$. Thus, by Theorem 8.2, (b*′*) $\Rightarrow$ (c*′*), applied to $V$, $\hat{x}$ must be regular with respect to ${V^{2}}$. As $\widehat{V}$ and $V^{\prime}$ are disjoint the Perron solution {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle P}\kern 0.0pt}\hss}{P}}}_{{V^{2}}}f within $\widehat{V}$ only depends on the boundary values on ${\partial^{2}}\widehat{V}$. Since $\hat{x}\notin{\overline{V^{\prime}}^{2}}$, it follows that $\hat{x}$ is regular also with respect to $\widehat{V}^{2}$. As $\widehat{V}\subset U$, it follows from Proposition 8.5 that $\hat{x}$ 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 ${\partial^{2}}\Omega$ is resolutive (and almost the only available way) is to show that continuous functions on ${\partial^{2}}\Omega$ can be uniformly approximated (on ${\partial^{2}}\Omega$) by functions in

[TABLE]

from which the resolutivity (and even Sobolev-resolutivity) of ${\partial^{2}}\Omega$ follows by Theorem 6.4 and Proposition 7.3 in Björn–Björn–Sjödin [17]. If one instead require that continuous functions on ${\overline{\Omega}^{2}}$ can be uniformly approximated (on ${\overline{\Omega}^{2}}$) by functions in $A$, then not only ${\partial^{2}}\Omega$ is resolutive, but also ${\partial^{2}}\Omega^{\prime}$ for any open $\Omega^{\prime}\subset\Omega$. To see this one just need to take restrictions to ${\overline{\Omega^{\prime}}^{2}}$, and apply the same resolutivity results. Note however that it is not trivial that the restriction of a {{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}\hss}{C}}}_{p}}(\,\cdot\,;{\Omega^{2}})-quasicontinuous function is {{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}\hss}{C}}}_{p}}(\,\cdot\,;{(\Omega^{\prime})^{2}})-quasicontinuous, since the conditions (3.1) and (3.2) are different, but this follows from the fact that {{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}\hss}{C}}}_{p}}(\,\cdot\,;{\Omega^{2}}) is an outer capacity, by Proposition 4.2 in [17].

In particular, this is true for the Mazurkiewicz boundary ${\partial^{M}}\Omega$, if $\Omega$ is finitely connected at the boundary, since Lipschitz functions on ${\overline{\Omega}\mspace{1.0mu}}^{M}$ belong to $N^{1,p}(\Omega)$. In this case, the resolutive of all subboundaries also follows from Theorem 11.2 in Björn–Björn–Shanmugalingam [15].

• 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. ∎

10 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 $\mathbf{R}^{2}$. To simplify notation we will consider $\mathbf{R}$ to be embedded into $\mathbf{R}^{2}$ in the usual way.

The boundaries in Examples 10.1–10.4 are all resolutive (and thus obviously regularity and resolutive-regularity coincide), while the boundary in Example 10.5 is not.

Example 10.1**.**

Let $\Omega=B(0,2)\setminus\{0,1\}\subset\mathbf{R}^{2}$ (unweighted) with $1<p\leq 2$. Then [math] and $1$ are both irregular with respect to $\mathbf{R}^{2}$. Let ${\overline{\Omega}^{1}}$ be ${\overline{\Omega}}$ with [math] and $1$ identified, $\hat{0}$ be the identified point, and $\Phi:{\overline{\Omega}}\to{\overline{\Omega}^{1}}$ be the projection. It follows from Theorem 4.3 that ${\partial^{1}}\Omega$ is resolutive. Let $\hat{f}=\chi_{\hat{0}}$ which is continuous on ${\partial^{1}}\Omega$ and $f=\chi_{\{0,1\}}$ which is continuous on $\partial\Omega$. By Theorem 4.3,

[TABLE]

which shows that $\hat{0}$ is irregular with respect to ${\partial^{1}}\Omega$, but $\Phi^{-1}(\hat{0})=\{0,1\}$ contains two irregular boundary points.

This shows that even though $\hat{0}$ splits nicely and ${\partial^{1}}\Omega$ is resolutive, 2 $\not\Rightarrow$ (d) and (c) $\not\Rightarrow$ (d) in Theorem 6.2 for ${\partial^{1}}\Omega\prec\partial\Omega$. It also shows that (a*′*) $\not\Rightarrow$ (c*′*) and (b*′*) $\not\Rightarrow$ (c*′*) in Theorem 8.2 for ${\partial^{1}}\Omega\prec\partial\Omega$.

Example 10.2**.**

Let $p=3$, $w(x)=|x|$ be a weight on $\mathbf{R}^{2}$, and $d\mu=w\,dx$. Then $w$ is a Muckenhoupt $A_{3}$-weight, see Section 1.6 in Heinonen–Kilpeläinen–Martio [25]. Moreover ${C_{p}}(\{x\})>0$ if $x\neq 0$, while ${C_{p}}(\{0\})=0$, see Example 2.22 in [25].

Let $\Omega$, ${\overline{\Omega}^{1}}$, $f$, $\hat{f}$ and $\Phi$ be as in Example 10.1. It again follows from Theorem 4.3 that ${\partial^{1}}\Omega$ is resolutive. This time

[TABLE]

by Lemma 3.4 and a straightforward calculation. Moreover

[TABLE]

by Theorem 4.3 and Björn–Björn–Shanmugalingam [14, Corollary 6.2] (or [10, Corollary 10.22]).

The set $\Omega\cup\{0\}$ is regular by the Kellogg property (for the given metric boundary). Hence $\lim_{x\to 1}u(x)=1$ and $\lim_{x\to 2}u(x)=0$, showing that $u$ is nonconstant. Thus by the strong maximum principle, $0<u<1$ in $\Omega\cup\{0\}$. In particular

[TABLE]

showing that $\hat{0}$ is irregular. Hence both the weak and the usual Kellogg properties fail for ${\Omega^{1}}$, and also the conclusion in Theorem 7.3 fails even though ${\partial^{1}}\Omega$ is q.e.-invariant (as the empty set is the only boundary set with zero capacity).

Example 10.3**.**

Let $p=3$, $X=[0,8]\times[0,1]\subset\mathbf{R}^{2}$ and equip it with the measure $d\mu=w\,dx$, where $w(x)=|x|$. Also let $\Omega=X\setminus\{0,8\}$. As in Example 10.2 we have ${C_{p}}(\{0\})=0$ and ${C_{p}}(X\setminus\Omega)={C_{p}}(\{8\})>0$. Then [math] is irregular, while $8$ is regular with respect to $\Omega$. Let ${\overline{\Omega}^{1}}$ be ${\overline{\Omega}}$ with [math] and $8$ identified as $\hat{0}$, and $\Phi:{\overline{\Omega}}\to{\overline{\Omega}^{1}}$ be the projection. It follows from Theorem 4.3 that ${\partial^{1}}\Omega$ is resolutive. Moreover, $\hat{0}$ is regular with respect to ${\Omega^{1}}$ (as it is the sole boundary point), while $\Phi^{-1}(\{\hat{0}\})$ contains exactly one irregular boundary point.

This shows that even though $\hat{0}$ splits nicely and ${\partial^{1}}\Omega$ is resolutive, (d) $\not\Rightarrow$ 2 and (c) $\not\Rightarrow$ 2 in Theorem 6.2 and (c*′*) $\not\Rightarrow$ (a*′*) and (b*′*) $\not\Rightarrow$ (a*′*) in Theorem 8.2 for ${\partial^{1}}\Omega\prec\partial\Omega$. It also shows that Proposition 8.1 cannot be generalized to the case when $\Phi^{-1}(x_{0})$ is finite.

Let $G=(\Omega\setminus B(4,2))\cup\{\hat{0}\}$, which is a ${\tau^{1}}$-neighbourhood of $\hat{0}$, $U=\{x\in G\cap\Omega:|x|<4\}$ and $f=\chi_{{\partial^{1}}U\setminus\{\hat{0}\}}\in C({\partial^{1}}G)$. Since [math] is irregular with respect to $G$, we see that $P_{G^{1}}f=\chi_{U}$ and thus $\hat{0}$ is irregular with respect to $(G\cap\Omega)^{1}$. 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 10.4**.**

Let $\Omega=B(0,1)\setminus\{0\}\subset\mathbf{R}^{2}$ (unweighted). We want to create a compactification of $\Omega$ such that the outer boundary $\partial\Omega\setminus\{0\}$ is as for the given metric, but such that the boundary point [math] corresponds to an interval $I=[-1,1]$. To do so, we let $h(x)=\sin(1/|x|)$. Let also $h_{j}(x_{1},x_{2})=x_{j}$ be the coordinate functions, and $Q=\{h,h_{1},h_{2}\}$.

We want ${\overline{\Omega}^{2}}$ to be the smallest compactification such that the three functions $h$, $h_{1}$ and $h_{2}$ have continuous extensions to ${\overline{\Omega}^{2}}$. Such an extension exists and is called the $Q$-compactification ${\overline{\Omega}\mspace{1.0mu}}^{Q}$ of $\Omega$, see Björn–Björn–Sjödin [17], (it is unique up to homeomorphism). Note that ${\overline{\Omega}}\prec{\overline{\Omega}^{2}}$, and that ${\overline{\Omega}^{2}}$ is metrizable, by Theorem 2.10 in [17]. Let $\Phi:{\overline{\Omega}^{2}}\to{\overline{\Omega}}$ be the corresponding projection. A neighbourhood base for $a\in I$ is given by

[TABLE]

with $0<\varepsilon<1$.

Assume that $1<p\leq 2$. Then [math] is irregular with respect to $\Omega$, and {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle P}\kern 0.0pt}\hss}{P}}}_{\Omega}f\equiv 0 for $f=\chi_{\{0\}}$. Then, by Theorem 4.3, $P_{{\Omega^{2}}}({f\circ\Phi})=P_{\Omega}f\equiv 0$, showing that all points in $I=\Phi^{-1}(\{0\})$ are irregular. Moreover, in the terminology of Björn [4] (or [10, Chapter 13]), [math] is semiregular and the Perron solutions with respect to $\Omega$ ignore the value at [math] for continuous boundary data. Hence, if $f\in C({\partial^{2}}\Omega)$ and we let

[TABLE]

then, by Theorem 4.3,

[TABLE]

from which we conclude that $f$ is resolutive and thus ${\partial^{2}}\Omega$ is resolutive. This shows that 2 $\not\Rightarrow$ (d) and (c) $\not\Rightarrow$ (d) in Theorem 6.2 when $x_{0}$ does not split nicely, even if ${\partial^{2}}\Omega$ is assumed to be resolutive.

Example 10.5**.**

Assume now that $\Omega$ and ${\overline{\Omega}^{2}}$ are as in Example 10.4, but this time with $p>2$. In this case ${C_{p}}(\{0\})>0$ and thus [math] is regular with respect to $\Omega$. Let again $f\in C({\partial^{2}}\Omega)$ and let $f_{1}$ and $f_{2}$ be given by (10.2). If $a<\sup_{I}f$ then, because of (10.1), any function $u\in\mathcal{U}_{f}({\Omega^{2}})$ is necessarily greater than $a$ on concentric circles which are arbitrarily close to [math], and hence it is greater than $a$ in a neighbourhood of [math], 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 $u\in\mathcal{U}_{f_{2}}(\Omega)$. Conversely, any function in $\mathcal{U}_{f_{2}}(\Omega)$ necessarily belongs to $\mathcal{U}_{f}({\Omega^{2}})$. We therefore conclude that

[TABLE]

Similarly, {\mathchoice{\hbox to0.0pt{\underline{\phantom{\displaystyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\underline{\phantom{\textstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\underline{\phantom{\scriptstyle P}\kern 0.0pt}\hss}{P}}{\hbox to0.0pt{\underline{\phantom{\scriptscriptstyle P}\kern 0.0pt}\hss}{P}}}_{{\Omega^{2}}}f=P_{\Omega}f_{1}. As $P_{\Omega}f_{1}\equiv P_{\Omega}f_{2}$ if and only if $f$ is constant on $I$, only such $f$ are resolutive with respect to ${\Omega^{2}}$, and thus ${\partial^{2}}\Omega$ is not resolutive.

From this we can easily conclude that all the points in $I$ are irregular, but resolutive-regular. As {{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}\hss}{C}}}_{p}}(I;{\Omega^{2}})={{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}\hss}{C}}}_{p}}(\{0\},\Omega)>0, by Lemma 3.4, we see that neither the weak nor the usual Kellogg property hold with respect to ${\Omega^{2}}$. On the other hand the resolutive Kellogg property does hold, as there are no resolutive-irregular boundary points. Moreover, this shows that (c) $\Rightarrow$ 2 in Theorem 6.2 can fail when $x_{0}$ does not split nicely, even if ${\partial^{2}}\Omega$ is assumed to be resolutive.

In fact, a similar argument (using concentric circles) shows that

[TABLE]

Hence ${\partial^{2}}\Omega$ is q.e.-invariant (as the only set $E\subset{\partial^{2}}\Omega$ with zero capacity is the empty set). Since any bounded $p\mspace{1.0mu}$-harmonic function on $\Omega$ has a limit as $x\to 0$ (see below), we also conclude from (10.3) that if $f\in C({\partial^{2}}\Omega)$ 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 $p\mspace{1.0mu}$-harmonic function $u\geq 0$ on $\Omega$ the limit $\lim_{x\to 0}u(x)$ exists. To this end, let

[TABLE]

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 $m:=\lim_{r\to 0}m(r)$ and $M:=\lim_{r\to 0}M(r)$ exist. By Harnack’s inequality, there is a constant $A$ such that M\bigl{(}\tfrac{1}{2}\bigr{)}\leq Am\bigl{(}\tfrac{1}{2}\bigr{)}, and by scaling invariance we can apply it also in smaller punctured balls. Let $\varepsilon>0$. Then there is $\rho>0$ such that for $0<r<\rho$ we have $u>m-\varepsilon$ in $B(0,2r)$. Applying the Harnack inequality to $u-(m-\varepsilon)$ shows that

[TABLE]

Letting first $r\to 0$ and then $\varepsilon\to 0$ shows that $M=m$.