S. K. Vodopyanov, A. O. Molchanova, “The boundary behavior of $\mathcal Q_{p,q}$-homeomorphisms”, Izv. Math., 87:4 (2023), 683

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Izvestiya: Mathematics, 2023, Volume 87, Issue 4, Pages 683–725
DOI: https://doi.org/10.4213/im9376e (Mi im9376)

This article is cited in 4 scientific papers (total in 4 papers)

The boundary behavior of

$\mathcal Q_{p,q}$ -homeomorphisms

S. K. Vodopyanov^a, A. O. Molchanova^b

^a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
^b University of Vienna, Vienna, Austria

English version PDF (1005 kB) HTML full-text Citations (4) Russian version article

References:

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DOI: https://doi.org/10.4213/im9376e

Abstract: This article studies systematically the boundary correspondence problem for

$\mathcal Q_{p,q}$ -homeomorphisms. The presented example demonstrates a deformation of the Euclidean boundary with the weight function degenerating on the boundary.

Keywords: quasiconformal analysis, Sobolev space, composition operator, capacity of a condenser, capacity metric, capacity boundary.

Funding agency	Grant number
Ministry of Science and Higher Education of the Russian Federation	FWNF-2022-0006
EU Framework Programme for Research and Innovation	847693
This article is prepared in fulfillment of the state contract of the Ministry of Education and Science of the Russian Federation for the Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences (Project no. FWNF-2022-0006). A. Molchanova was supported by the European Unions Horizon 2020 research and innovation programme under the Marie Składowska-Curie grant agreement no. 847693.

Received: 11.05.2022
Revised: 06.10.2022

Bibliographic databases:

Document Type: Article

UDC: 517.518+517.54

MSC: 30C65, 47B33, 46E35

Language: English

Original paper language: English

Introduction

In this section, we briefly survey the articles dealing with the boundary behaviour of mappings in quasiconformal analysis. Consider two domains $D,D' \subset \mathbb R^2$ bounded by Jordan curves and a conformal mapping $f\colon D \to D'$ . The classical result, established independently by Carathéodory [1] and Osgood and Taylor [2], asserts that $f$ extends to the boundary, giving a homeomorphism $\overline{f}\colon \overline{D} \to \overline{D'}$ . The Jordan condition for the boundary is necessary, which is easy to see in the example of a slit disc. Nevertheless, a homeomorphic extension is possible for some generalized boundary accounting for the geometry of the domain. This construction, introduced by Carathéodory [1] and called the prime end boundary, initiated intensive applications of the geometric approach to study the boundary behaviour of mappings.

Carathéodory’s prime end theory received developments on the plane $\mathbb{R}^2$ (see [3], [4]) and in the space $\mathbb{R}^n$ for $n> 2$ (see [5], [6]), in studying Dirichlet problems for elliptic equations [7], and in the theory of dynamical systems (see [8], [9]). For more detailed surveys of the available results and literature, see [10]–[13].

A natural development of these questions is to study the boundary behaviour of quasiconformal mappings in space. It requires a more refined analysis of the geometric properties of domains. Indeed, in the higher-dimensional case there exist a Jordan domain and a quasiconformal mapping admitting no homeomorphic extension to the boundary of this domain [14]. In some questions it turned out helpful to describe the geometric properties of domains using the concept of modulus of a curve family [15]. With that, a simple classification of boundary points was introduced: for instance, the properties of the boundary to be quasiconformally flat or quasiconformally accessible in [16], [17], or properties $P1$ and $P2$ of [18]. This approach became widely used in the last decade to study the geometric properties of mappings. Let us mention only some articles concerning the boundary correspondence of quasiconformal mappings [19], [20], $Q$ -homeomorphisms, see the book [21] and the articles [13], [22] (a more detailed discussion appears in § 4), as well as the mappings satisfying generalized modular inequalities [23].

An alternative functional-geometric approach to study the boundary behaviour of quasiconformal mappings is based on the relation between the Euclidean geometry of the domain and the functional space $L^1_n$ via the concept of the variational capacity of a condenser. This approach was founded in [24]–[26] and applied also to studying mappings which are not quasiconformal [27]. As [17] shows, the functional-geometric approach can be interpreted in the language of moduli of curve families.

The three main approaches to the boundary behaviour of mappings, using prime ends, geometric description, and functional-geometric definition, form an hierarchy, as each of them adequately describes the boundary behaviour of certain classes of mappings. This article studies the problem of boundary correspondence for $\mathcal Q_{p,q}$ -homeomorphisms, whose fundamental properties were established in [29]–[34]. To this end, we complete the domains in special capacity metrics on the image and the preimage, associated with the geometry of a suitable Sobolev class. The elements adjoined to the domain in the completion of the corresponding metric space constitute an improper boundary, which we call the capacity boundary $H_{\rho}$ .

In § 2 the study of the boundary behaviour of the homeomorphism $f\in\mathcal Q_{p,q}$ defined in § 1 consists in:

(1) continuing $f$ to the capacity boundary $H_{\rho}$ , with the main result stated as Theorem 2.22;

(2) establishing a connection between the elements of the capacity boundary and the points of the Euclidean boundary of the domain, see Theorem 2.37 and Corollaries 2.38 and 2.39.

In § 3, we compare the approaches stated in the languages of moduli and capacity. In § 4 we contrast the conclusions of this article with the main results of other approaches to the problem of boundary behaviour of mappings. Some applications of our results are given in § 5.

This article naturally enters the line of publications [28]–[36], preceded by the results of [37]–[39] and the articles cited in the bibliographies in [28]–[34] and arising on the crossroads of the theory of Sobolev function spaces [40], [41] and geometric theory of functions [18], [42]–[48]. Some results of this series of articles have found applications in nonlinear elasticity, see [49].

§ 1. Classes of $\mathcal Q_{p,q}$ -homeomorphisms

In what follows $D$ and $D'$ stand for domains (open connected sets) in $\mathbb{R}^n$ . The norm $|x|_p$ of a vector $x=(x_1,x_2,\dots,x_n)\in\mathbb R^n$ is defined as $|x|_p=\bigl(\sum_{k=1}^n|x_k|^p\bigr)^{1/p}$ for $p\in[1,\infty)$ and $|x|_\infty=\max_{k=1,\dots,n}|x_k|$ . A ball in the norm $|x|_2$ is a Euclidean ball, while in the norm $|x|_\infty$ it is a Euclidean cube.

1.1. Definitions of Sobolev spaces and the capacity of condensers

For the general theory of Sobolev spaces, the reader is referred to [40], [41]. We recall that a function $u\colon D\to\mathbb R$ is of Sobolev class $L^1_{p}(D)$ if $u\in L_{1,\mathrm{loc}}(D)$ , meaning that $u\in L_1(U)$ for every domain $U$ compactly embedded into $D$ , written $U\Subset D$ , and it has the generalized derivatives $\partial u/dx_j\in L_{1,\mathrm{loc}}(D)$ for every $j=1,\dots,n$ and finite seminorm

$\begin{equation*} \|u\mid L^1_{p}(D)\|=\biggl(\int_{D}|\nabla u(y)|^p\,dy\biggr)^{1/p},\qquad 1\leqslant p\leqslant \infty, \end{equation*} \notag$

where

$\nabla u(y)=(\partial u/dx_1,\partial u/dx_2,\dots,\partial u/dx_n)$ is the generalized gradient of

$u$ . A mapping

$\varphi=(\varphi_1,\dots, \varphi_n)\colon D \to \mathbb R^n$ belongs to the Sobolev class

$W^1_{p,\mathrm{loc}}(D; \mathbb{R}^n)$ whenever

$\varphi_j(x) \in L_{p,\mathrm{loc}}(D)$ and

$\partial\varphi_j/dx_i\in L_{p,\mathrm{loc}}(D)$ for all

$j,\,i=1,\dots,n$ .

We say that a mapping $\varphi\colon D\to \mathbb R^n$ of Sobolev class $W^1_{1,\mathrm{loc}}(D;\mathbb{R}^n)$ is a mapping with finite distortion whenever

$\begin{equation} D\varphi(x)=0\text{ almost everywhere (a.e.) on the set }Z=\{x\in D\colon \det D\varphi (x)=0\}. \end{equation} \tag{1.1}$

(Meaning

$\det D\varphi (x)=0$ at all points of

$Z$ except for a set of Lebesgue measure zero.)

Here, and henceforth, $D\varphi (x)=(\partial\varphi_j(x)/\partial x_i)_{i,j=1}^{n}$ stands for the Jacobi matrix of the mapping $\varphi$ at $x\in D$ , while $|D\varphi (x)|$ , for its Euclidean operator norm, and $\det D\varphi (x)$ , for its determinant, the Jacobian.

A locally integrable function $\omega\colon D'\to\mathbb R$ is called a weight whenever $0\,{<}\,\omega(y)\,{<}\,\infty$ for a.e. $y\in D'$ . A function $u\colon D'\to\mathbb R$ belongs to the weighted Sobolev class $L^1_{p}(D';\omega)$ , with $p\in[1,\infty)$ , if $u \in L_{1,\mathrm{loc}}(D')$ and $\partial u/\partial y_j \in L_{p}(D';\omega)$ for every $j=1,\dots,n$ . The seminorm of a function $u\in L^1_{p}(D';\omega)$ is then defined as

$\begin{equation} \|u\mid L^1_{p}(D';\omega)\|=\biggl(\int_{D'}|\nabla u(y)|^p\omega(y)\,dy\biggr)^{1/p}. \end{equation} \tag{1.2}$

In the case

$\omega\equiv 1$ , instead of

$L^1_{p}(D';1)$ we write simply

$L^1_{p}(D')$ .

Henceforth, the symbol $\operatorname{Lip}_{\mathrm{loc}}(D')$ stands for the space of locally Lipschitz functions on $D'$ . It is obvious that

$\begin{equation*} \operatorname{Lip}_{\mathrm{loc}}(D')=W^1_{\infty,\mathrm{loc}}(D')\cap C(D'), \end{equation*} \notag$

where

$W^1_{\infty,\mathrm{loc}}(D')$ is the space of locally bounded measurable functions on

$D'$ with locally bounded generalized derivative.

We say that a homeomorphism $\varphi \colon D \to D'$ induces the bounded composition operator

$\begin{equation*} \varphi^* \colon {L}^1_p(D';\omega) \cap \operatorname{Lip}_{\mathrm{loc}}(D')\to L^1_q(D), \qquad 1\leqslant q \leqslant p < \infty, \end{equation*} \notag$

acting as

$D\ni x\mapsto(\varphi^*u)(x)=u(\varphi(x))$ , whenever for some constant

$K_{q,p}<\infty$ the inequality

$\begin{equation*} \|\varphi^*u\mid L^1_q(D)\|\leqslant K_{q,p}\|u\mid L^1_p(D';\omega)\| \end{equation*} \notag$

holds for every function

$u\in L^1_p(D')\cap \operatorname{Lip}_{\mathrm{loc}}(D')$ .

1.2. Condensers and their capacity in Sobolev spaces

A condenser in a domain $D\subset \mathbb{R}^n$ is a pair $\mathcal E=(F_1,F_0)$ of connected compact sets (continua) $F_1$ , $F_0\subset D$ . For a continuum $F\subset U$ , where $U\Subset D$ is an open connected compactly embedded set, we denote the condenser $\mathcal E=(F,\partial U)$ by $\mathcal E=(F,U)$ .

A condenser $\mathcal E=(F, U)$ is called annular whenever the complement in $\mathbb R^n$ to the open set $U\setminus F$ consists of two closed sets each of which is connected: the bounded connected component is the continuum $F$ , and the unbounded component is $\mathbb R^n\setminus U$ .

A condenser $\mathcal E=(F, U)$ in $\mathbb R^n$ is called spherical whenever $U=B(x,R)=\{y\in\mathbb R^n \colon |y-x|_2< R\}$ and $F=\overline{B(x,r)}=\{y\in\mathbb R^n \colon |y-x|_2 \leqslant r\}$ , where $r<R$ , and cubical whenever $U=Q(x,R)=\{y\in\mathbb R^n \colon |y-x|_\infty< R\}$ and $F=\overline{Q(x,r)}=\{y\in\mathbb R^n \colon |y-x|_\infty\leqslant r\}$ , respectively.

Definition 1.1. A function $u\colon D\to\mathbb R$ of the class $W^1_{1,\mathrm{loc}}(D)$ is called admissible for a condenser $\mathcal E=(F_1,F_0)\subset D$ whenever

(1) $u$ is continuous,

(2) $u\equiv 1$ on $F_1$ , and

(3) $u \equiv 0$ on $F_0$ .

We denote the collection of admissible functions for a condenser $\mathcal E=(F_1,F_0)$ by $\mathcal A(\mathcal E)$ .

The capacity of a condenser $\mathcal E=(F_1,F_0)$ in the space $L^1_q(D)$ with $q\in[1,\infty)$ is defined as

$\begin{equation} \operatorname{cap}\bigl(\mathcal E; L^1_q(D)\bigr)=\inf_{u}\|u\mid L^1_{q}(D)\|^q, \end{equation} \tag{1.3}$

where the infimum is taken over all admissible functions

$u\in \mathcal A(\mathcal E)\cap L^1_{q}(D)$ for the condenser

$\mathcal E=(F_1,F_0)\subset D$ .

Let us now define the weighted capacity of a condenser $\mathcal E=(F_1,F_0)\subset D'$ in the space $L^1_p(D';\omega)$ by analogy with (1.3):

$\begin{equation*} \operatorname{cap}\bigl(\mathcal E; L^1_p(D';\omega)\bigr)=\inf_{u} \|u\mid L^1_{p}(D';\omega)\|^p, \end{equation*} \notag$

where the infimum is over all admissible functions

$u \in\mathcal A(\mathcal E)\cap \operatorname{Lip}_{\mathrm{loc}}(D')\cap L^1_{p}(D';\omega)$ for the condenser

$\mathcal E=(F_1,F_0)$ .

See the books [41], [44], which present the properties of capacity in Sobolev spaces. For more details on the properties of weighted capacity (for a special class of admissible weights), see [50], Chap. 2.

The definition of capacity yields the following property.

Property 1.2 (subordination principle). Consider two condensers $\mathcal E'\,{=}\,(F'_1,F'_0)$ and $\mathcal E=(F_1,F_0)$ in a domain $D'$ with the plates of the first condenser included in those of the second one, $F'_1\subset F_1$ and $F'_0\subset F_0$ . Then

$\begin{equation*} \operatorname{cap}\bigl(\mathcal E'; L^1_p(D';\omega)\bigr)\leqslant \operatorname{cap}\bigl(\mathcal E; L^1_p(D';\omega)\bigr). \end{equation*} \notag$

1.3. A quasi-additive set function and its properties

Denote by ${\mathcal O}(D)$ a system of open sets in $D$ with the following properties:

(1) $D\in{\mathcal O}(D)$ and if the closure of an open ball $B$ (cube $Q$ ) lies in $D$ , then $B\in{\mathcal O}(D)$ ( $Q\in{\mathcal O}(D)$ );

(2) if $U_1,\dots,U_k\in{\mathcal O}(D)$ is a disjoint system of open sets, then $\bigcup_{i=1}^kU_i\in{\mathcal O}(D)$ , where $k\in \mathbb N$ is an arbitrary number.

The choice of a ball or cube in this definition depends on the choice of a system of elementary sets with respect to which the set function is differentiated, see (1.6).

Definition 1.3. A mapping $\Phi\colon {\mathcal O}(D)\to[0,\infty]$ is called a quasi-additive set function if

(1) for every point $x\in D$ there exists a number $\delta(x)\in(0,\infty)$ such that $\overline{B(x,\delta(x))}\subset D$ and $0<\Phi(B(x,\delta))<\infty$ for all $\delta\in(0, \delta(x))$ , and the ball in this condition can be replaced with a cube;

(2) every finite tuple $\{U_i\in{\mathcal O}(D)\}$ , for $i=1,\dots,l$ , of disjoint open sets with

$\begin{equation} \bigcup_{i=1}^lU_i \subset U,\quad \text{where }U\in{\mathcal O}(D), \text{ satisfies } \sum_{i=1}^{l}\Phi(U_i)\leqslant \Phi(U). \end{equation} \tag{1.4}$

If every finite tuple $\{U_i\in{\mathcal O}(D)\}$ of pairwise disjoint open sets satisfies

$\begin{equation} \sum_{i=1}^{n}\Phi(U_i)=\Phi\biggl(\bigcup_{i=1}^{n} U_i \biggr), \end{equation} \tag{1.5}$

then this set function is called finitely additive, while if (1.5) holds for every countable tuple

$\{U_i\in{\mathcal O}(D)\}$ of disjoint open sets, then this set function is called countably additive. The function

$\Phi$ is monotone whenever

$\Phi(U_1)\leqslant \Phi(U_2)$ as soon as

$U_1\subset U_2 \subset D$ with

$U_1,U_2\in{\mathcal O}(D)$ . Every quasi-additive set function is obviously monotone. A quasi-additive set function

$\Phi\colon {\mathcal O}(D)\to[0,\infty]$ is called a bounded quasi-additive set function whenever

$D\in {\mathcal O}(D)$ and

$\Phi(D)<\infty$ .

It is known (see [51]–[53] for instance) that every quasi-additive set function $\Phi$ defined on some system ${\mathcal O}(D')$ of open subsets of a domain $D'$ is differentiable in the following sense: for a.e. point $y\in D'$ there exists the finite derivative¹:

$\begin{equation} \lim_{\delta\to 0,\, y\in B_\delta}\frac{\Phi(B_\delta)}{\mathcal{H}^n(B_{\delta})}=\Phi'(y); \end{equation} \tag{1.6}$

and, for every open set

$U\in \mathcal O(D')$ ,

$\begin{equation} \int_{U}\Phi'(y)\,dy \leqslant \Phi(U). \end{equation} \tag{1.7}$

1.4. Definition of the class of $\mathcal Q_{p,q}(D',\omega;D)$ -homeomorphisms and their properties

Denote by $\mathcal O_{\mathrm c}(D')$ the minimal system of open sets in $D'$ , which contains:

(1) $D'$ ;

(2) every open cube $Q$ whenever $\overline Q\subset D'$ ;

(3) the complement $Q_2\setminus \overline Q_1$ whenever $Q_1\subset Q_2$ are two cubes with a common center and $\overline Q_2\subset D'$ .

In Definition 1.4 and Theorem 1.6, we consider the mapping $\Phi\colon \mathcal O_{\mathrm c}(D')\to[0,\infty)$ as the bounded quasi-additive set function.

Definition 1.4 [31]. Given two domains $D$ , $D'\subset\mathbb R^n$ , for $n\geqslant2$ , we say that a homeomorphism $f\colon D'\to D$ is of class² $\mathcal{CRQ}_{p, q}(D',\omega;D)$ , where $1< q\leqslant p<\infty$ for $n\geqslant3$ and $1\leqslant q\leqslant p<\infty$ for $n=2$ , while $\omega\in L_{1,\mathrm{loc}}(D')$ is a weight function, if there exist

(1) a constant $K_p>0$ for $q=p$ or

(2) a bounded quasi-additive function $\Psi_{p,q}$ defined on the system $\mathcal O_{\mathrm c}(D')$ of open sets in $D'$ for $q<p$

such that for every cubical condenser $\mathcal E=(\overline{Q(x,r)}, Q(x,R))\subset D'$ with $0<r<R$ with the image $f(\mathcal E)=(f(\overline{Q(x,r)}), f(Q(x,R))\subset D$ we have

$\begin{equation} \begin{cases} \operatorname{cap}^{1/p}\bigl(f(\mathcal E); L^1_p(D)\bigr) \leqslant K_p\operatorname{cap}^{1/p}\bigl(\mathcal E; L^1_p(D';\omega)\bigr), & q=p, \\ \operatorname{cap}^{1/q}\bigl(f(\mathcal E); L^1_q(D)\bigr) \leqslant \Psi_{p,q}(Q(x,R)\setminus \overline{Q(x,r)})^{1/\sigma} \operatorname{cap}^{1/p}\bigl(\mathcal E; L^1_p(D';\omega)\bigr), &q<p, \end{cases} \end{equation} \tag{1.8}$

where

$1/\sigma=1/q-1/p$ .

Definition 1.5 (see [31], [32]). Let $D$ and $D'$ be open sets in $\mathbb R^n$ with $n\geqslant 2$ , $1< q\leqslant p<\infty$ for $n\geqslant3$ and $1\leqslant q\leqslant p<\infty$ for $n=2$ , and $\omega\in L_{1,\mathrm{loc}}(D')$ be a weight function. We say that a homeomorphism $\varphi \colon D\to D'$ belongs to the class $\mathcal{Q}_{p, q}(D',\omega;D)$ , whenever each condenser $\mathcal E=(F_1,F_0)$ in $D'$ with the preimage $\varphi^{-1}(\mathcal E)=(\varphi^{-1}(F_1),\varphi^{-1}(F_0))$ in $D$ satisfies

$\begin{equation} \begin{aligned} \, &\operatorname{cap}^{1/q}\bigl(\varphi^{-1}(\mathcal E); L^1_q(D)\bigr) \nonumber \\ &\qquad\leqslant \begin{cases} \widetilde K_p \operatorname{cap}^{1/p}\bigl(\mathcal E; L^1_p(D';\omega)\bigr), &1<q=p<\infty, \\ \widetilde\Psi(D'\setminus(F_0\cup F_1))^{1/\sigma} \operatorname{cap}^{1/p}\bigl(\mathcal E; L^1_p(D';\omega)\bigr), &1<q<p<\infty, \end{cases} \end{aligned} \end{equation} \tag{1.9}$

where

$1/\sigma=1/q-1/p$ , while

$\widetilde\Psi$ is some bounded quasi-additive set function defined on open subsets of

$D'$ .

It is easy to see that if $\varphi\in\mathcal{Q}_{p, q}(D',\omega;D)$ , then $f=\varphi^{-1}\in\mathcal{CRQ}_{p, q}(D',\omega;D)$ .

The following Theorem 1.6 gives an analytic description of the mappings with inverses of class $\mathcal{CRQ}_{p, q}(D',\omega;D)$ .

Theorem 1.6 (see [33], Theorem 1). A homeomorphism $f\colon D' \to D$ belongs to the class $\mathcal{CRQ}_{p, q}(D',\omega;D)$ with $1<q\leqslant p<\infty$ for $n\geqslant 3$ and $1\leqslant q\leqslant p<\infty$ for $n=2$ if and only if the inverse homeomorphism $\varphi=f^{-1}\colon D\to D'$ enjoys one of the following properties:

(1) the composition operator $\varphi^*\colon {L}^1_p(D';\omega) \cap \operatorname{Lip}_{\mathrm{loc}}(D')\to L^1_q(D)$ , with $1< q \leqslant p<\infty$ , is bounded;

(2) the homeomorphism $\varphi\colon D\to D'$ is of class $\mathcal{Q}_{p, q}(D',\omega;D)$ in the sense of Definition 1.5, with some bounded quasi-additive set function $\widetilde\Psi$ defined on open subsets of $D'$ ;

(3) a homeomorphism $\varphi \colon D \to D'$

- (a) is of Sobolev class $W^1_{q, \mathrm{loc}}(D)$ ,
- (b) has finite distortion in the sense of (1.1), and
- (c) the operator distortion function
  $\begin{equation} D\ni x \mapsto K^{1,\omega}_{q,p}(x,\varphi) = \begin{cases} \dfrac{|D\varphi(x)|}{|{\det D\varphi (x)}|^{1/p}\omega^{1/p}(\varphi(x))} &\textit{if } \det D\varphi (x)\neq 0, \\ 0 &\textit{if } \det D\varphi (x) = 0 \end{cases} \end{equation} \tag{1.10}$
  belongs to $L_{\sigma}(D)$ , where $1/\sigma=1/q-1/p$ if $1< q<p<\infty$ and $\sigma=\infty$ if $q=p$ ;

(4) if $n=2$ , then claims (1)–(3) also hold in the case $1=q \leqslant p<\infty$ .

Note that Theorem 1.6 is a consequence of [29], Theorem 1, [30], and [31], [32], Theorem 1, see details in [33], Theorem 1. The smallest quantities $K_p$ and $\widetilde K_p$ (quasiadditive functions $\Psi$ and $\widetilde\Psi$ ) in (1.8), (1.9) satisfy

$\begin{equation} \text{for }q=p \quad \|\varphi^*\| =\|K^{1,\omega}_{p,p}(\,{\cdot}\,)\mid L_\infty(D)\|= K_p=\widetilde K_p \end{equation} \tag{1.11}$

$\begin{equation} \bigl(\text{for }q<p\quad \|\varphi^*_W\|^\sigma =\|K^{1,\omega}_{q,p}(\,{\cdot}\,)\mid L_\sigma(\varphi^{-1}(W))\|^\sigma= \Psi(W)=\widetilde\Psi(W)\bigr) \end{equation} \tag{1.12}$

for an open set

$W\subset D'$ , where

$\|\varphi^*_W\|$ is the norm of the restriction

$\begin{equation*} \varphi_W\colon {L}^1_p(W;\omega) \cap \mathring{\mathrm{Lip}}_{\mathrm{loc}}(W)\to L^1_q(D); \end{equation*} \notag$

here,

$\mathring{\mathrm{Lip}}_{\mathrm{loc}}(W)$ stands for the space of locally Lipschitz functions vanishing on the boundary of

$W$ , see [34], Theorem 4.

Let us formulate the following corollary of Theorem 1.6.

Corollary 1.7. A homeomorphism $f\colon D\to D'$ is of class $\mathcal{CRQ}_{p, q}(D',\omega;D)$ with $1<q\leqslant p<\infty$ for $n\geqslant 3$ and $1\leqslant q\leqslant p<\infty$ for $n=2$ if and only if $\varphi=f^{-1}$ is also of class $\mathcal{Q}_{p, q}(D',\omega;D)$ .

Therefore, from now on, we use only $\mathcal{Q}_{p, q}(D',\omega;D)$ to refer to both classes $\mathcal{CRQ}_{p, q}(D',\omega;D)$ and $\mathcal{Q}_{p, q}(D',\omega;D)$ .

The differential properties of mappings of the classes $\mathcal{Q}_{p, q}(D',\omega;D)$ are established in [30] and [31], Theorem 2.

Remark 1.8. The homeomorphisms $\varphi\colon D \to D'$ with $f=\varphi^{-1}\in \mathcal{Q}_{p, q}(D',\omega;D)$ in the cases

(1) $q=p=n$ and $\omega\equiv1$ coincide with quasiconformal mappings [18], [42]–[45];

(2) $1<q=p<\infty$ and $\omega\equiv1$ were studied in [28];

(3) $1<q<p<\infty$ and $\omega\equiv1$ were studied in [28], [37]–[39].

Let us extract from Theorem 1.6 and Corollary 1.7 the following two examples of $Q_{p,q}$ -homeomorphisms.

Example 1.9 (see [29], [32]). If a homeomorphism $\varphi\colon D \to D'$ induces a bounded composition operator $\varphi^*\colon {L}^1_p(D';\omega) \cap \operatorname{Lip}_{\mathrm{loc}}(D')\to L^1_q(D)$ , with $1<q\leqslant p<\infty$ for $n\geqslant 3$ and $1\leqslant q\leqslant p<\infty$ for $n=2$ , then the inverse homeomorphism $f=\varphi^{-1}\colon D'\to D$ is of class $\mathcal{Q}_{p, q}(D',\omega;D)$ .

Example 1.10 (see [29], [32]). Consider a homeomorphism $\varphi \colon D \to D'$ of the Sobolev class $W^1_{q, \mathrm{loc}}(D)$ with finite distortion (1.1) and the operator function distortion (1.10) of class $L_{\sigma}(D)$ , where $1/\sigma=1/q-1/p$ for $1\leqslant q<p<\infty$ and $\sigma=\infty$ for $q=p$ .

If $1<q\leqslant p<\infty$ for $n\geqslant 3$ and $1\leqslant q\leqslant p<\infty$ for $n=2$ , then the inverse homeomorphism $f=\varphi^{-1}\colon D'\to D$ is of class $\mathcal{Q}_{p, q}(D',\omega;D)$ .

In addition to Examples 1.9 and 1.10, other classes of mappings in the family $\mathcal Q_{p,q}(D',\omega;D)$ were considered in [31]. Let us present some of them.

Example 1.11 (see [31], Example 3)). Consider a homeomorphism $\varphi\colon D\to D'$ of Sobolev class $W^1_{p,\mathrm{loc}}(D)$ , where $1<p<\infty$ for $n\geqslant3$ and $1\leqslant p<\infty$ for $n=2$ , with finite distortion. The inverse homeomorphism $f=\varphi^{-1}\colon D'\to D$ is of class $\mathcal Q_{p,p}(D',\omega;D)$ with the constant $K_p=1$ and the weight function

$\begin{equation} D'\ni y\mapsto \omega(y)= \begin{cases} \dfrac{|D\varphi(\varphi^{-1}(y))|^p}{|{\det D\varphi (\varphi^{-1}(y))}|} &\text{if } y\in D'\setminus (Z'\cup\Sigma'), \\ 1 &\text{otherwise.} \end{cases} \end{equation} \tag{1.13}$

Remark 1.12. As Theorem 5 in [31] shows, the weight function (1.13) is locally integrable.

Example 1.13 (see [31], Example 4). For $n-1< s<\infty$ , consider a homeomorphism $f \colon D' \to D$ of open domains $D',D\subset \mathbb{R}^n$ , where $n\geqslant 2$ , such that

(1) $f\in W^1_{n-1, \mathrm{loc}}(D')$ ;

(2) the mapping $f$ has finite distortion;

(3) the outer distortion function

$\begin{equation} D'\ni y \mapsto K^{1,1}_{n-1,s}(y,f) = \begin{cases} \dfrac{|Df(y)|}{|{\det Df (y)}|^{1/s}} &\text{if } \det Df (y)\neq 0, \\ 0 &\text{if }\det Df (y) = 0 \end{cases} \end{equation} \tag{1.14}$

lies in

$L_{\sigma}(D)$ , where

$\sigma=(n-1)p$ with

$p=s/(s-(n-1))$ .

Then the inverse homeomorphism $\varphi=f^{-1}\colon D\to D'$ has the properties

(4) $\varphi\in W^1_{p, \mathrm{loc}}(D)$ , $p=s/(s-(n-1))$ ;

(5) $\varphi$ has finite distortion;

while the homeomorphism $f\colon D'\to D$

(6) is of class $\mathcal Q_{p,p}(D',\omega;D)$ with the constant $K_p=1$ and the weight function $\omega\in L_{1,\mathrm{loc}}(D')$ defined as

$\begin{equation} \omega(y)= \begin{cases} \dfrac{|{\operatorname{adj} Df(y)}|^{p}}{|{\det Df (y)}|^{p-1}} &\text{if }y\in D'\setminus Z', \\ 1 &\text{otherwise}, \end{cases} \end{equation} \tag{1.15}$

where

$Z'=\{y\in D'\colon Df(y)=0\}$ .

Say that a mapping $f\in W^1_{1, \mathrm{loc}}(D')$ has finite codistortion if the adjoint matrix $\operatorname{adj} Df(y)$ of the differential equals $0$ a.e. on the zero set of the Jacobian

$\begin{equation*} Z=\{y\in D' \mid \det Df(y)=0\}. \end{equation*} \notag$

Example 1.14 (see [31], Example 5). For $n-1< s<\infty$ , consider a homeomorphism $f \colon D' \to D$ of domains $D',D\subset \mathbb{R}^n$ , with $n\geqslant 2$ , such that

(1) $f\in W^1_{n-1, \mathrm{loc}}(D')$ ;

(2) the mapping $f$ has finite codistortion;

(3) the inner distortion function

$\begin{equation} D'\ni y \mapsto \mathcal K^{1,1}_{n-1,s}(y,f) = \begin{cases} \dfrac{|{\operatorname{adj} Df(y)}|}{|{\det Df (y)}|^{(n-1)/s}} &\text{if }\det Df (y)\neq 0, \\ 0 &\text{if }\det Df (y) = 0 \end{cases} \end{equation} \tag{1.16}$

belongs to

$L_{p}(D')$ , where

$p=s/(s-(n-1))$ and

$n-1<s<\infty$ .

Then the inverse homeomorphism $\varphi=f^{-1}\colon D\to D'$ has the properties

(4) $\varphi\in W^1_{p, \mathrm{loc}}(D)$ and $p=s/(s-(n-1))$ ;

(5) $\varphi$ has finite distortion;

and the homeomorphism $f\colon D'\to D$

(6) is of class $\mathcal Q_{p,p}(D',\omega;D)$ with the constant $K_p=1$ and the weight function (1.15);

(7) has finite distortion for $n-1< s<n+1/(n-2)$ .

Example 1.15 (see [35], Definition 11, Theorem 34). A homeomorphism $f\colon D'\,{\to}\, D$ is called a homeomorphism with inner bounded $\theta$ -weighted $(s,r)$ -distortion, or of class $\mathcal{ID}(D';s,r;\theta,1)$ , where $n-1< s\leqslant r<\infty$ , whenever:

(1) $f\in W^1_{n-1,\mathrm{loc}}(D')$ ;

(2) the mapping $f$ has finite codistortion;

(3) the function of local $\theta$ -weight $(s,r)$ -distortion

$\begin{equation} D' \ni x\mapsto \mathcal K_{s,r}^{\theta,1}(x,f) =\begin{cases} \dfrac{\theta^{(n-1)/s}(x)|{\operatorname{adj}} D f(x)|}{|{\det D f(x)}|^{(n-1)/r}} &\text{if } \det D f(x)\ne0, \\ 0 &\text{otherwise} \end{cases} \end{equation} \tag{1.17}$

belongs to

$L_{\varrho}(\Omega)$ , where

$\varrho$ can be found from the condition

$1/\varrho = (n-1)/s-(n- 1)/r$ , and

$\varrho= \infty$ for

$s=r$ .

Hence, under the condition $n-1< s\leqslant r<\infty$ and the local summability of the function $\omega(x)=\theta^{-(n-1)/(s-(n-1))}(x)$ , the homeomorphism $f\colon D'\to D$ belongs to $\mathcal Q_{p,q}(D',\omega;D)$ , where $q=r/(r-(n-1))$ and $p=s/(s-(n-1))$ , for $1<q\leqslant p<\infty$ . Furthermore, the factors on the right-hand side of (1.8) are equal to $K_p=\|\mathcal K_{r,r}^{\theta,1}(\,{\cdot}\,,f)\mid L_{\infty}(\Omega)\|$ for $q=p$ and

$\begin{equation*} \Psi_{p,q}(Q(x,R)\setminus \overline{Q(x,r)})^{1/\sigma}= \bigl\|\mathcal K_{s,r}^{\theta,1}(\,{\cdot}\,,f)\bigm| L_{\varrho}(Q(x,R)\setminus \overline{Q(x,r)})\bigr\| \quad \text{for} \quad q<p, \end{equation*} \notag$

where

$1/\sigma=1/q-1/p=1/\varrho$ .

Example 1.16 (see [36], Definition 3, Theorem 19). A homeomorphism $f\colon D'\to D$ is of class $\mathcal{OD}(D';s,r;\theta,1)$ , with $n-1< s\leqslant r<\infty$ , and is called a mapping with outer bounded $\theta$ -weighted $(s,r)$ -distortion, whenever:

(1) $f\in W^1_{n-1,\mathrm{loc}}(D')$ ;

(2) the mapping $f$ has finite distortion;

(3) the function of local $\theta$ -weighted $(s,r)$ -distortion

$\begin{equation*} D' \ni x\mapsto K_{s,r}^{\theta,1}(x,f)= \begin{cases} \dfrac{\theta^{1/s}(x)|D f(x)|}{|{\det D f(x)}|^{1/r}} &\text{if }\det D f(x)\ne0, \\ 0 &\text{otherwise} \end{cases} \end{equation*} \notag$

belongs to

$L_{\rho}(D')$ , where

$\rho$ can be found from the conditions

$1/\rho = 1/s-1/r$ and

$\rho = \infty$ for

$s=r$ .

Hence, under the condition $n-1< s\leqslant r<\infty$ and the local summability of $\omega(x)=\theta^{-(n-1)/(s-(n-1))}(x)$ , the homeomorphism $f\colon D'\to D$ belongs to $\mathcal Q_{p,q}(D',\omega;D)$ , where $q=r/(r-(n-1))$ and $p=s/(s-(n-1))$ with $1<q\leqslant p<\infty$ . The factors on the right-hand side of (1.8) are equal to $K_p=\|K_{r,r}^{\theta,1}(\,{\cdot}\,,f)\mid L_{\infty}(D')\|^{n-1}$ for $q=p$ and

$\begin{equation*} \Psi_{p,q}(Q(x,R)\setminus \overline{Q(x,r)})^{1/\sigma}= \bigl\|K_{s,r}^{\theta,1}(\,{\cdot}\,,f)\bigm| L_{\rho}(Q(x,R)\setminus \overline{Q(x,r)})\bigr\|^{n-1} \end{equation*} \notag$

for

$q<p$ , where

$1/\sigma=1/q-1/p=(n-1)/\varrho$ .

It is shown in [36], Theorem 8, that the inclusion

$\begin{equation*} \mathcal{OD}(D';s,r;\theta,1) \subset \mathcal{ID}(D';s,r;\theta,1) \end{equation*} \notag$

holds under the condition

$n-1< s\leqslant r<\infty$ . Moreover, for every homeomorphism

$f\colon D'\to D$ of class

$\mathcal{OD}(D';s,r;\theta,1)$ , with

$n-1< s\leqslant r<\infty$ , we have

$\begin{equation*} \|\mathcal K_{s,r}^{\theta,1}(\,{\cdot}\,,f)\mid L_{\sigma}(D')\|\leqslant \| K_{s,r}^{\theta,1}(\,{\cdot}\,,f)\mid L_\rho(D')\|^{n-1}, \end{equation*} \notag$

where the numbers

$\rho$ and

$\sigma$ are defined in Examples 1.15 and 1.16.

More examples of $\mathcal{OD}(D'; s, r; \theta, 1)$ -homeomorphisms in $\mathbb R^2$ can be found in [54].

§ 2. Behaviour of mappings with respect to the capacity metric

We fix two domains $D, D'\subset \mathbb R^n$ , a locally integrable weight function $\omega\colon D'\to\mathbb R$ on $D'$ , and a mapping $f\in\mathcal Q_{p,q}(D',\omega;D)$ with $n-1<q\leqslant p<\infty$ .

Recall that Corollary 1.7 guarantees that $f$ satisfies (1.9) for every condenser $\mathcal E=(F_1,F_0)$ in $D'$ .

We also fix some continuum $F_0\subset D'$ with non-empty interior such that the open set $D'\setminus F_0$ is connected.

2.1. Capacity metric functions in domains for the homeomorphisms of class $\mathcal Q_{p,q}(D',\omega;D)$ for $n-1<q\leqslant p\leqslant n$

Observe that in the case $n-1<q\leqslant n$ the left-hand side of (1.9) is non-zero as long as the continuum $f(F_1)$ is distinct from a point. Indeed, we have the following proposition.

Lemma 2.1. In a domain $D\subset \mathbb R^n$ , let $B_0\Subset D$ and $B_1\Subset D$ be two balls satisfying $\overline{B_0}\cap\overline{B_1}=\varnothing$ . Then, for $n-1<q\leqslant n$ , a fixed continuum $T_0\subset B_0$ , and an arbitrary continuum $T_1\subset \overline{B_1}$ , the relation

$\begin{equation} \operatorname{cap}^{1/q}\bigl((T_1,T_0); L^1_q(D)\bigr)\to 0 \end{equation} \tag{2.1}$

holds³^[x]³In other words, the left-hand side of (1.9) is small if and only if

$\operatorname{diam} T_1$ is small (under the condition that the continuum

$T_1$ lies in some ball

$B_1\Subset D$ with

$\overline{B_0}\cap\overline{B_1}=\varnothing$ ). if and only if

$\operatorname{diam} T_1\to 0$ .

Proof. Let us present the scheme of the proof of Lemma 2.1.

Necessity. By [48], Lemma 3, there is a John domain $\Omega\in J(\alpha,\beta)$ (see [48], Definition 8) compactly embedded into $D$ , with some positive parameters $\alpha$ and $\beta$ depending on $D$ and the balls $B_0$ and $B_1$ , which includes the closures of both balls. On the domain $\Omega$ under the conditions $1\leqslant q < n$ and $q\leqslant q^*\leqslant nq/(n-q)$ we have the following Poincaré inequality [55], Theorems 4 and 9:

$\begin{equation} \|u - c_u\mid L_{q^*}(\Omega)\| \leqslant C_\Omega\biggl(\frac{\alpha}{\beta}\biggr)^{n}(\operatorname{diam}\Omega)^{1-n/q+n/q^*} \|\nabla u \mid L_q(\Omega)\|, \end{equation} \tag{2.2}$

where

$c_u$ and

$C_\Omega$ are constants, with

$C_\Omega>0$ independent of

$u$ ,

$\alpha$ , and

$\beta$ . By (2.1) there exists a sequence of continua

$T_{1,k}\subset B_1$ and admissible functions

$u_k\in C(\Omega)\cap L^1_q(\Omega)$ for the capacity

$\operatorname{cap}((T_{1,k}, T_0); L^1_q(\Omega))$ such that

$\begin{equation} u_k\vert_{T_{1,k}}= 1,\quad u_k\vert_{T_0}= 0,\quad 0\leqslant u_k\leqslant 1\quad \text{and} \quad \|\nabla u_k \mid L_q(\Omega)\|\to 0\quad \text{as}\quad k\to\infty. \end{equation} \tag{2.3}$

Now inequality (2.2) implies that

$\|u_k - c_{u_k}\mid L_{q^*}(\Omega)\|\to 0$ as

$k\to\infty$ . Note that the sequence of numbers

$\{c_{u_k}\}$ is bounded. Indeed, if

$\{c_{u_k}\}$ is not bounded, then, since

$0\leqslant u_k\leqslant 1$ , the left-hand side of (2.2) is also not bounded, which contradicts the right convergence in (2.3). Therefore, we may assume that

$c_{u_k}$ converges to some number

$c_0$ , and up to subsequence

$u_{k} - c_{u_{k}}\to 0$ for a.e.

$x\in \Omega$ as

$k\to\infty$ . Hence,

$u_{k} \to c_0$ for a.e.

$x\in \Omega$ as

$k\to\infty$ , and due to

$u_{k}|_{B_0}\equiv0$ we deduce

$c_0=0$ . In addition,

$\Omega$ is a bounded domain, and the Lebesgue dominated convergence theorem shows that

$\begin{equation} \|u_l \mid L_{q}(\Omega)\|\to 0\quad\text{as}\quad l\to\infty. \end{equation} \tag{2.4}$

From (2.3) and (2.4) we infer that

$\|u_l \mid W^1_{q}(\Omega)\|\to 0$ as

$l\to\infty$ . We can extend the restrictions

$u_l|_{B_1}$ to the functions

$\widetilde u_l\in W^1_{q}(\mathbb R^n)$ so that the extension operator is bounded. Therefore,

$\begin{equation*} \|\widetilde u_l \mid W^1_{q}(\mathbb R^n)\|\to 0\quad \text{as}\quad l\to\infty. \end{equation*} \notag$

We obtain then that the capacity of the continua

$T_{1,l}$ in the space

$W^1_{q}(\mathbb R^n)$ of Bessel potentials is positive and tends to

$0$ as

$l\to\infty$ . For

$n-1< q < n$ the latter is possible only if

$\operatorname{diam} T_{1,l}\to 0$ as

$l\to\infty$ ; see the details in [56], [41].

The case $q =n$ reduces to the previous one using Hölder’s inequality.

Sufficiency. By Property 1.2 we have

$\begin{equation*} \operatorname{cap}\bigl((T_1,T_0); L^1_q(D)\bigr) \leqslant \operatorname{cap}\bigl((T_1,\overline{B_0}); L^1_q(D)\bigr), \end{equation*} \notag$

and so, it suffices to prove that

$\operatorname{cap}((T_1,\overline{B_0}); L^1_q(D))\to0$ as

$\operatorname{diam} T_1\to 0$ .

We put $R=\operatorname{dist}(B_0,B_1)$ and suppose that the continuum $T_1$ satisfies $r_{T_1}<R$ . Then we may assume that every admissible function for the condenser $(\overline{B(x,r_{T_1})}, B(x,R))$ is also admissible for the condenser $(T_1,T_0)$ , and so

$\begin{equation*} \operatorname{cap}\bigl((T_1,T_0); L^1_q(D)\bigr)\leqslant \operatorname{cap}\bigl(\bigl(\overline{B(x,r_{T_1})},B(x,R)\bigr); L^1_q(B(x,R))\bigr). \end{equation*} \notag$

From Example 2.7 below for $\alpha=0$ , we conclude

$\begin{equation*} \begin{aligned} \, &\operatorname{cap}\bigl(\bigl(\overline{B(0,r)},B(0,R)\bigr);L^1_q(B(0,R))\bigr) \\ &\qquad= \begin{cases} \sigma_{n-1}\biggl(\dfrac{n-q}{n-1}\biggr)^{q-1} (r^{(q-n)/(q-1)}-R^{(q-n)/(q-1)})^{1-q} &\text{for }q<n, \\ \sigma_{n-1}\biggl(\ln\dfrac{R}r\biggr)^{1-n} &\text{for }q=n, \end{cases} \end{aligned} \end{equation*} \notag$

where

$r\in(0,R)$ , while

$\sigma_{n-1}$ is the measure of the unit

$(n-1)$ -dimensional sphere in the space

$\mathbb R^n$ . Thus,

$\begin{equation*} \operatorname{cap}\bigl(\bigl(\overline{B(x,r_{T_1})},B(x,R)\bigr); L^1_q(B(x,R)\bigr) \to0 \quad \text{as}\quad r_{T_1}\to 0, \end{equation*} \notag$

and the proof of Lemma 2.1 is complete.

Corollary 2.2. For $n-1\,{<}\,q\,{\leqslant}\, n$ , the existence of a mapping $f\,{\in}\,\mathcal Q_{p,q}(D',\omega;D)$ is ensured by the condition

$\begin{equation} \operatorname{cap}^{1/p}\bigl(\mathcal E; L^1_p(D';\omega)\bigr)\ne0 \end{equation} \tag{2.5}$

for an arbitrary condenser

$\mathcal E=(\gamma,F_0)$ , where

$\gamma\colon[a,b]\to D'\setminus F_0$ is an arbitrary closed curve with distinct endpoints

$x=\gamma(a)$ and

$y=\gamma(b)$ .

Proof. Since the continuum

$F_0\subset D'$ has non-empty interior, there exists a closed ball

$\overline{B_0'}\subset F_0$ and a closed ball

$\overline{B_1'}\subset D'$ centered on

$\gamma$ such that

$\overline{B_1'}\cap \overline{B_0'}=\varnothing$ . Consider the condenser

$\mathcal E=(\gamma\cap\overline{B_1'}, \overline{B_0'})$ . By (1.8), it suffices to show that

$\begin{equation} \operatorname{cap}\bigl(f(\mathcal E); L^1_q(D)\bigr)\ne0. \end{equation} \tag{2.6}$

The latter follows from Lemma 2.1. Indeed, there are closed disjoint balls

$\overline{B_0''}\subset f(\overline{B_0'})$ and

$\overline{B_1''}\subset f(\overline{B_1'})$ whose intersection

$\gamma\cap\overline{B_1''}$ is a nondegenerate continuum. Then, Lemma 2.1 and (1.8) yield

$\begin{equation*} 0 \ne \operatorname{cap}\bigl(\bigl(\gamma\cap\overline{B_1'}, \overline{B_0'}\bigr);L^1_q(D)\bigr)\leqslant \operatorname{cap}\bigl(f(\mathcal E); L^1_q(D)\bigr). \end{equation*} \notag$

This justifies Corollary 2.2.

With (2.5) we can define a metric function similar to the one introduced in [25], [26], Chap. 5, in the unweighted case.

Definition 2.3. The capacity $(\omega,p)$ -metric function between two distinct points $x,y \in D' \setminus F_0$ with respect to $F_0$ is defined as

$\begin{equation} \rho^{\omega}_{p,F_0}(x,y) = \inf_{\overline{xy}} \operatorname{cap}^{1/p}\bigl((\overline{xy}, F_0); L^1_p(D';\omega)\bigr), \end{equation} \tag{2.7}$

where the infimum is over all curves

$\overline{xy}$ in

$D'\setminus F_0$ with endpoints

$x,y\in D'\setminus F_0$ .

By analogy, we define the capacity $q$ -metric function $\rho_{q,f(F_0)}(a,b)$ between two points $a,b \in D \setminus f(F_0)$ with respect to the continuum $f(F_0)$ in the image $D'$ :

$\begin{equation} \rho_{q,f(F_0)}(a,b) = \inf_{\overline{ab}} \operatorname{cap}^{1/q}\bigl((\overline{ab}, f(F_0)); L^1_q(D)\bigr). \end{equation} \tag{2.8}$

Proposition 2.4. If a homeomorphism $f\colon D' \to D$ belongs to $\mathcal{Q}_{p, q}(D',\omega;D)$ , where $n-1<q\leqslant p\leqslant n$ for $n\geqslant 3$ and $1\leqslant q\leqslant p\leqslant2$ for $n=2$ , then the capacity metric functions satisfy

$\begin{equation} \begin{cases} \rho_{p,f(F_0)}\bigl(f(x),f(y)\bigr) \leqslant K_p\rho^{\omega}_{p,F_0}(x,y) &\textit{if } q=p, \\ \rho_{q,f(F_0)}\bigl(f(x),f(y)\bigr) \leqslant \Psi_{p,q}(D'\setminus F_0)^{1/\sigma} \rho^{\omega}_{p,F_0}(x,y) &\textit{if } q<p, \end{cases} \end{equation} \tag{2.9}$

for all points

$x,y\in D'\setminus F_0$ , where

$1/\sigma=1/q-1/p$ .

Proof. Take

$\mathcal E=(\overline{xy},F_0)$ in

$D'$ , then from (1.9) it follows that

$\begin{equation*} \begin{aligned} \, \rho_{q,f(F_0)}(f(x),f(y))& \leqslant\operatorname{cap}^{1/q}\bigl((f(\overline{xy}),f(F_0)); L^1_q(D)\bigr) \\ &\leqslant \Psi_{p,q}(D'\setminus F_0)^{1/\sigma} \operatorname{cap}^{1/p}\bigl((\overline{xy},F_0); L^1_p(D';\omega)\bigr) \end{aligned} \end{equation*} \notag$

provided that

$q<p$ . Passing to the infimum over all curves

$\overline{xy}\subset D'\setminus F_0$ with endpoints

$x$ and

$y$ , we arrive at the second inequality in (2.9).

The case $q=p$ is similar.

The proposition is proved.

Proposition 2.5. In the case $n-1<q\leqslant p\leqslant n$ , for $n\geqslant 3$ and $1\leqslant q\leqslant p\leqslant2$ for $n=2$ , the capacity $(\omega,p)$ -metric function $\rho^{\omega}_{p,F_0}(x,y)$ enjoys the properties

(1) $\rho^{\omega}_{p,F_0}(x,y)=\rho^{\omega}_{p,F_0}(y,x)$ for all points $x,y\in D' \setminus F_0$ ;

(2) $\rho^{\omega}_{p,F_0}(x,z)\leqslant \rho^{\omega}_{p,F_0}(x,y)+\rho^{\omega}_{p,F_0}(y,z)$ for all points $x,y,z\in D' \setminus F_0$ .

Proof. Property (1) is obvious.

To verify the second property, consider the case $x\ne z$ , $x\ne y$ , $y\ne z$ ; otherwise property (2) obviously holds. We fix $\varepsilon>0$ and some curves $\overline{xy}$ and $\overline{yz}$ with endpoints $x$ , $y$ and $y$ , $z$ , respectively, such that

$\begin{equation} \operatorname{cap}^{1/p}\bigl((\overline{xy}, F_0); L^1_p(D';\omega)\bigr) < \rho^{\omega}_{p,F_0}(x,y)+\frac{\varepsilon}{4}, \end{equation} \tag{2.10}$

$\begin{equation} \operatorname{cap}^{1/p}\bigl((\overline{yz}, F_0); L^1_p(D';\omega)\bigr) < \rho^{\omega}_{p,F_0}(y,z)+\frac{\varepsilon}{4}. \end{equation} \tag{2.11}$

Take two functions

$u_1$ and

$u_2$ admissible for the capacities

$\operatorname{cap}((\overline{xy}, F_0); L^1_p(D';\omega))$ and

$\operatorname{cap}((\overline{yz}, F_0); L^1_p(D';\omega))$ such that

$\begin{equation} \biggl(\int_{D'}|\nabla u_1|^p(y)\omega(y)\,dy\biggr)^{1/p} < \operatorname{cap}^{1/p}\bigl((\overline{xy}, F_0); L^1_p(D';\omega)\bigr)+\frac{\varepsilon}{4}, \end{equation} \tag{2.12}$

$\begin{equation} \biggl(\int_{D'}|\nabla u_2|^p(y)\omega(y)\,dy\biggr)^{1/p} < \operatorname{cap}^{1/p}\bigl((\overline{yz}, F_0); L^1_p(D';\omega)\bigr)+\frac{\varepsilon}{4}. \end{equation} \tag{2.13}$

It is easy to see that

$u_1+u_2$ is admissible for the capacity

$\operatorname{cap}^{1/p}((\overline{xy}\cup\overline{yz}, F_0); L^1_p(D';\omega))$ . Hence, from (2.10)–(2.13), we obtain

$\begin{equation*} \begin{aligned} \, \rho^{\omega}_{p,F_0}(x,z) &\leqslant \operatorname{cap}^{1/p}\bigl((\overline{xy}\cup\overline{yz}, F_0); L^1_p(D';\omega)\bigr) \leqslant \biggl(\int_{D'}|\nabla (u_1+u_2)|^p(y)\omega(y)\,dy\biggr)^{1/p} \\ &\leqslant \biggl(\int_{D'}|\nabla u_1|^p(y)\omega(y)\,dy\biggr)^{1/p}+ \biggl(\int_{D'}|\nabla u_2|^p(y)\omega(y)\,dy\biggr)^{1/p} \\ &<\rho^{\omega}_{p,F_0}(x,y)+\rho^{\omega}_{p,F_0}(y,z)+\varepsilon. \end{aligned} \end{equation*} \notag$

Since

$\varepsilon>0$ is arbitrarily, the triangle inequality is verified, proving the proposition.

Recall that the metric function $\rho^{\omega}_{p,F_0}$ is defined in (2.7) for distinct points $x\ne y$ of the open set $D' \setminus F_0$ . If $x=y\in D' \setminus F_0$ , we put

$\begin{equation} \rho^{\omega}_{p,F_0}(x,x)=\operatorname{cap}^{1/p}\bigl((\{x\}, F_0); L^1_p(D';\omega)\bigr). \end{equation} \tag{2.14}$

For the capacity metric function

$\rho^{\omega}_{p,F_0}$ to be a metric, we must ensure that

$\begin{equation} \rho^{\omega}_{p,F_0}(x,x)=0 \end{equation} \tag{2.15}$

for every point

$x\in D'\setminus F_0$ .

Proposition 2.6. Given $x\in D'\setminus F_0$ , condition (2.15) holds if and only if

$\begin{equation} \lim_{r\to0} \operatorname{cap}\bigl(\bigl(\overline{B(x,r)}, F_0\bigr); L^1_p(D';\omega)\bigr)=0. \end{equation} \tag{2.16}$

Proof. Since the condenser

$(\{x\}, F_0)$ is a part of the condenser

$(\overline{B(x,r)}, F_0)$ , Property 1.2 yields

$\begin{equation*} \rho^{\omega}_{p,F_0}(x,x)\leqslant \lim_{r\to0}\operatorname{cap}^{1/p} \bigl(\bigl(\overline{B(x,r)}, F_0\bigr); L^1_p(D';\omega)\bigr). \end{equation*} \notag$

Granted (2.16), this implies (2.15).

Suppose now that (2.15) holds: $\rho^{\omega}_{p,F_0}(x,x)=\operatorname{cap}^{1/p}((\{x\}, F_0); L^1_p(D';\omega))=0$ . By the definition of capacity, for every $\varepsilon\in(0,1/2)$ , there exists a function $u_\varepsilon\in \operatorname{Lip}_{\mathrm{loc}}(D')$ such that $u_\varepsilon(y)\in [0,1]$ for all $y\in D'$ , while $u_\varepsilon\vert_{F_0}=0$ , $u_\varepsilon(x)=1$ , and

$\begin{equation} \int_{D'}|\nabla u_\varepsilon |^p(y)\omega(y)\,dy<\varepsilon. \end{equation} \tag{2.17}$

Since

$x$ is an interior point of

$\{y\in D'\colon u_\varepsilon(y)>1-\varepsilon\}$ , we have

$B(x,r_0)\subset \{y\in D'\colon u_\varepsilon(y)>1-\varepsilon\}$ for some ball

$B(x,r_0)$ . Consequently, the function

$\begin{equation*} \frac{\min(u_\varepsilon(y),1-\varepsilon)}{1-\varepsilon} \end{equation*} \notag$

is admissible for the capacity of the condenser

$(\overline{B(x,r)}, F_0)$ provided that

$r\in (0,r_0)$ . Therefore,

$\begin{equation*} \begin{aligned} \, \operatorname{cap}\bigl(\bigl(\overline{B(x,r)}, F_0\bigr); L^1_p(D';\omega)\bigr) &\leqslant \frac1{(1-\varepsilon)^p}\int_{D'} \bigl|\nabla\bigl(\min(u_\varepsilon(y),1-\varepsilon)\bigr)\bigr|^p\omega(y)\,dy \\ &\leqslant \frac1{(1-\varepsilon)^p}\int_{D'} |\nabla u_\varepsilon|^p(y)\omega(y)\,dy\leqslant \frac{\varepsilon}{(1-\varepsilon)^p}< 2^p\varepsilon \end{aligned} \end{equation*} \notag$

by (2.17). Since

$\varepsilon\in(0,1/2)$ is arbitrary, (2.16) is justified.

This completes the proof of Proposition 2.6.

Observe that (2.16) always holds in the case $q \leqslant p \leqslant n$ and $\omega\equiv1$ . In the case of a non-trivial weight function condition (2.15) need not hold, see Examples 2.7 and 2.8.

Example 2.7 (see [50], Example 2.22). Consider the domain $D' = B(0,2)$ with the weight $\omega(x) = |x|^{\alpha}$ , where $\alpha> -n$ , and $p>1$ . The capacity of the condenser $\mathcal E=(\overline{B(0,r)}, B(0,1))$ with $0<r<1$ in the space $L_p(D';\omega)$ , where the weight function $\omega$ belongs to the special class of weight functions called admissible in [50], is

$\begin{equation*} \begin{aligned} \, &\operatorname{cap}\bigl(\bigl(\overline{B(0,r)},B(0,1)\bigr);L^1_p(D';\omega)\bigr) \\ &\qquad= \begin{cases} c(n,p,\alpha) |1-r^{(p-n-\alpha)/(p-1)}|^{1-p} &\text{for }p-n-\alpha\ne0, \\ \sigma_{n-1}\biggl(\ln\dfrac1r\biggr)^{1-p} &\text{for }p-n-\alpha=0, \end{cases} \end{aligned} \end{equation*} \notag$

where

$\sigma_{n-1}$ is the measure of the unit

$(n-1)$ -dimensional sphere in

$\mathbb R^n$ , while

$c(n,p,\alpha)$ is a constant depending only on

$n$ ,

$p$ , and

$\alpha$ . Since

$\begin{equation*} \operatorname{cap}\bigl(\bigl(\overline{B(0,r)},B(0,1)\bigr);L^1_p(D';\omega)\bigr) \to \operatorname{cap}\bigl(\bigl(\{0\},B(0,1)\bigr);L^1_p(D';\omega)\bigr)\quad \text{as}\quad r\to 0, \end{equation*} \notag$

the definition of the capacity metric function yields

$\rho^{\omega}_{p,S(0,1)}(0,0)\neq 0$ if

$p-n-\alpha>0$ .

In the following example, we will construct a weight function for which condition (2.15) is violated on a countable dense subset of $D'$ .

Example 2.8. Consider an arbitrary bounded domain $D' \subset\mathbb R^n$ , a continuum $F_0$ , and a number $\alpha$ satisfying $p-n-\alpha>0$ . With each point $x_i$ of some countable dense subset of $D'$ , we associate the function

$\begin{equation*} D'\ni x\mapsto \omega_i(x)= \begin{cases} \omega(x-x_i) &\text{if }x\in B(x_i,2)\cap D', \\ 2^{\alpha} &\text{if } x\in D'\setminus B(x_i,2), \end{cases} \end{equation*} \notag$

where

$\omega$ is the weight function of Example 2.7. As the weight function on the domain

$D'$ , we consider

$\begin{equation*} D'\ni x\mapsto \sigma(x)= \sum_{i=1}^\infty \frac{1}{2^i}\omega_i(x). \end{equation*} \notag$

It is not difficult to check that the function

$\sigma$ is integrable on

$D'$ . We fix an index

$j\in\mathbb N$ and a function

$u\in\operatorname{Lip}_{\mathrm{loc}}(D')\cap L^1_{p}(D';\sigma)$ admissible for the capacity

$\operatorname{cap}\bigl((\{x_j\}, F_0); L^1_p(D';\omega)\bigr)$ . In view of the inequality

$\begin{equation*} \frac{1}{2^{ip}}\int_{D'}|\nabla u(x)|^p\omega_i(x)\,dx\leqslant \int_{D'}|\nabla u(x)|^p\sigma(x)\,dx, \end{equation*} \notag$

which holds for every admissible function

$u$ mentioned above, the left-hand side of the last inequality is separated from zero by some constant independent of

$u$ . Therefore,

$\begin{equation*} \rho^{\sigma}_{p,F_0}(x_j,x_j)=\operatorname{cap}^{1/p}\bigl((\{x_j\}, F_0); L^1_p(D';\sigma)\bigr) \ne0 \end{equation*} \notag$

for every index

$j\in\mathbb N$ .

Example 2.9. Consider a bounded domain $D' \subset\mathbb R^n$ , a point $x\in D'$ , a continuum $F_0 \subset D' \setminus B(x, e^{-1})$ , and a weight $\omega\colon D' \to [1,\infty)$ with $\omega \in \mathrm{BMO}(D')$ . For $0<r<e^{-2}$ , define the function

$\begin{equation*} u_r(y) = \begin{cases} 0 &\text{ if } y\in D' \setminus B(x, e^{-1}), \\ \dfrac{\log(\log (1/|y|))}{\log(\log(1/r))} &\text{ if } y\in D'\cap (B(x, e^{-1})\setminus B(x, r)), \\ 1 &\text{ if } y\in D' \cap B(x, r). \end{cases} \end{equation*} \notag$

It is not difficult to verify that

$u_r$ belongs to the class of admissible functions

$\mathcal{A}(B(x,r)\cap D', F_0)$ . Now by the definition of the capacity

$\begin{equation*} \begin{aligned} \, \rho^{\omega}_{n,F_0}(x,x) &= \operatorname{cap}\bigl((\{x\}, F_0); L^1_{n}(D'; \omega)\bigr) = \lim_{r\to 0}\operatorname{cap}\bigl( \bigl(B(x,r)\cap D', F_0\bigr); L^1_{n}(D';\omega)\bigr) \\ &\leqslant \lim_{r\to 0} \int_{D'} |\nabla u_r(y)|^n \omega(y) \,dy = 0. \end{aligned} \end{equation*} \notag$

The last equality holds thanks to the following estimate for

$\omega \in \mathrm{BMO}(B(x,1))$ (see Lemma 5.2 in [21]):

$\begin{equation*} \begin{aligned} \, \int_{D'} |\nabla u_r(y)|^n \omega(y) \,dy &\leqslant \frac{1}{\log(\log(1/r))} \int_{B(x, e^{-1}) \setminus B(x, r)} \frac{\omega(y) \,dy}{|y|^n (\log(1/|y|))^n} \\ &\leqslant \frac{C}{\log(\log(1/r))}, \end{aligned} \end{equation*} \notag$

where the constant

$C$ depends only on

$n$ and

$\omega$ , but is independent of

$r$ .

Examples 2.7–2.9 show that condition (2.15) depends on the properties of the weight function $\omega$ .

We let $d(x,y)$ denote the Euclidean distance between two points $x,y\,{\in}\,\mathbb R^n$ .

Proposition 2.10. Consider a homeomorphism $f\colon D' \to D$ belonging to the class $\mathcal{Q}_{p, q}(D',\omega;D)$ , where $n-1<q\leqslant p\leqslant n$ for $n\geqslant 3$ and $1\leqslant q\leqslant p\leqslant2$ for $n=2$ .

(1) If $y\in D'\setminus F_0$ and $\rho^{\omega}_{p,F_0}(z_m,y)\to 0$ as $m\to\infty$ in the domain $D'\setminus F_0$ , then

- (a) one of conditions (2.15) and (2.16) is met at the point $y$ ;
- (b) $d(z_m,y)\to0$ as $m\to\infty$ .

(2) Provided with (2.15) at $y\in D'\setminus F_0$ , the convergence $d(z_m,y)\to0$ as $m\to\infty$ implies the convergence $\rho^{\omega}_{p,F_0}(z_m,y)\to 0$ with respect to the capacity $(\omega,p)$ -metric function $\rho^{\omega}_{p,F_0}$ in the domain $D'\setminus F_0$ .

Proof. (1) By Definition 2.3, for each

$m\in \mathbb N$ , there exists a continuous curve

$\gamma_m\colon [0,1]\to D'\setminus F_0$ with endpoints

$z_m=\gamma_m(0)$ ,

$y=\gamma_m(1)\in D'\setminus F_0$ such that

$\begin{equation} \operatorname{cap}^{1/p}\bigl((\overline{\gamma_m}, F_0); L^1_p(D';\omega)\bigr) \leqslant 2\rho^{\omega}_{p,F_0}(z_m,y), \end{equation} \tag{2.18}$

where

$\overline{\gamma_m}=\gamma_m([0,1])$ stands for the image of the curve

$\gamma_m\colon [0,1]\to D'\setminus F_0$ . Using the inequality

$\begin{equation*} \operatorname{cap}^{1/p}\bigl((\{y\}, F_0); L^1_p(D';\omega)\bigr) \leqslant \operatorname{cap}^{1/p}\bigl((\overline{z_my}, F_0); L^1_p(D';\omega)\bigr), \end{equation*} \notag$

valid for all

$m\in \mathbb N$ , from (2.18) and the condition

$\rho^{\omega}_{p,F_0}(z_m,y)\to 0$ as

$m\to\infty$ in the domain

$D'\setminus F_0$ , we infer that

$\begin{equation*} \operatorname{cap}^{1/p}\bigl((\{y\}, F_0); L^1_p(D';\omega)\bigr)=0. \end{equation*} \notag$

Furthermore, from (2.9) and the condition

$\rho^{\omega}_{p,F_0}(z_m,y)\to 0$ as

$m\to\infty$ we find that

$\rho_{q,f(F_0)}(f(z_m),f(y))\to 0$ as

$m\to\infty$ . By Lemma 2.1, the latter is possible if and only if

$f(z_m)\to f(y)$ as

$m\to\infty$ . Hence

$z_m\to y$ as

$m\to\infty$ .

(2) Assume that condition (2.15) holds at $y\in D'\setminus F_0$ and $d(z_m,y)\to0$ as $m\to\infty$ for some sequence $z_m\in D'\setminus F_0$ . On assuming condition (2.15), Proposition 2.6 implies that

$\begin{equation} \lim_{r\to0} \operatorname{cap}^{1/p}\bigl(\bigl(\overline{B(y,r)}, F_0\bigr); L^1_p(D';\omega)\bigr)=0. \end{equation} \tag{2.19}$

For

$z_m\in B(y,r)$ , from the properties of capacity, we infer that

$\begin{equation*} \rho^{\omega}_{p,F_0}(z_m,y)\leqslant \operatorname{cap}^{1/p}\bigl(\bigl(\overline{B(y,r)}, F_0\bigr); L^1_p(D';\omega)\bigr), \end{equation*} \notag$

and hence

$\rho^{\omega}_{p,F_0}(z_m,y)\to 0$ as

$m\to \infty$ .

This proves Proposition 2.10.

Given a set $B\subset\mathbb R^n$ , we let $\operatorname{dist}(y,B):=\inf_{z\in B}d(y,z)$ denote the distance from a point $y\in\mathbb R^n$ to $B$ , where $d(\,{\cdot}\,,{\cdot}\,)$ is the Euclidean distance. The following proposition generalizes Proposition 2.10.

Proposition 2.11. Consider a homeomorphism $f\colon D' \to D$ belonging to the class $\mathcal{Q}_{p, q}(D',\omega;D)$ , where $n-1<q\leqslant p\leqslant n$ for $n\geqslant 3$ and $1\leqslant q\leqslant p\leqslant2$ for $n=2$ . If $\{y_l\in D'\setminus F_0\}$ , for $l\in \mathbb N$ , is a fundamental sequence with respect to the metric function $\rho^{\omega}_{p,F_0}$ , while $y$ is one of its partial limits in the topology of the extended space $\overline{\mathbb R^n}$ , then the following claims hold:

(1) if $y\in D'\setminus F_0$ , then $d(y_l,y)\to 0$ as $l\to \infty$ ;

(2) if $y\in F_0$ , then $d(y_l,y)\to 0$ as $l\to \infty$ ;

(3) if $y\in \partial D'$ and $\{y_{l}\in D'\}$ is bounded, then $\operatorname{dist}(y_l, \partial D')\to 0$ as $l\to \infty$ ;

(4) if $\{y\}=\overline{\mathbb R^n}\setminus \mathbb R^n$ , either $y_l\to y$ as $l\to \infty$ in the topology of $\overline{\mathbb R^n}$ , or $\varliminf_{\,l\to \infty}d(y_l,0)\,{<}\,\infty$ and $\lim_{k\to \infty}\operatorname{dist}(y_{l_k}, \partial D')=0$ for every subsequence $\{y_{l_k}\in D'\}$ bounded in $\mathbb R^n$ .

Proof. Let us prove the claims of Proposition 2.11 one by one.

(1) Take a fundamental sequence $\{y_l\in D'\setminus F_0\}$ , for $l\in \mathbb N$ , with respect to the metric function $\rho^{\omega}_{p,F_0}$ and its subsequence $\{y_{l_k}\in D'\setminus F_0\}$ , for $k\in \mathbb N$ , converging in the topology of the Euclidean space $\mathbb R^n$ to some point $y\in D'\setminus F_0$ as $k\to \infty$ . By (2.9), the sequence $\{f(y_l)\in D\setminus f(F_0)\}$ , $l\in \mathbb N$ , is also fundamental with respect to $\rho_{q,f(F_0)}$ . In addition, since $f$ is continuous at $y\in D'\setminus F_0$ , we have the convergence $f(y_{l_k})\to f(y)$ as $k\to \infty$ . Lemma 2.1 implies the convergence $\rho_{q,f(F_0)}(f(y_{l_k}),f(y))\to 0$ as $k\to \infty$ . Since the sequence $\{f(y_l)\in D\setminus f(F_0)\}$ , for $l\in \mathbb N$ , is fundamental with respect to the metric function $\rho_{q,f(F_0)}$ , we see that $\rho_{q,f(F_0)}(f(y_{l}),f(y))\to 0$ as $l\to \infty$ . Moreover, $f(y_{l})\to f(y)$ as $l\to \infty$ , again by Lemma 2.1. Since $f^{-1}$ is continuous at $f(y)$ , we infer that $y_{l}\to y$ as $l\to \infty$ .

(2) Take a fundamental sequence $\{y_l\in D'\setminus F_0\}$ , $l\in \mathbb N$ , with respect to the metric function $\rho^{\omega}_{p,F_0}$ and its subsequence $\{y_{l_k}\in D'\setminus F_0\}$ , $k\in \mathbb N$ , converging in the topology of the Euclidean space $\mathbb R^n$ to some point $y\in F_0$ as $k\to\infty$ . The second claim will be justified once we verify that the stated properties contradict the existence of a subsequence $\{y_{l_j}\}$ , $j\in \mathbb N$ , such that $d(y_{l_j},y)\geqslant 1/\beta$ for all $j\in \mathbb N$ , where $\beta>1$ is some number. Indeed, if such a subsequence exists, then

$\begin{equation} d(f(y_{l_j}),f(y))\geqslant \frac{1}{\beta'} \end{equation} \tag{2.20}$

for all

$j\in \mathbb N$ , where

$\beta'>1$ is some number, whose existence is ensured by the locally uniform continuity of the homeomorphism

$f$ . On the other hand, the sequence

$\{f(y_l)\in D\setminus f(F_0)\}$ , for

$l\in \mathbb N$ , is fundamental with respect to the metric function

$\rho_{q,f(F_0)}$ . Applying the subordination principle, see Property 1.2, we infer that this sequence is also fundamental with respect to the metric function

$\rho_{q,K}$ for an arbitrary compact set

$K\subset \operatorname{int}f(F_0)$ with non-empty interior. By Lemma 2.1, the sequence

$f(y_{l})$ converges to

$f(y)\notin K$ as

$l\to \infty$ . The latter contradicts (2.20).

(3) Take a partial limit $y=\lim_{j\to \infty} y_{l_j}\in \partial D'$ and assume on the contrary that there exists a subsequence $\{y_{l_k}\}$ , for $k\in \mathbb N$ , such that $\operatorname{dist}(y_{l_k}, \partial D')\geqslant \beta_0>0$ for all $k\in \mathbb N$ , where $\beta_0$ is some number. By the latter property, since $\{y_{l}\}$ is bounded, we may assume that the subsequence $\{y_{l_k}\}$ converges to some $z\in D'$ . Consequently, the hypotheses of the first claim are fulfilled, and so $y_l\to z$ as $l\to\infty$ , which contradicts the property $\lim_{j\to \infty} y_{l_j}=y \in \partial D'$ .

(4) If under the condition $\{y\}=\overline{\mathbb R^n}\setminus \mathbb R^n$ we have $\varliminf_{\,l\to \infty}d(y_l,0)=\infty$ , then $y_l\to y$ as $l\to \infty$ in the topology of $\overline{\mathbb R^n}$ .

Assume that if $\varliminf_{\,l\to \infty}d(y_l,0)<\infty$ , then $\varlimsup_{k\to \infty}\operatorname{dist}(y_{l_k}, \partial D')> 0$ for some bounded subsequence $\{y_{l_k}\}$ , for $k\in \mathbb N$ . Then some subsequence $y_{l_{k_j}}\to z\in D'$ as $j\to\infty$ . The first claim yields $y_{l}\to z\in D'$ as $l\to\infty$ , which contradicts the hypotheses of claim (4). Proposition 2.11 is proved.

Remark 2.12. Below, we consider the fundamental sequences with respect to the metric function $\rho^{\omega}_{p,F_0}$ which satisfy just one of claims (1), (3), and (4) of Proposition 2.11.

2.2. Capacity metric and completion of the domain

Definition 2.13. Let $D'_{\rho,p}$ be the collection of points $\{y\in D'\setminus F_0\}$ with the capacity metric function $\rho^{\omega}_{p,F_0}$ .

Definition 2.14. Two fundamental sequences $\{y_{l}\in D'_{\rho,p}\}$ and $\{z_{l}\in D'_{\rho,p}\}$ , $l\in \mathbb N$ , with respect to the capacity metric function $\rho^{\omega}_{p,F_0}$ are called equivalent whenever $\rho^{\omega}_{p,F_0}(y_l,z_l)\to 0$ as $l\to \infty$ .

Let us define a new metric space $({\widetilde D}'_{\rho,p}, \widetilde\rho^{\,\omega}_{p,F_0})$ :

(1) its elements are the classes of equivalent fundamental sequences, and

(2) the distance between two elements $X, Y\in {\widetilde D}'_{\rho,p}$ equals

$\begin{equation} \widetilde\rho^{\,\omega}_{p,F_0}(X,Y)=\lim_{l\to\infty}\rho^{\omega}_{p,F_0}(x_l,y_l), \end{equation} \tag{2.21}$

where

$\{x_l\}$ and

$\{y_l\}$ are fundamental sequences in

$X$ and

$Y$ , respectively.

Assume henceforth that the metric space $({\widetilde D}'_{\rho,p}, \widetilde\rho^{\,\omega}_{p,F_0})$ is non-empty.

By analogy with the Hausdorff completion theorem, see [57], Chap. 2, § 6, and [58], § 21.3, for instance, we can prove the following result.

Proposition 2.15. The following claims hold:

(1) the metric function (2.21) is independent of the choice of fundamental sequences $\{x_l\}$ in the class $X$ and $\{y_l\}$ in the class $Y$ ;

(2) the metric function (2.21) in Definition 2.14 satisfies on ${\widetilde D}'_{\rho,p}$ the axioms of a metric space;

(3) the space $({\widetilde D}'_{\rho,p}, \widetilde\rho^{\,\omega}_{p,F_0})$ includes a subset isometric to the metric space

$\begin{equation*} \{y\in D'\setminus F_0\mid \rho^{\omega}_{p,F_0}(y,y)=0\} \end{equation*} \notag$

with the metric

$\rho^{\omega}_{p,F_0}$ .

Proof. Recall how we identify the points of

$\{y\in D'\setminus F_0\mid \rho^{\omega}_{p,F_0}(y,y)=0\}$ with the metric

$\rho^{\omega}_{p,F_0}$ and those of some subset in

$({\widetilde D}'_{\rho,p},\widetilde\rho^{\,\omega}_{p,F_0})$ .

With a point $y\in {D}'_{\rho,p}$ we associate the equivalence class $i(y)\in {\widetilde D}'_{\rho,p}$ containing the constant sequence $\{y,y,\dots, y,\dots\}$ . It is obvious that

$\begin{equation*} \widetilde\rho^{\,\omega}_{p,F_0}(i(x),i(y))=\rho^{\omega}_{p,F_0}(x,y), \end{equation*} \notag$

so that the embedding

$\begin{equation*} i\colon {D}'_{\rho,p}\to {\widetilde D}'_{\rho,p} \end{equation*} \notag$

is an isometry. Proposition 2.15 is proved.

Definition 2.16. Refer to the metric space $({D}'_{\rho,p}, \rho^{\omega}_{p,F_0})$ to the subset $\{y\in D'\setminus F_0\mid \rho^{\omega}_{p,F_0}(y,y)=0\}$ with the metric $\rho^{\omega}_{p,F_0}$ .

Proposition 2.17. Consider a homeomorphism $f\colon D' \to D$ from $\mathcal{Q}_{p, q}(D',\omega;D)$ , where $n-1<q\leqslant p\leqslant n$ for $n\geqslant 3$ and $1\leqslant q\leqslant p\leqslant2$ for $n=2$ . We fix an equivalence class $h\in {\widetilde D}'_{\rho,p}$ and take an arbitrary fundamental sequence $\{y_l\}$ in this class. Then the following behaviour of $\{y_l\}$ is possible:

(1) (a) $y_l\to y\in D'\setminus F_0$ as $l\to\infty$ in the Euclidean metric and the limit $y$ is unique, meaning that it is independent of the choice of sequence in $h$ ;

(b) $y_l\to y\in F_0$ as $l\to\infty$ in the Euclidean metric and the limit $y$ is unique;

(2) otherwise, depending on the choice of fundamental sequence in $h$ , the following cases are possible:

(a) $\varlimsup_{l\to \infty}d(y_l,0)<\infty$ and then $\operatorname{dist}(y_l, \partial D')\to 0$ as $l\to \infty$ ;

(b) $\varlimsup_{l\to \infty}d(y_l,0)=\infty$ and $\varliminf_{\,l\to \infty}d(y_l)<\infty$ , and then

$\begin{equation*} \lim_{l\to \infty}\operatorname{dist}(y_{l_k}, \partial D')=0 \end{equation*} \notag$

for every bounded subsequence

$\{y_{l_k}\in D'\}$ of

$\mathbb R^n$ ;

Proof. The fundamental sequence

$\{y_l\}$ of class

$h\in {\widetilde D}'_{\rho,p}$ bounded in

$\mathbb R^n$ satisfies the hypotheses of Proposition 2.11, and so its claims (1)–(4) can hold for it. It remains to verify that the same claims hold for every bounded sequence

$\{z_l\}$ of the class

$h\in {\widetilde D}'_{\rho,p}$ .

Indeed, the sequence $y_1,z_1,y_2,z_2,\dots,y_n,z_n,\dots$ is fundamental with respect to the metric function $\rho^{\omega}_{p,F_0}$ , bounded in $\mathbb R^n$ , and has an accumulation point $y$ , which lies either in $D'$ or in $\partial D'$ .

In the first case by claim (1) of Proposition 2.11 some subsequence of the sequence

$\begin{equation} y_1,\ z_1,\ y_2,\ z_2,\ \dots,\ y_n,\ z_n,\ \dots \end{equation} \tag{2.22}$

converges to

$y\in D'$ . Hence, both sequences (2.22) and

$\{z_l\}$ converge to

$y$ as

$l\to\infty$ . In the second case, no subsequence

$\{z_{l_k}\}$ of the sequence

$\{z_l\}$ can converge to any point

$z\in D'$ , because similar arguments would yield the impossible coincidence

$y=z$ . So, if the sequence

$\{z_l\}$ is bounded, then claim (3) of Proposition 2.11 shows that

$\operatorname{dist}(z_l, \partial D')\to 0$ as

$l\to \infty$ .

If some sequence $\{y_l\}$ of class $h\in {\widetilde D}'_{\rho,p}$ is not bounded, then we should apply claim (4) of Proposition 2.11 to justify claims (2)(b) and (2)(c) of Proposition 2.17.

Now we take another fundamental sequence $\{z_l\}$ , $l\in \mathbb N$ , in the same class $h\in {\widetilde D}'_{\rho,p}$ . Applying Proposition 2.11 to it, we conclude that $z_l$ cannot converge to any point $z\in D'$ , as otherwise $y_l$ would also converge to $z\in D'$ as $l\to \infty$ . Thus, for the sequence $z_l$ , only claims (3) or (4) of Proposition 2.11 can hold, which proves Proposition 2.17.

The following example shows that each of the possibilities (a), (b) and (c) of part 2 of Proposition 2.17 can be realized in various sequences of the same class.

Example 2.18 (ridge domain). In [18], [26], and [45], one can find an example of a simply-connected domain with non-trivial boundary elements, although the domain is locally connected at all boundary points of the Euclidean boundary. For $q=p=n=3$ and $\omega \equiv 1$ , consider the ridge domain

$\begin{equation*} D' = \{x=(x_1,x_2,x_3) \colon |x_2| < x_1^{\alpha},\, \alpha >2, \, 0<x_1<1, \, 0<x_3<\infty\}. \end{equation*} \notag$

We take the sequences

$\begin{equation*} y^1_l = \biggl(\frac{1}{l},\frac{1}{2l^{\alpha}},1\biggr), \qquad y^3_l = \biggl(\frac{1}{l},\frac{1}{2l^{\alpha}},l\biggr), \end{equation*} \notag$

and define the sequence

$\{y^2_l\}$ by alternating

$\{y^1_l\}$ and

$\{y^3_l\}$ :

$\begin{equation*} y^2_{2l} = \biggl(\frac{1}{l},\frac{1}{2l^{\alpha}},1\biggr)\quad \text{and}\quad y^2_{2l+1} = \biggl(\frac{1}{l},\frac{1}{2l^{\alpha}},l\biggr). \end{equation*} \notag$

Consequently,

$\{y^1_l\}$ ,

$\{y^2_l\}$ and

$\{y^3_l\}$ satisfy conditions (2)(a), (2)(b) and (2)(c) of Proposition 2.17, respectively, since

$y_l^1,y_{2l}^1 \to (0,0,1)$ and since

$\lim_{l\to\infty} d(y_{2l+1}^2,0) =\lim_{l\to\infty} d(y_l^3,0) = \infty$ . In addition, the chosen sequences lie in the same equivalence class

$h \in \widetilde{D}'_{\rho,3}$ . Here, the metric

$\rho^{\omega}_{p,F_0}$ is defined with respect to the Sobolev space

$L^1_3(D')$ and

$F_0\subset D'$ is an arbitrary continuum with non-empty interior.

With the new notation and concepts, we can interpret Proposition 2.4 as follows.

Theorem 2.19 (extension of $\mathcal Q_{p,q}$ -homeomorphisms). Consider a homeomorphism $f\colon D' \to D$ of the class $\mathcal{Q}_{p, q}(D',\omega;D)$ , where $n-1<q\leqslant p\leqslant n$ for $n\geqslant 3$ and $1\leqslant q\leqslant p\leqslant 2$ for $n=2$ . Then

(1) the mapping $f\colon D' \to D$ induces the Lipschitz mapping

$\begin{equation*} f\colon \bigl({D}'_{\rho,p},\widetilde\rho^{\,\omega}_{p,F_0}\bigr)\to \bigl({D}_{\rho,q},\widetilde\rho_{q,f(F_0)}\bigr) \end{equation*} \notag$

of metric spaces, with the estimate for metric distances

$\begin{equation} \begin{cases} \widetilde\rho_{p,f(F_0)}\bigl(f(x),f(y)\bigr) \leqslant K_p\widetilde\rho^{\,\omega}_{p,F_0}(x,y) &\textit{if } q=p, \\ \widetilde\rho_{q,f(F_0)}\bigl(f(x),f(y)\bigr) \leqslant \Psi_{p,q}(D'\setminus F_0)^{1/\sigma} \widetilde\rho^{\,\omega}_{p,F_0}(x,y) &\textit{if } q<p, \end{cases} \end{equation} \tag{2.23}$

for all points

$x,y\in {D}'_{\rho,p}$ , where

$1/\sigma=1/q-1/p$ ;

(2) the mapping $f\colon D' \to D$ induces the Lipschitz mapping

$\begin{equation*} \widetilde f\colon \bigl({\widetilde D}'_{\rho,p},\widetilde\rho^{\,\omega}_{p,F_0}\bigr) \to \bigl({\widetilde D}_{\rho,q},\widetilde\rho_{q,f(F_0)}\bigr) \end{equation*} \notag$

of the “completed” metric spaces: with each element

$X\in( {\widetilde D}'_{\rho,p},\widetilde\rho^{\,\omega}_{p,F_0})$ associate the element

$\widetilde f(X)\in ({\widetilde D}_{\rho,q},\widetilde\rho_{q,f(F_0)})$ containing the fundamental sequence

$\{f(x_l)\}$ , where

$\{x_l\}\in X$ , with the estimate for metric distances

$\begin{equation} \begin{cases} \widetilde\rho_{p,f(F_0)}\bigl(\widetilde f(X),\widetilde f(Y)\bigr) \leqslant K_p\widetilde\rho^{\,\omega}_{p,F_0}(X,Y) &\textit{if } q=p, \\ \widetilde\rho_{q,f(F_0)}\bigl(\widetilde f(X),\widetilde f(Y)\bigr) \leqslant \Psi_{p,q}(D'\setminus F_0)^{1/\sigma} \widetilde\rho^{\,\omega}_{p,F_0}(X,Y) &\textit{if } q<p, \end{cases} \end{equation} \tag{2.24}$

for

$x,y\in {\widetilde D}'_{\rho,p}$ .

Proof. Claim (1) and (2.23) follow directly from Proposition 2.4, while (2.24) is secured by Definition (2.21) of the metric distance between the elements of “completed” spaces. Indeed, if a sequence

$\{x_l\}$ belongs to

$X\in {\widetilde D}'_{\rho,p}$ , then by (2.23) the sequence

$\{f(x_l)\}$ is fundamental with respect to the metric function

$\widetilde\rho_{q,f(F_0)}$ . We call the class of equivalent sequences containing

$\{f(x_l)\}$ the image of the class

$X$ , and denote the resulting mapping by

$\widetilde f$ . Deducing that

$\begin{equation*} \widetilde\rho_{p,f(F_0)}\bigl(\widetilde f(X),\widetilde f(Y)\bigr) =\lim_{l\to\infty}\widetilde\rho_{p,f(F_0)}\bigl(\widetilde f(x_l),\widetilde f(y_l)\bigr) \end{equation*} \notag$

and using Definition (2.21), as well as (2.23), we obtain the claim.

Therefore, Proposition 2.19 determines the extended mapping $\widetilde f$ .

Definition 2.20. Let $f\colon D'{\to}\, D$ be a homeomorphism of the class $\mathcal{Q}_{p, q}(D',\omega;D)$ , where $n-1<q\leqslant p\leqslant n$ for $n\geqslant 3$ and $1\leqslant q\leqslant p\leqslant 2$ for $n=2$ . Denote by $\widetilde f\colon ({\widetilde D}'_{\rho,p},\widetilde\rho^{\,\omega}_{p,F_0})\to({\widetilde D}_{\rho,q},\widetilde\rho_{q,f(F_0)})$ the extension of $f$ to the “completed” metric spaces: to each $X\in({\widetilde D}'_{\rho,p},\widetilde\rho^{\,\omega}_{p,F_0})$ we associate $\widetilde f(X)\in ({\widetilde D}_{\rho,q},\widetilde\rho_{q,f(F_0)})$ containing the fundamental sequence $\{f(x_l)\}$ .

2.3. Capacity boundary. Boundary correspondence of mappings

By Proposition 2.17, in the topology of the extended space $\mathbb R^n$ , the limit points of the fundamental sequence $\{y_l\}$ , $l\in \mathbb N$ , of some class $h\in {\widetilde D}'_{\rho,p}$ can be

(1a) the points $y\in D'\setminus F_0$ : in this case, $y_l\to y\in D'\setminus F_0$ as $l\to\infty$ in the Euclidean metric;

(1b) the points $y\in F_0$ : in this case, $y_l\to y\in F_0$ as $l\to\infty$ in the Euclidean metric.

Otherwise, depending on the choice of fundamental sequence $\{y_l\}$ , $l\in \mathbb N$ , of class $h$ , the possible variants are

(2a) the points $y\in \partial D'$ ;

(2b) the point $y=\infty$ .

Clearly, in case (1a) we can identify the class $h\in {\widetilde D}'_{\rho,p}$ with some point $y\in D'\setminus F_0$ , while in case (1b), with some point $y\in F_0$ .

With this observation at hand, we define the concept of the capacity boundary. By claim (3) of Proposition 2.15, the points of the metric space $({D}'_{\rho,p},\rho^{\omega}_{p,F_0})$ are identified with those in some subset of $({\widetilde D}'_{\rho,p},\widetilde\rho^{\,\omega}_{p,F_0})$ so that the embedding

$\begin{equation*} i\colon {D}'_{\rho,p}\to {\widetilde D}'_{\rho,p} \end{equation*} \notag$

is an isometry. Henceforth, we identify

${D}'_{\rho,p}$ with the image

$i({D}'_{\rho,p})$ in

${\widetilde D}'_{\rho,p}$ .

Definition 2.21. The complement

$\begin{equation*} H^{\omega}_{\rho,p}(D') = {\widetilde D}'_{\rho,p} \setminus {D}'\quad \bigl(H_{\rho,q}(D) = {\widetilde D}_{\rho,q} \setminus {D}\bigr) \end{equation*} \notag$

is called the capacity boundary of

$D'$ (respectively,

$D$ ). The metric on the boundary is induced from the ambient space. The capacity boundary elements of the domain

$D'$ or

$D$ are the points of the capacity boundary

$H^{\omega}_{\rho,p}(D')$ or

$H_{\rho,q}(D)$ .

Theorem 2.22 (boundary correspondence of $\mathcal Q_{p,q}$ -homeomorphisms). Let $f\colon D' \to D$ be a homeomorphism of the class $\mathcal{Q}_{p, q}(D',\omega;D)$ , where $n-1<q\leqslant p\leqslant n$ for $n\geqslant 3$ and $1\leqslant q\leqslant p\leqslant 2$ for $n=2$ . Then the restriction $\widetilde f\mid_{H^\omega_{\rho,p}(D')}$ is a Lipschitz mapping

$\begin{equation} \widetilde f\mid_{H^\omega_{\rho,p}(D')}\colon \bigl(H^\omega_{\rho,p}(D'), \widetilde\rho^{\,\omega}_{p,F_0}\bigr) \to \bigl(H_{\rho,q}(D), \widetilde\rho_{q,f(F_0)}\bigr) \end{equation} \tag{2.25}$

of capacity boundaries.

Proof. Let

$\widetilde f\colon ({\widetilde D}'_{\rho,p},\widetilde\rho^{\,\omega}_{p,F_0})\to({\widetilde D}_{\rho,q},\widetilde\rho_{q,f(F_0)})$ be the mapping from Theorem 2.19. Then the restriction

$\widetilde f\mid_{H^\omega_{\rho,p}(D')}$ is the Lipschitz mapping

$\begin{equation} \widetilde f\mid_{H^\omega_{\rho,p}(D')}\colon \bigl(H^\omega_{\rho,p}(D'), \widetilde\rho^{\,\omega}_{p,F_0}\bigr)\to \bigl({\widetilde D}_{\rho,q},\widetilde\rho_{q,f(F_0)}\bigr). \end{equation} \tag{2.26}$

To prove the claim, it remains to verify that the image of this mapping lies in

$(H_{\rho,q}(D), \widetilde\rho_{q,f(F_0)})$ .

Assume on the contrary that there exists a boundary element $h\,{\in}\, (H^\omega_{\rho,p}(D'), \widetilde\rho^{\,\omega}_{p,F_0})$ such that $\widetilde f(h)= y\in (D,\widetilde\rho_{q,f(F_0)})$ . Then there exists a sequence $\{x_l\}\in h$ , where $h\in ({\widetilde D}'_{\rho,p},\widetilde\rho^{\,\omega}_{p,F_0})$ , such that $f(x_l)\to y$ in the metric space $(D_{\rho,q},\widetilde\rho_{q,f(F_0)})$ . By Proposition 2.10, the sequence $f(x_l)$ converges to $y\in D$ in the Euclidean metric as well. Therefore, $f^{-1}( f(x_l))=x_l$ converges to $\varphi(y)\in D'$ in $\mathbb{R}^n$ . Proposition 2.17 shows that every sequence $\{z_l\}\in h$ converges to $\varphi(y)\in D'$ in the Euclidean metric, and so in the metric space $({\widetilde D}'_{\rho,p},\widetilde\rho^{\,\omega}_{p,F_0})$ as well, see Proposition 2.10, which obviously contradicts the initial assumption. Theorem is proved.

2.4. Support of a boundary element

In this section, we fix an arbitrary number $p$ satisfying $n-1< p\leqslant n$ for $n\geqslant 3$ and $1\leqslant p\leqslant 2$ for $n=2$ .

Definition 2.23. Given a domain $D'$ in $\mathbb R^n$ , the support $\mathcal{S}_h$ of a boundary element $h \in H^\omega_{\rho,p}(D')$ is the set of all accumulation points in the topology of the extended space $\overline{\mathbb R^n}$ of all fundamental sequences with respect to the capacity metric lying in the equivalence class defining $h$ .

Remark 2.24. Proposition 2.17 and Definition 2.21 show that no accumulation point of a sequence in $h \in H^\omega_{\rho,p}(D')$ fundamental with respect to the capacity metric belongs to $D'$ . Therefore,

$\begin{equation*} \mathcal{S}_h\subset \partial D'\cup\{\infty\}. \end{equation*} \notag$

Proposition 2.25. If $D'$ is a domain in $\mathbb R^n$ , then

(1) the support $\mathcal{S}_h$ of a boundary element $h \in H^\omega_{\rho,p}(D')$ coincides with the intersection $\bigcap_{\varepsilon > 0} \overline{B_{\rho}(h,\varepsilon) \cap D'}$ ,

$\begin{equation} \mathcal{S}_h= \bigcap_{\varepsilon > 0} \overline{B_{\rho}(h,\varepsilon) \cap D'}, \end{equation} \tag{2.27}$

where the closure is taken in the topology of the extended space

$\overline{\mathbb R^n}$ ;

(2) if $\rho^{\omega}_{p,F_0}(h_1,h_2)=0$ for two boundary elements $h_1,h_2 \in H^\omega_{\rho,p}(D')$ , then $\mathcal{S}_{h_1}= \mathcal{S}_{h_2}$ .

Proof. The proof is in three steps.

(1) We fix a boundary element $h \in H^\omega_{\rho,p}(D')$ . Let us verify the inclusion

$\begin{equation} \mathcal{S}_h \subset \bigcap_{\varepsilon > 0} \overline{B_{\rho}(h,\varepsilon) \cap D'}. \end{equation} \tag{2.28}$

By the definition of a boundary element

$h \in H^\omega_{\rho,p}(D')$ , there exists a fundamental sequence

$\{y_l\}\in h$ with respect to the

$(\omega,p)$ -metric function with

$\rho^{\omega}_{p,F_0}(y_l,h)\to 0$ as

$l\to\infty$ . For the sequence

$\{y_l\in D'_{\rho,p}\}$ and its subsequences, only the behaviour described in Proposition 2.17 is possible:

(a) $y_l\to y\in D'\setminus F_0$ or $y_l\to y\in F_0$ as $l\to\infty$ in the Euclidean metric, and the limit $y$ is unique, meaning independent of the choice of sequence in $h$ ;

(b) $\varlimsup_{l\to \infty}d(y_l,0)<\infty$ and then $\operatorname{dist}(y_l, \partial D')\to 0$ as $l\to \infty$ ;

$\begin{equation*} \lim_{l\to \infty}\operatorname{dist}(y_{l_k}, \partial D')=0 \end{equation*} \notag$

for every subsequence

$\{y_{l_k}\in D'\}$ bounded in

$\mathbb R^n$ ;

(d) if $d(y_l,0)\to\infty$ , then $\infty\in \mathcal{S}_h$ .

Definition 2.21 excludes case (a). In cases (b)–(d), we have

$\begin{equation*} \mathcal{S}_h\subset \partial D'\cup\{\infty\}. \end{equation*} \notag$

In these cases, for every

$\varepsilon > 0$ the elements of the sequence

$\{y_l\in D'\}$ starting with some index

$l_0$ lie in

$B_{\rho}(h,\varepsilon)\cap D'$ for all

$l\geqslant l_0$ . Thus the accumulation points of

$\{y_l\in D'\}$ lie in the closure

$\overline{B_{\rho}(h,\varepsilon) \cap D'}$ in the topology of the extended space

$\overline{\mathbb R^n}$ . Since we choose the fundamental sequence

$\{y_l\}\in h$ for the boundary element

$h$ arbitrarily, it follows that

$\mathcal{S}_h \subset\overline{B_{\rho}(h,\varepsilon) \cap D'}$ . The inclusion (2.28) is established as

$\varepsilon > 0$ is arbitrary.

(2) In the case $\rho^{\omega}_{p,F_0}(h_1,h_2)=0$ , the equivalence classes of fundamental sequences for the boundary elements $h_1$ and $h_2$ coincide. Hence, we conclude that the supports of $h_1$ and $h_2$ coincide.

(3) To justify (2.27), it remains to verify the reverse inclusion to (2.28):

$\begin{equation} \bigcap_{\varepsilon > 0} \overline{B_{\rho}(h,\varepsilon) \cap D'} \subset \mathcal{S}_h. \end{equation} \tag{2.29}$

Indeed, if

$x\,{\in} \bigcap_{\varepsilon > 0} \overline{B_{\rho}(h,\varepsilon) \cap D'}$ , then, for each

$l\in\mathbb N$ , there exists

$x_l\in B_{\rho}(h,1/l) \cap D'$ such that simultaneously

$\rho^{\omega}_{p,F_0}(x_l,h)\to 0$ as

$l\to\infty$ and (using Proposition 2.17 and extracting a subsequence if necessary)

$x_l\to x$ in the topology of the extended space

$\overline{\mathbb R^n}$ . Therefore, the fundamental sequence

$\{x_l\}$ with respect to the capacity metric determines a boundary element, which coincides with

$h$ . Thus,

$x\in \mathcal{S}_h$ and (2.29) is established. The inclusions (2.28) and (2.29) are equivalent to (2.27). This proves the proposition.

Proposition 2.26. The support $\mathcal{S}_h$ of each boundary element $h\in H^\omega_{\rho,p}(D')$ is connected in the topology of the space $\overline{\mathbb R^n}$ .

Proof. Assume on the contrary that, for some boundary element

$h\in H^\omega_{\rho,p}(D')$ , there are two disjoint open sets

$V,W\subset\overline{\mathbb R^n}$ with

$\mathcal{S}_h\subset V\cup W$ , while

$\mathcal{S}_h\cap V\ne\varnothing$ and

$\mathcal{S}_h\cap W\ne\varnothing$ . Take two points

$x\in\mathcal{S}_h\cap V$ and

$y\in\mathcal{S}_h\cap W$ and fundamental sequences

$\{x_m\},\{y_m\}\in h$ with respect to the capacity metric such that

$x_m\to x$ and

$y_m\to y$ as

$m\to\infty$ . There is a curve

$\gamma_m\subset D'$ with endpoints

$x_m$ and

$y_m$ such that

$\operatorname{cap}((\gamma_m, F_0); L^1_p(D';\omega))\to 0$ as

$m\to\infty$ . For all big enough

$m$ , starting with some there exists a point

$z_m\in \gamma_m$ satisfying

$z_m\notin V\cup W$ . We emphasize that the sequence

$\{z_m\}$ , fundamental with respect to the capacity metric, belongs to the equivalence class

$h$ . Extracting a subsequence, we may assume that

$z_m\to z_0$ , where

$z_0\in \overline{D'}\setminus (V\cup W)$ ; here, the closure is taken in the topology of the extended space

$\overline{\mathbb R^n}$ . Since

$z_0\notin \mathcal{S}_h$ , we arrive at a contradiction with the definition of the support of a boundary element. Proposition 2.26 is proved.

Proposition 2.27. Consider the support $\mathcal{S}_h$ of $h\in H^\omega_{\rho,p}(D')$ . For every sequence $\{x_m\}\in h$ we have the convergence $x_m\to \mathcal{S}_h$ as $m \to \infty$ in the topology of the extended space $\overline{\mathbb R^n}$ .

Proof. Proposition 2.25 excludes the possibility that

$\mathcal{S}_h\cap D'\ne\varnothing$ .

Suppose that $\mathcal{S}_h$ is bounded in $\mathbb R^n$ and $\mathcal{S}_h\subset \partial D'$ . Suppose that there exists a subsequence $\{x_{m_k}\in D'\}$ , for $k\in \mathbb N$ , of some fundamental sequence $\{x_m\}\in h$ such that $d(x_{m_k},\mathcal{S}_h) \geqslant\alpha> 0$ for all $k\in \mathbb N$ , where $\alpha$ is some constant. Then the sequence $\{x_{m}\}$ has an accumulation point at some positive distance from $\mathcal{S}_h$ . This point must lie in the support of the boundary element $h$ , which contradicts the connectedness of $\mathcal{S}_h$ .

However, if the support $\mathcal{S}_h$ is unbounded and the sequence $x_m$ does not converge to $\mathcal{S}_h$ in the topology of the extended space $\overline{\mathbb R^n}$ , then $\varliminf_{\,m\to\infty}x_m<\infty$ . Consequently, there exists a finite accumulation point at some positive distance from $\mathcal{S}_h$ . As in the previous case, we arrive at a contradiction with the connectedness of $\mathcal{S}_h$ . Proposition 2.27 is proved.

Proposition 2.28 (criterion for singleton support). Given a boundary element $h\in H^\omega_{\rho,p}(D')$ of the domain $D'$ , the support $\mathcal{S}_h$ amounts to a single point if and only if for all fundamental sequences $\{x_m\},\{y_m\}\in h$ with respect to the capacity metric there exist curves $\overline{x_m y_{m}}$ , for $m\in \mathbb N$ , with $\operatorname{diam}(\overline{x_m y_{m}}) \to 0$ as $m\to \infty$ .

Proof. Necessity. Suppose that

$\mathcal{S}_h=\{x_0\}$ . Assume on the contrary that there exist fundamental sequences

$\{x_m\}$ and

$\{y_m\}$ of class

$h$ with respect to the capacity metric converging to

$x_0$ , curves

$\gamma_{m} = \overline{x_m y_{m}}$ with

$\begin{equation} \operatorname{cap}^{1/p}\bigl((\gamma_{m}, F_0); L^1_p(D';\omega)\bigr)\to 0 \quad\text{as}\quad m\to \infty, \end{equation} \tag{2.30}$

and a number

$\alpha>0$ such that

$\begin{equation*} \operatorname{diam} \gamma_{m}\geqslant \alpha > 4d(x_m, y_{m})\quad\text{for all}\quad m\in\mathbb N \end{equation*} \notag$

because

$x_m\to x_0$ and

$y_m\to x_0$ as

$m\to\infty$ . Then, for each

$m\in\mathbb N$ , there exists a point

$z_{m} \in \gamma_{m}$ such that, on the one hand,

$\begin{equation} d(x_m,z_{m})> \frac{\alpha}{4}, \qquad d(y_{m},z_{m})> \frac{\alpha}{4} \end{equation} \tag{2.31}$

and on the other hand, (2.7) and (2.30) yield

$\rho^{\omega}_{p,F_0} (z_{m}, x_{m}) \to 0$ as

$m\to \infty$ . Hence, we infer that the sequence

$\{z_{m}\}$ , for

$m\in \mathbb N$ , is fundamental with respect to the capacity metric and belongs to the boundary element

$h$ . On the other hand, there exists a subsequence

$\{z_{m_i}\}$ , for

$i\in \mathbb N$ , converging to some point

$z_0$ ; moreover, (2.31) implies that

$z_0 \neq x_0$ . Since

$z_0\in \mathcal{S}_h$ by the definition of support, we arrive at a contradiction with its being a singleton.

Sufficiency. Assume on the contrary that there are two sequences $\{x_m\}$ , $\{y_m\} \in h$ fundamental with respect to the capacity metric and converging to distinct points $x$ and $y$ of the support $\mathcal{S}_h$ . By the hypotheses, there exist curves $\gamma_m = \overline{x_m y_m}$ such that $\operatorname{diam} \gamma_m \to 0$ as $m \to \infty$ . In particular, $\operatorname{diam} \gamma_m\,{\geqslant}\, d(x_m, y_m)\,{\to}\, d(x, y)\,{>}\,0$ as $m \to \infty$ , which, evidently, contradicts the property $\operatorname{diam} \gamma_m \to 0$ as $m \to \infty$ inferred from the assumption, proving the proposition.

2.5. Continuous extension of mappings of class $\mathcal{Q}_{p, q}(D',\omega;D)$ to the Euclidean boundary

In this section, we fix arbitrary numbers $q$ and $p$ satisfying $n-1<q\leqslant p\leqslant n$ for $n\geqslant 3$ and $1\leqslant q\leqslant p\leqslant 2$ for $n=2$ .

In what follows, we define domains $\mu$ -connected at boundary points.

Definition 2.29 (connectedness properties [16], [18]). (1) A domain $D'$ is called locally connected at $x \in \partial D'$ if for every neighbourhood $U$ of $x$ there is a neighbourhood $V \subset U$ of this point such that $V \cap D'$ is connected.

(2) An unbounded domain $D'$ is called locally connected at $\infty$ if for every neighbourhood $U$ of $\infty$ there is a neighbourhood $V \subset U$ of this point such that $V \cap D'$ is connected.

(3) A domain $D'$ is called locally $\mu$ -connected at $x \in \partial D'$ , where $\mu\in\mathbb N$ , if for every neighbourhood $U$ of $x$ there is a neighbourhood $V \subset U$ of this point such that $V\cap D'$ consists of $\mu$ connected components, each of which is locally connected at $x$ . Observe that a domain $D'$ locally $1$ -connected at $x \in \partial D'$ is precisely the domain $D'$ locally connected at $x \in \partial D'$ .

(4) An unbounded domain $D'$ is called locally $\mu$ -connected at $\infty$ , where $\mu\in\mathbb N$ , if for every neighbourhood $U$ of $\infty$ there is a neighbourhood $V \subset U$ of this point such that $V \cap D'$ consists of $\mu$ connected components, each of which is locally connected at $\infty$ . In the case $\mu=1$ we obtain the domain $D'$ locally connected at $\infty$ .

(5) A domain $D'$ is called finitely connected at $x \in \partial D'$ or $x=\infty$ whenever it is $\mu$ -connected at $x$ for some $\mu\in\mathbb N$ .

The following example demonstrates the appearance of domains which are multiply connected at boundary points.

Example 2.30 (slit ball). Let $D' = B(0,1) \setminus (\{0\}\times[0,1)^{n-1})$ . It is not difficult to see that $D'$ is locally $2$ -connected at each point $x\in \{0\}\times(0,1)^{n-1}$ . If $\omega= 1$ is the trivial weight and $p=n$ , then condition (2.37) is met for every point $x\in \{0\}\times(0,1)^{n-1}$ , and $x$ lies in the support of two distinct boundary elements $h_{+},h_{-} \in H_{\rho,n}(D')$ .

Let us present the methods of [16], Theorem 1.10, for describing connectedness alternative to Definition 2.29 and useful below.

Proposition 2.31. Given a domain $D'\in \mathbb R^n$ and its boundary point $x \in \partial D'$ , the following statements are equivalent:

(1) $D'$ is locally $\mu$ -connected at $x$ ;

(2) for every neighbourhood $U$ of $x$ there exists a neighbourhood $V \subset U$ of this point such that $V \cap D'$ consists of $\mu$ connected components, the boundary of each of which contains $x$ ;

(3) $\mu$ is the smallest integer for which the following condition holds: given $\mu + 1$ sequences $\{x_ {1, k}\}, \dots, \{x_ {\mu+1, k}\}$ of points in $D'$ converging to $x$ , if $V$ is some neighbourhood of $x$ , then there exists a connected component of $V \cap D'$ including subsequences of two distinct sequences.

To obtain similar properties at $\infty$ , we should use the stereographic projection to map the domain $D'$ onto the unit sphere in $\mathbb R^{n+1}$ with the point $\infty$ going into the north pole, on which the property of local $\mu$ -connectedness at $\infty$ can be stated by analogy with the above.

Example 2.32. On the plane $\mathbb R^2$ , consider the complement

$\begin{equation*} B(0, 4) \setminus \{x=(x_1,x_2)\in B(0, 2)\mid x_1\cdot x_2=0 \} \end{equation*} \notag$

as the domain

$D'$ . We fix two numbers

$\alpha>-2$ and

$p\in (1,2]$ with

$p-2>\alpha$ , as well as a continuum

$F_0\subset B(0, 4)\setminus \overline{B(0, 2)}$ with non-empty interior. As the weight function

$\sigma\colon B(0, 4)\to(0,\infty)$ , we take

$\begin{equation*} D'\ni x\mapsto \sigma(x)= \begin{cases} \omega(x) &\text{if }x\in B(0,2)\cap D'\text{ and }x_1\cdot x_2>0, \\ 2^{\alpha} &\text{otherwise}, \end{cases} \end{equation*} \notag$

where

$\omega$ is the weight function of example 2.7.

The domain $D'$ is obviously $4$ -connected at $0$ : each intersection $B(0,r)\cap D'$ , for $r\in (0,2)$ , consists of $4$ connected components. We denote them by $V_1$ and $V_3$ if $x_1\cdot x_2>0$ and by $V_2$ and $V_4$ otherwise.

It is natural to define the weighted capacity of the condenser $\mathcal E=(\{0\},F_0)\subset D'$ in the space $L^1_p(D';\sigma)$ with respect to the connected component $V_i$ as

$\begin{equation} \operatorname{cap}\bigl((\{0\}, F_0); L^1_p(V_i,D';\omega)\bigr) =\inf_{u}\|u\mid L^1_{p}(D';\omega)\|^p, \end{equation} \tag{2.32}$

where the infimum is over all functions

$u\in\operatorname{Lip}_{\mathrm{loc}}(D')\cap L^1_{p}(D';\omega)$ such that

$u|_{B(0,r)\cap V_i}\equiv1$ for some

$r>0$ , depending on

$u$ , and

$u|_{F_0}\equiv0$ .

On account of Example 2.7, the capacity of the point $0$ with respect to $V_1$ and $V_3$ is positive, and with respect to $V_2$ and $V_4$ it vanishes.

This example motivates the following definition.

Definition 2.33. Suppose that a domain $D'$ is locally $\mu$ -connected at a boundary point $x\in \partial D'$ and denote by $V_1,V_2,\dots,V_\mu$ the distinct connected components of $B(x,r)\cap D'$ , where $r\in(0,r_0)$ for sufficiently small $r_0>0$ , whose boundaries contain $x$ . Next, we define the weighted capacity of the condenser $\mathcal E=(\{x\},F_0)\subset D'$ in the space $L^1_p(D';\omega)$ with respect to the connected component $V_i$ by

$\begin{equation} \operatorname{cap}\bigl((\{x\}, F_0); L^1_p(V_i,D';\omega)\bigr) =\inf_{u}\|u\mid L^1_{p}(D';\omega)\|^p, \end{equation} \tag{2.33}$

where the infimum is over all functions

$u\in\operatorname{Lip}_{\mathrm{loc}}(D')\cap L^1_{p}(D';\omega)$ such that

$u|_{B(x,r)\cap V_i}\equiv1$ for some

$r\in (0,r_0)$ , depending on

$u$ , and

$u|_{F_0}\equiv 0$ .

If $\mu=1$ , then instead of notation (2.33) we will simply write

$\begin{equation*} \operatorname{cap}\bigl((\{x\}, F_0); L^1_p(D';\omega)\bigr). \end{equation*} \notag$

In the case $x=\infty$ , the lower bound in (2.33) is taken over all functions $u\in\operatorname{Lip}_{\mathrm{loc}}(D')\cap L^1_{p}(D';\omega)$ such that $u|_{(\mathbb R^n\setminus B(x,r))\cap V_i}\equiv1$ for some $r>0$ , depending on $u$ , and $u|_{F_0}\equiv0$ , and denoted by

$\begin{equation} \operatorname{cap}\bigl((\{\infty\}, F_0); L^1_p(V_i,D';\omega)\bigr). \end{equation} \tag{2.34}$

A boundary point $x\in \partial D'$ is called a point of zero capacity with respect to the connected component $V_i$ whenever

$\begin{equation} \operatorname{cap}\bigl((\{x\}, F_0); L^1_p(V_i,D';\omega)\bigr)=0. \end{equation} \tag{2.35}$

If condition (2.35) is independent of the choice of continuum

$F_0$ , we simply write

$\begin{equation} \operatorname{cap}\bigl((\{x\}); L^1_p(V_i,D';\omega)\bigr)=0. \end{equation} \tag{2.36}$

Proposition 2.28 yields the following corollary.

Corollary 2.34. The following claims hold.

(1) If the domain $D'$ is locally connected at $x_0$ and the condition

$\begin{equation} \operatorname{cap}\bigl((\{x_0\}, F_0); L^1_p(D';\omega)\bigr)=0 \end{equation} \tag{2.37}$

holds at

$x_0$ , then the boundary elements

$h_1$ and

$h_2\in H^\omega_{\rho,p}(D')$ of the domain

$D'$ whose supports

$\mathcal{S}_{h_1}$ and

$\mathcal{S}_{h_2}$ meet at

$x_0$ cannot be distinct:

$h_1=h_2$ .

(2) Suppose that the domain $D'$ is locally $\mu$ -connected at $x_0$ , and that at $x_0$ condition (2.35)

$\begin{equation*} \operatorname{cap}\bigl((\{x_0\}, F_0); L^1_p(V_i,D';\omega)\bigr)=0 \end{equation*} \notag$

holds for all

$i=1,\dots,\mu$ . Then the boundary elements

$h_1,h_2,\dots,h_\mu,h_{\mu+1} \in H^\omega_{\rho,p}(D')$ of

$D'$ whose supports

$\mathcal{S}_{h_1},\mathcal{S}_{h_2},\dots,\mathcal{S}_{h_\mu},\mathcal{S}_{h_{\mu+1}}$ share the point

$x_0$ cannot be distinct: at least two of them coincide.

Proof. (1) Suppose that the supports

$\mathcal{S}_{h_1}$ and

$\mathcal{S}_{h_2}$ of two boundary elements

$h_1,h_2\in H^\omega_{\rho,p}(D')$ of

$D'$ meet at

$x_0$ . Take two arbitrary sequences

$\{x_k\}\in h_1$ and

$\{y_k\}\in h_2$ fundamental with respect to the metric

$\rho^{\omega}_{p,F_0}$ such that

$x_k\to x_0$ and

$y_k\to x_0$ as

$k\to\infty$ . Since

$D'$ is locally connected at

$x_0$ , we can connect

$x_k$ and

$y_k$ with curves

$\gamma_k = \overline{x_k y_k}$ such that

$\operatorname{diam}\gamma_k \to 0$ as

$k\to\infty$ . Since

$D'$ is locally connected at

$x$ , condition (2.37) also yields

$\begin{equation*} \operatorname{cap}\bigl((\gamma_k, F_0); L^1_p(D';\omega)\bigr)\to0\quad\text{as}\quad k\to\infty. \end{equation*} \notag$

Hence, we see that the sequence

$\{x_k\}$ and

$\{y_k\}$ are equivalent, which implies

$h_1\,{=}\,h_2$ .

(2) Assume that the supports $\mathcal{S}_{h_1},\mathcal{S}_{h_2},\dots,\mathcal{S}_{h_{\mu+1}}$ of some boundary elements $h_1,h_2,\dots,h_{\mu+1}\,{\in}\, H^\omega_{\rho,p}(D')$ , for $\mu\in \mathbb N$ , of $D'$ meet at $x_0$ . Take an arbitrary fundamental sequence $\{x_{ik}\}\in h_i$ with respect to the metric $\rho^{\omega}_{p,F_0}$ such that $x_{ik}\to x_0$ as $k\to\infty$ , for $i=1,\dots,\mu+1$ . By claim (3) of Proposition 2.31, since $D'$ is locally $\mu$ -connected at $x_0$ , there exists a connected component $V_{i_0}$ , for $1\leqslant i_0\leqslant \mu_0$ , of the intersection $B(x_0,r)\cap D'$ containing subsequences, for instance, $x_{1k_j}$ and $x_{2l_j}$ , for $j\in \mathbb N$ , of two distinct sequences $x_{1k}$ and $x_{2k}$ , for $k\in \mathbb N$ . Since the connected component $V_{i_0}$ is locally connected at $x_0$ and

$\begin{equation*} \operatorname{cap}\bigl((\{x_0\}, F_0); L^1_p(V_{i_0},D';\omega)\bigr)=0, \end{equation*} \notag$

the hypotheses of claim 1 hold, which yields

$h_1=h_2$ . This proves the corollary.

Definition 2.35 (associated support and connected components). Consider some boundary element $h \in H^\omega_{\rho,p}(D')$ whose support $\mathcal{S}_h$ contains $x\in \partial D'$ such that the domain $D'$ is $\mu$ -connected at $x$ , while $\{y_{m}\}$ is a fundamental sequence with respect to the metric $\rho^{\omega}_{p,F_0}$ belonging to the boundary element $h$ and converging to $x$ in the topology of $\overline{\mathbb R^n}$ . Since $D'$ is $\mu$ -connected at $x$ , there exists at least one connected component $V_i$ of the intersection $B(x,r)\cap D'$ , where $r>0$ is a sufficiently small number, which contains some subsequence $\{y_{m_k}\}$ , for $k\in\mathbb N$ . In this case, say that the support $\mathcal{S}_h$ of the boundary element $h$ and the connected component $V_i$ are associated with each other at $x\in \mathcal{S}_h$ .

Proposition 2.36. The following claims hold.

(1) If $D'$ is a locally $\mu$ -connected domain at $x$ , the support $\mathcal{S}_h$ of some boundary element $h \in H^\omega_{\rho,p}(D')$ contains $x\in \partial D'$ and is associated with the connected component $V_i$ at $x$ , while the weighted capacity of $x$ with respect to the connected component $V_i$ vanishes,

$\begin{equation*} \operatorname{cap}\bigl((\{x\}, F_0); L^1_p(V_i,D';\omega)\bigr)=0, \end{equation*} \notag$

then, for every sequence

$\{x_m \in V_i\cap D'\}$ of points,

$d(x_m,x) \to 0$ implies that

$\{x_m\}\in h$ and

$\begin{equation} \rho_{q,f(F_0)} \bigl(f(x_m), \widetilde f(h)\bigr)\to 0\quad \textit{as}\quad m \to \infty. \end{equation} \tag{2.38}$

(2) If $D'$ is a locally $\mu$ -connected domain at $\infty$ , the support $\mathcal{S}_h$ of some boundary element $h \in H^\omega_{\rho,p}(D')$ contains $\infty$ and is associated with the connected component $V_i$ at $\infty$ , while the weighted capacity of the point $\infty$ with respect to some connected component $V_i$ vanishes,

$\begin{equation*} \operatorname{cap}\bigl((\{\infty\}, F_0); L^1_p(V_i,D';\omega)\bigr)=0, \end{equation*} \notag$

then, for every sequence

$\{ x_m \in V_i\cap D'\}$ of points,

$d(x_m,0) \to \infty$ implies that

$\{x_m\}\,{\in}\, h$ and (2.38) holds.

Proof. (1) Choose

$x\,{\in}\, \partial D'$ and a sequence

$\{x_m{\in}\, V_i\,{\cap}\, D'\}$ such that

$d(x_m,x)\,{\to}\, 0$ as

$m \to \infty$ . Since

$V_i\cap D'$ is locally connected at

$x$ , see claim (2) of Proposition 2.31, we infer the existence of curves

$\overline{x_mx_{m+k}}$ with endpoints

$x_m$ and

$x_{m+k}$ , for

$k\geqslant1$ , such that

$\operatorname{diam} \overline{x_mx_{m+k}} \to 0$ as

$m,k\to \infty$ . Since

$\operatorname{cap}((\{x\}, F_0); L^1_p(V_i,D';\omega))=0$ , Definition 2.33 yields

$\rho^{\omega}_{p,F_0} (x_m, x_{m+k})\to 0$ as

$m,k\to \infty$ . Thus, on the one hand the sequence

$\{x_{m}\}$ is fundamental with respect to the metric

$\rho^{\omega}_{p,F_0}$ , and on the other,

$d(x_m,x) \to 0$ as

$m\to \infty$ .

Now we take an arbitrary sequence $\{y_{m}\in V_i\cap D'\}$ , for $m\in\mathbb N$ , fundamental with respect to the metric $\rho^{\omega}_{p,F_0}$ , belonging to some boundary element $h$ , and converging $x$ in the Euclidean metric. We claim that each fundamental sequence $\{x_{m}\}$ with respect to the metric $\rho^{\omega}_{p,F_0}$ satisfies

$\begin{equation} \rho^{\omega}_{p,F_0} (x_m, y_m)\to 0 \quad\text{as} \quad m\to \infty. \end{equation} \tag{2.39}$

As in the previous argument, we conclude that

$\rho^{\omega}_{p,F_0} (x_m, y_m)\to 0$ as

$m\to \infty$ . Thus, property (2.39) and property

$\{x_m\}\in h$ together with it are justified.

Applying (2.9), we deduce (2.38): indeed, the sequences $\{f(x_m)\}$ and $\{f(y_m)\}$ are equivalent with respect to the capacity metric function $\rho_{q,f(F_0)}$ in the domain $D$ . Hence, $\{f(x_m)\}\in \widetilde f(h)$ and $\rho_{q,f(F_0)} (f(x_m), \widetilde f(h))\to 0$ as $m \to \infty$ .

(2) The second claim can be justified similarly.

Proposition 2.36 is proved.

Theorem 2.37 (boundary behaviour of homeomorphisms). Consider a homeomorphism $f \colon D'\to D$ of class $\mathcal{Q}_{p, q}(D',\omega;D)$ , where $n-1<q\leqslant p\leqslant n$ for $n\geqslant 3$ and $1\leqslant q\leqslant p\leqslant 2$ for $n=2$ , as well as a weight function $\omega\in L_{1,\mathrm{loc}}(D')$ .

Suppose that the domain $D'$

(1) is locally $\mu$ -connected at some boundary point $y\in\partial D'$ ,

(2) the support $\mathcal{S}_{h}$ of some boundary element $h\in H^\omega_{\rho,p}(D')$ contains $y$ ,

(3) $\operatorname{cap}((\{y\}, F_0); L^1_p(V_i,D';\omega))=0$ , where $V_i$ is the connected component associated with the support $\mathcal{S}_{h}$ at $y$ .

Then the boundary behaviour of the mapping $f \colon D'\to D$ at $x\in\partial D'$ is

$\begin{equation*} f(z)\to \mathcal{S}_{\widetilde f(h)}\quad \textit{as}\quad z\to y,\quad z\in V_i\cap D', \end{equation*} \notag$

in the topology of the extended space

$\mathbb R^n$ .

Proof. We take a sequence

$\{y_m\in V_i\cap D'\}$ converging to

$y\in \partial D'$ as

$m\to\infty$ . Proposition 2.36 shows that

$\rho^{\omega}_{q,f(F_0)} (f(y_m),\widetilde f(h))\to 0$ as

$m \to \infty$ . In addition, by Proposition 2.27 the sequence

$\{f(y_m)\}$ converges to the support

$\mathcal{S}_{\widetilde f( h)}$ in the topology of the extended space

$\mathbb R^n$ . The proof of Theorem 2.37 is complete.

Corollary 2.38 (continuous extension to boundary points). Consider a homeomorphism $f \colon D'\to D$ of the class $\mathcal{Q}_{p, q}(D',\omega;D)$ , where $n-1< q\leqslant p\leqslant n$ for $n\geqslant3$ and $1\leqslant q\leqslant p\leqslant 2$ for $n=2$ , as well as a weight function $\omega\in L_{1,\mathrm{loc}}(D')$ .

Suppose also that

(1) the domain $D'$ is locally $\mu$ -connected at some boundary point $y\in\partial D'$ ;

(2) the support $\mathcal{S}_{h}$ of a boundary element $h\in H^\omega_{\rho,p}(D')$ contains $y$ ;

(3) $\operatorname{cap}((\{y\}, F_0); L^1_p(V_i,D';\omega))=0$ , where $V_i$ is the connected component associated with the support $\mathcal{S}_{h}$ at $y$ ;

(4) the support $\mathcal{S}_{\widetilde f(h)}$ of the boundary element $\widetilde f(h)$ amounts to a singleton: $\mathcal{S}_{\widetilde f(h)}=\{x\}\in \partial D$ .

Then the mapping $f \colon D'\to D$ extends by continuity to $y\in\partial D'$ and

$\begin{equation*} \lim_{z\to y,\,z\in V_i\cap D'}f(z)=x. \end{equation*} \notag$

Proof. We take a sequence

$\{y_m\in V_i\cap D'\}$ converging to

$y\in \partial D'$ as

$m\to\infty$ . Theorem 2.37 shows that

$\begin{equation*} f(z)\to \mathcal{S}_{\widetilde f(h)}\quad \text{as}\quad z\to y,\quad z\in V_i\cap D' \end{equation*} \notag$

in the topology of the extended space

$\mathbb R^n$ . Since by the assumption the support

$\mathcal{S}_{\widetilde f( h)}$ of the boundary element

$\widetilde f(h)$ is a singleton,

$\mathcal{S}_{\widetilde f( h)}=\{x\}\in \partial D$ , the above implies that the sequence

$\{f(y_m)\}$ converges to

$x\in \partial D$ . The proof of Corollary 2.38 is complete.

From Corollary 2.38 we have

Corollary 2.39 (continuous extension to the Euclidean boundary). Consider a homeomorphism $f \colon D'\to D$ of the class $\mathcal{Q}_{p, q}(D',\omega;D)$ , where $n-1< q\leqslant p\leqslant n$ for $n\geqslant3$ and $1\leqslant q\leqslant p\leqslant 2$ for $n=2$ , as well as a weight function $\omega\in L_{1,\mathrm{loc}}(D')$ . The following claims hold:

(1) if $D'$ is locally connected at $y\in\partial D'$ and $\operatorname{cap}((\{y\}, F_0); L^1_p(D';\omega))=0$ , then $y$ lies in the support $\mathcal{S}_{h}$ of some boundary element $h\in H^\omega_{\rho,p}(D')$ ;

(2) if the support $\mathcal{S}_{\widetilde f(h)}$ of the boundary element $\widetilde f(h)$ is a singleton, $\mathcal{S}_{\widetilde f( h)}=\{x\}\in \partial D$ , then the mapping $f\colon D'\to D$ extends by continuity to $y\in\mathcal{S}_{h}$ of the boundary element $h\in H^\omega_{\rho,p}(D')$ , and

$\begin{equation} \lim_{z\to y,\,z\in D'} f(z)=x \quad \textit{for every point}\quad y\in \mathcal{S}_{h}. \end{equation} \tag{2.40}$

Proof. All hypotheses of Proposition 2.36 are obviously met, and so

$y$ lies in some boundary element

$h\in H^\omega_{\rho,p}(D')$ . The argument above and the hypotheses of the corollary ensure the fulfillment of the conditions of Corollary 2.38 for

$\mu=1$ . It shows that the mapping

$f \colon D'\to D$ extends by continuity to

$y\in\mathcal{S}_{h}$ , and the limit equals (2.40). Corollary 2.39 is proved.

Example 2.40 (a domain with non-trivial boundary elements). Let us consider $D=(0,1)^2\setminus\bigcup_{k\in\mathbb{N}}I_k\subset \mathbb{R}^2$ , where $I_k=[1/2,1) \times \{1/2^k\}$ determine the cuts. It is not difficult to see that $I=[1/2,1)\times\{0\}$ is the support of a boundary element for $p=2$ and $\omega\equiv1$ .

Example 2.41. For the domain from Example 2.18, the edge of the ridge

$\begin{equation*} E = \{x=(x_1,x_2,x_3) \colon x_1 = x_2 = 0, \, 0 \leqslant x_3 \leqslant \infty\} \end{equation*} \notag$

is indeed the support of a boundary element.

Remark 2.42. For the weight $\omega$ and the domain $D'$ such that the collection $H^\omega_{\rho,p}(D')$ of boundary elements is independent of the choice of the continuum $F_0$ , the support $\mathcal{S}_h$ of an arbitrary boundary element $h\in H^\omega_{\rho,p}(D')$ is independent of the choice of $F_0$ , and consequently, all statements of this section are absolute.

§ 3. Moduli of curve families and homeomorphisms of class $\mathcal Q_{p,q}(D',\omega)$

Consider a domain $D'$ in $\mathbb{R}^{n}$ , where $n \geqslant 2$ , a weight function $\omega\colon D' \to (0, \infty)$ of class $L_{1,\mathrm{loc}}$ , and a family $\Gamma$ of (continuous) curves or paths $\gamma\colon[a,b]\to D'$ .

Recall that, given a curve family $\Gamma$ in $D'$ and a real number $p\geqslant 1$ , the weighted $p$ -modulus of $\Gamma$ is defined as

$\begin{equation*} \operatorname{mod}^{\omega}_{p}(\Gamma)=\inf_\rho \int_{D'} \rho^{p}(x)\omega(x)\, dx, \end{equation*} \notag$

where the infimum is over all nonnegative Borel functions

$\rho\colon D' \to [0, \infty]$ with

$\begin{equation} \int_{\gamma} \rho\, ds \geqslant 1 \end{equation} \tag{3.1}$

for all (locally) rectifiable curves

$\gamma \in \Gamma$ . In the case of trivial weight

$\omega \equiv 1$ we write

$\operatorname{mod}_{p}(\Gamma)$ instead of

$\operatorname{mod}^1_{p}(\Gamma)$ . Recall that the integral in (3.1) for a rectifiable curve

$\gamma\colon[a,b]\to D'$ is defined as

$\begin{equation*} \int_{\gamma} \rho\, ds = \int_{0}^{l(\gamma)} \rho(\widetilde{\gamma}(t))\, dt, \end{equation*} \notag$

where

$l(\gamma)$ is the length of

$\gamma\colon[a,b]\to D'$ , while

$\widetilde{\gamma}\colon[0,l(\gamma)]\to D'$ is its natural parametrization, that is, the unique continuous mapping with

$\gamma=\widetilde{\gamma}\circ S_{\gamma}$ , where

$S_{\gamma}\colon[a,b]\to[0,l(\gamma)]$ is the length function, defined at

$t\in [a,b]$ as

$S_{\gamma}(t)=l(\gamma\vert_{[a,t]})$ . If

$\gamma$ is only a locally rectifiable curve, then we put

$\begin{equation*} \int_{\gamma} \rho\,ds = \sup \int_{\gamma'} \rho\,ds \end{equation*} \notag$

with the least upper bound taken over all rectifiable subcurves

$\gamma'\colon [a', b'] \to D'$ of

$\gamma$ , where

$[a', b']\subset(a,b)$ and

$\gamma'= \gamma_{[a', b']}$ .

The functions $\rho$ satisfying (3.1) are called admissible functions, or metrics, for the family $\Gamma$ .

An equivalent description of the mappings of classes $\mathcal Q_{p,q}(D',\omega;D)$ is obtained in [33] in the modular language: to this end, we should replace capacity in the definition of $\mathcal Q_{p,q}(D',\omega;D)$ by the modulus of the curve family whose endpoints lie on the plates of the condenser.

Remark 3.1. It is observed in [32], § 4.4, that in the case $q=p=n$ ( $n-1<q=p< n$ ) the class of homeomorphisms $\mathcal Q_{n,n}(D',\omega;D)$ ( $\mathcal Q_{p,p}(D',\omega;D)$ ) is included into the class of $\omega$ -homeomorphisms ( $(p, \omega)$ -homeomorphisms)⁴ [21] (and [59]), defined via a controlled variation of the modulus of the curve family.

We will verify that, actually, the class $\mathcal Q_{n,n}(D',\omega;D)$ coincides with the family of $\omega$ -homeomorphisms of [21], § 4.1. Consider two domains $D'$ and $D$ in $\mathbb{R}^{n}$ , where $n \geqslant 2$ , and a function $\omega\colon D' \to [1, \infty)$ of class $L_{1,\mathrm{loc}}$ . Recall that a homeomorphism $f\colon D' \to D$ is called an $\omega$ -homeomorphism whenever

$\begin{equation} \operatorname{mod}_n(f \Gamma) \leqslant \int_{D'} \omega(x) \cdot \rho^{n}(x)\,dx \end{equation} \tag{3.2}$

for each family

$\Gamma$ of paths in

$D'$ and every admissible function

$\rho$ for

$\Gamma$ . By [33], Theorem 19, the homeomorphisms satisfying (3.2) coincide with the homeomorphisms

$f\colon D'\to D$ of class

$\mathcal Q_{n,n}(D',\omega; D)$ .

Some properties of the homeomorphisms of class $\mathcal Q_{p,q}(D',\omega)$ were studied in [27] (for $n-1<q<p=n$ , the value $\Psi_{q,n}(U)$ instead of $\Psi_{q,n}(U\setminus F)$ , and $\omega\equiv 1$ ), [21], [60]–[64] (all for $q=p=n$ and $\omega=Q$ ), [65], [66] (for $1<q=p<n$ and $\omega=Q$ ), and many others. In all articles mentioned except [27] the distortion of the geometry of condensers is stated in the language of moduli of curve families, which in a series of cases is a more restrictive characteristic than capacity as far as meaningful applications are concerned.

§ 4. Geometry the boundary

In this section, we consider geometric concepts and the main results of other approaches to the boundary behaviour problem.

Definition 4.1. The boundary $\partial D'$ of a domain $D'$ is called $(p,\omega)$ -weakly flat at $x_0 \in \partial D'$ , where $p > 1$ , if for every neighbourhood $U$ of $x_0$ and every number $\lambda > 0$ , there is a neighbourhood $V \subset U$ of $x_{0}$ such that for all continua⁵ $F_0$ and $F_1$ in $D'$ , intersecting $\partial U$ and $\partial V$ , the capacity of the condenser $\mathcal E=(F_1,F_0)$ satisfies $\operatorname{cap} (\mathcal E; L_p(D',\omega)) \geqslant \lambda$ . The boundary $\partial D'$ is called $(p,\omega)$ -weakly flat whenever it is $(p,\omega)$ -weakly flat at each of its points.

A point $x_0 \in \partial D'$ is called $(p,\omega)$ -strongly accessible, where $p > 1$ , if for every neighbourhood $U$ of $x_0$ , there exist a neighbourhood $V \subset U$ of this point, a compact set $F_0 \subset D'$ , and a number $\delta > 0$ , such that for all continua $F_1$ in $D'$ intersecting $\partial U$ and $\partial V$ the capacity of the condenser $\mathcal E=(F_1,F_0)$ is bounded from below: $\operatorname{cap} (\mathcal E; L_p(D',\omega)) \geqslant \delta$ . The boundary $\partial D'$ is called $(p,\omega)$ -strongly accessible whenever each of its points is $(p,\omega)$ -strongly accessible.

In the unweighted case, for $p = n$ , the properties of the boundary to be weakly flat and strongly accessible are introduced in [21], § 3.8, in terms of moduli of curve families. These conditions generalize properties $P1$ and $P2$ of [18], § 17, and the properties of the boundary to be quasiconformally flat and quasiconformally accessible [16]. The case of arbitrary $p >n-1$ is considered, for instance, in [67].

Proposition 4.2. Suppose that $1\leqslant p <\infty$ . If a domain $D'\subset \mathbb{R}^n$ , where $n\geqslant 2$ , has $(p,\omega)$ -weakly flat boundary and $\omega\in L_{1,\mathrm{loc}}(D')$ , then

(1) the boundary $\partial D'$ is $(p,\omega)$ -strongly accessible;

(2) $D'$ is locally connected at the boundary points.

Proof. The proof follows is similar to that of Proposition 3.1 and Lemma 3.15 of [21] with obvious adjustments.

Remark 4.3. In the unweighted case the modulus and capacity coincide [68]–[70], and hence the properties of the boundary to be weakly flat and strongly accessible of [21] precisely coincide with the case of trivial weight and $p = n$ in Definition 4.1 of $(n,1)$ -weakly flat and $(n,1)$ -strongly accessible boundary.

Moreover, a point $x_0\in \partial D'$ is $(n,1)$ -strongly accessible whenever it is quasiconformally accessible [16], Definition 1.7: given a neighbourhood $U$ of $x_0$ , there are a continuum $F_0 \subset D'$ and a number $\delta > 0$ such that $\operatorname{cap} ((F_1,F_0); L^1_p(D',\omega)) \geqslant \delta$ for all connected sets $F_1$ in $D'$ satisfying $x_0\in\overline F_1$ and $F_1 \cap \partial U \neq \varnothing$ .

Note the following connection between the singleton support of a boundary element and the above conditions on the geometry of the boundary.

Proposition 4.4. Given a weight $\omega$ and a domain $D'$ satisfying Remark 2.42, let $h\in H^\omega_{\rho,p}(D')$ a boundary element and let a point $x_0\in S_h$ be $(p,\omega)$ -strongly accessible in the sense of Definition 4.1. Then $\mathcal{S}_h = \{x_0\}$ .

Proof. Assume on the contrary that

$x_0$ is

$(p,\omega)$ -strongly accessible and there exists a point

$y_0 \in \mathcal{S}_h$ with

$d(x_0,y_0)\geqslant \alpha>0$ . By the definition of the support of a boundary element, there exist fundamental sequences

$\{x_m \,{\in}\, D'_{\rho,p}\}$ and

$\{y_m \,{\in}\, D'_{\rho,p}\}$ with respect to the metric

$\rho^{\omega}_{p,F_0}$ such that

$x_m \to x_0$ and

$y_m\to y_0$ in the topology of the extended Euclidean space. We fix a neighbourhood

$V \subset U = B(x_0,\alpha/3)$ of

$x_0$ , a compact set

$F_0 \subset D'$ , and a number

$\delta > 0$ according to Definition 4.1. Find a number

$m_0$ such that

$x_m \in V$ and

$y_m \in B(y_0,\alpha/3)$ for all

$m\geqslant m_0$ . It is obvious that for

$m\geqslant m_0$ every curve

$\overline{x_m y_m}$ crosses

$\partial V$ and

$\partial U$ , and so, since the image of the curve is a continuum, the definition of strong accessibility yields

$\operatorname{cap}((\overline{x_m y_m},F_0); L_p(D',\omega)) \geqslant \delta$ .

By the definition of the capacity metric (2.7), among the mentioned continua with endpoints $x_m \in V$ and $y_m \in B(y_0,\alpha/3)$ . there is $\gamma_m = \overline{x_m y_m}$ such that

$\begin{equation} \rho^{\omega}_{p,F_0}(x_m, y_m) \geqslant \operatorname{cap}\bigl((\gamma_m,F_0); L_p(D',\omega)\bigr) - \frac{\delta}{2^m} \geqslant \delta\biggl(1-\frac{1}{2^m}\biggr). \end{equation} \tag{4.1}$

On the other hand,

$x_0$ ,

$y_0 \in \mathcal{S}_h$ implies that the sequences

$\{x_m \in D'_{\rho,p}\}$ and

$\{y_m \in D'_{\rho,p}\}$ are equivalent. Therefore,

$\rho^{\omega}_{p,F_0}(x_m, y_m) \to 0$ , which contradicts (4.1). Proposition 4.4 is proved.

Corollary 4.5 of Theorem 2.19 ([25], [26], Chap. 5, Theorem 1.3, [17], Theorem 10.4). Consider two domains $D$ and $D'$ in $\mathbb{R}^n$ , where $n \geqslant 2$ . Every quasiconformal mapping $f\colon D' \to D$ admits a homeomorphic extension to the capacity boundary

$\begin{equation*} \widetilde f\mid_{H_{\rho,n}(D')}\colon \bigl(H_{\rho,n}(D'), \widetilde\rho_{n,F_0}\bigr)\to \bigl(H_{\rho,n}(D), \widetilde\rho_{n,f(F_0)}\bigr). \end{equation*} \notag$

Proof. By Definition 1.4, the quasiconformal mapping belongs to

$\mathcal{Q}_{n,n}(D',1;D)$ . The claim follows directly from Theorem 2.22.

Corollary 4.6 of Theorem 2.38. Consider two domains $D$ and $D'$ in $\mathbb{R}^n$ , where $n \geqslant 2$ , and a homeomorphism $f\colon D' \to D$ satisfying one of the following conditions:

(1) $f$ is quasiconformal, $D'$ is locally connected on the boundary, and $\partial D$ is quasiconformally accessible [16], Theorem 2.4.

(2) $f\in\mathcal Q_{n,n}(D',\omega;D)$ , in particular, $f$ is an $\omega$ -homeomorphism in the sense of Remark 3.1, for⁶ $\omega\in \mathrm{BMO}(\overline{D'})$ , $D'$ is locally connected on the boundary, and $\partial D$ is $(n,1)$ -strongly accessible [21], Lemma 5.3.

Then $f$ admits a continuous extension $\overline f \colon \overline{D'} \to \overline{D}$ to the boundary.

Proof. Verify that the hypotheses of Corollary 2.39 hold in both cases, and so

$f\colon D' \to D$ extends by continuity to the closure

$\overline{D'}$ .

In case (1), for every point $x\in \overline{D'}$ we have $\operatorname{cap}((\{x\}, F_0); L^1_n(D'))=0$ . Since every quasiconformal mapping is of class $\mathcal{Q}_{n,n}(D',1;D)$ , it remains to verify that if $x\in\mathcal{S}_{h}$ and $h\in H_{\rho,n}(D')$ , then the support $\mathcal{S}_{\widetilde f(h)}$ of the boundary element $\widetilde f(h)$ is a singleton, where $\widetilde f$ is the extension of $f$ of Theorem 2.19. The latter follows from the quasiconformal accessibility of $\partial D$ , Proposition 4.4, and Remark 4.3. The possibility of extending the mapping $f$ by continuity to $\partial D'$ follows from Corollary 2.39.

In case (2), we observe first that $\operatorname{cap}((\{x\}, F_0); L^1_p(D';\omega))\,{=}\,0$ for every boundary point $x\in \partial D'$ (see Example 2.9), and this property is local. Hence, it is independent of the continuum $F_0$ . Moreover, by Remark 3.1, the $\omega$ -homeomorphism $f$ belongs to $\mathcal{Q}_{n,n}(D',\omega;D)$ . As above, Proposition 4.4 shows that the support $\mathcal{S}_{\widetilde f(h)}$ of the boundary element $\widetilde f(h)$ is a singleton, and Corollary 2.39 guarantees the required result. Corollary 4.6 is proved.

Remark 4.7. In the planar case, $n=2$ , the capacity boundary $H_{\rho,2}$ with respect to the Sobolev class $L^1_2$ is homeomorphic to the boundary of prime ends, see [71], for instance. In the space $\mathbb{R}^n$ , where $n\geqslant 3$ , it is known that for the domains quasiconformally equivalent to a domain with locally quasiconformal boundary, called regular domains, the completion in the prime ends topology is equivalent to the completion in the modular [17] and capacity [26] metrics.

Example 4.8. Consider the domain $D' = [0,1]^3 \subset \mathbb{R}^3$ , the weight $\omega(y) = y_1^{\beta}$ with $\beta>-3$ , and the ridge domain from Example 2.18:

$\begin{equation*} D = \{x=(x_1,x_2,x_3) \colon |x_2| < x_1^{\alpha},\, 0<x_1, x_3<1\}\subset \mathbb{R}^3, \qquad \alpha>2. \end{equation*} \notag$

Next, consider the mapping

$f$ whose inverse

$\varphi(x)=f^{-1}(x)$ is defined as

$\begin{equation*} \varphi(x)= \begin{pmatrix} x_1 \\ x_2 x_1^{\alpha}\\ x_3 \end{pmatrix} \colon D \to D'. \end{equation*} \notag$

It is not difficult to verify that

$\begin{equation*} \begin{gathered} \, |D\varphi(x)| \approx \max \{1, \alpha x_2x_1^{\alpha-1}, x_1^{\alpha}\} \approx 1 \quad \text{and} \quad \det J(x,f) = x_1^{\alpha}, \\ K^{1,\omega}_{3,3}(x,\varphi) \approx x_1^{-(\beta+\alpha)/3}\in L_{\infty}(D) \quad\text{for} \quad \beta+\alpha\leqslant 0. \end{gathered} \end{equation*} \notag$

Theorem 1.6 shows that

$f$ is of the class

$\mathcal{Q}_{3,3}(D',\omega;D)$ and Theorem 2.19 can be applied to it: there exists a continuous extension

$f\colon (\widetilde{D}'_{\rho,3},\widetilde\rho^{\,\omega}_{3,F_0}) \to(\widetilde{D}_{\rho,3},\widetilde\rho_{3,f(F_0)})$ .

As far as the authors are aware, this example cannot be handled in the framework of other articles concerning boundary correspondence. For instance, [13], [22] require that the boundary of the domain $D$ be $(n,1)$ -strongly accessible. In the case of $D$ under consideration, the ridge is neither $(n,1)$ -weakly flat nor $(n,1)$ -strongly accessible for $\alpha>2$ . Indeed, [16], Example 5.5, shows that the points on the ridge are quasiconformally accessible if and only if $1<\alpha<2$ and are not quasiconformally flat for any $\alpha>1$ . In addition, it is not difficult to verify that necessary conditions for the ridge to be quasiconformally flat and quasiconformally accessible are also necessary for the ridge to be $(n,1)$ -weakly flat and $(n,1)$ -strongly accessible, see [16], Theorems 5.3, 5.4.

§ 5. Applications

In this section, we apply the results on boundary behaviour to the homeomorphisms of certain classes $\mathcal Q_{p,q}(D',\omega;D)$ considered in the examples of this article.

5.1. The homeomorphism of Example 1.13

The following mapping is considered in [31].

For $n-1< s<\infty$ , consider a homeomorphism $f \colon D' \to D$ of open domains $D'$ , $D\subset \mathbb{R}^n$ , where $n\geqslant 2$ , such that

(1) $f\in W^1_{n-1, \mathrm{loc}}(D')$ ;

(2) the mapping $f$ has finite distortion;

(3) the outer distortion function

$\begin{equation} D'\ni y \mapsto K^{1,1}_{n-1,s}(y,f) = \begin{cases} \dfrac{|Df(y)|}{|{\det Df (y)}|^{1/s}} &\text{if }\det Df (y)\neq 0, \\ 0 &\text{if }\det Df (y) = 0 \end{cases} \end{equation} \tag{5.1}$

belongs to

$L_{\sigma}(D)$ , where

$\sigma=(n-1)p$ and

$p=s/(s-(n-1))$ .

According to Theorem 4 in [28], the inverse homeomorphism $\varphi=f^{-1}\colon D\to D'$ has the following properties:

(4) $\varphi\in W^1_{p, \mathrm{loc}}(D)$ , $p=s/(s-(n-1))$ ;

(5) $\varphi$ has finite distortion.

The original homeomorphism $f\colon D'\to D$ has the following properties:

(6) it is of class $\mathcal Q_{p,p}(D',\omega;D)$ with the constant $K_p=1$ [31], Corollary 26, and the weight function $\omega\in L_{1,\mathrm{loc}}(D')$ defined as

see [31], formula (37), where

$Z'=\{y\in D'\colon Df(y)=0\}$ ;

(7) if $p>n-1$ (which corresponds to $s<n+1/(n-2)$ ), then the composition operator

$\begin{equation*} f^*\colon L^1_{p'}(D)\cap \operatorname{Lip}_{\mathrm{loc}}(D)\to L^1_{p'}(D';\theta) \end{equation*} \notag$

is bounded, where

$p'=p/(p-(n-1))$ and

$\theta(y)=\omega^{-(n-1)/(p-(n-1))}(y)$ .

Proposition 5.1. The results of this article concerning the boundary behaviour of homeomorphisms, namely, Theorems 2.19 and 2.37, Corollaries 2.38 and 2.39, are applicable to the mapping $f$ of § 5.1.

Explicitly, for $n\leqslant s < n+1/(n-2)$ the homeomorphism $f$ introduced above has the following properties:

(1) the mapping $f$ induces a Lipschitz mapping $f\colon ({D}'_{\rho,p},\widetilde\rho^{\,\omega}_{p,F_0}) \to({D}_{\rho,p},\widetilde\rho_{p,f(F_0)})$ of metric spaces: $\widetilde\rho_{p,f(F_0)}(f(x),f(y))\leqslant \widetilde\rho^{\,\omega}_{p,F_0}(x,y)$ for all points $x,y\in {D}'_{\rho,p}$ ;

(2) the mapping $f$ induces a Lipschitz mapping $\widetilde f\colon ({\widetilde D}'_{\rho,p},\widetilde\rho^{\,\omega}_{p,F_0}) \to({\widetilde D}_{\rho,p},\widetilde\rho_{p,f(F_0)})$ of “completed” metric spaces:

to $X\in( {\widetilde D}'_{\rho,p},\widetilde\rho^{\,\omega}_{p,F_0})$ associate $\widetilde f(X)\in ({\widetilde D}_{\rho,q},\widetilde\rho_{q,f(F_0)})$ , which contains the fundamental sequence $\{f(x_l)\}$ , where $\{x_l\}\in X$ :

$\begin{equation*} \widetilde\rho_{p,f(F_0)}\bigl(\widetilde f(X),\widetilde f(Y)\bigr)\leqslant \widetilde\rho^{\,\omega}_{p,F_0}(X,Y) \end{equation*} \notag$

for

$X,Y\in {\widetilde D}'_{\rho,p}$ ;

(3) the restriction $\widetilde f\mid_{H^\omega_{\rho,p}(D')}\colon (H^\omega_{\rho,p}(D'), \widetilde\rho^{\,\omega}_{p,F_0})\to (H_{\rho,p}(D), \widetilde\rho_{p,f(F_0)})$ is a Lipschitz mapping of capacity boundaries;

(4) if the domain $D'$ is locally $\mu$ -connected at a boundary point $y\in\partial D'$ , the support $\mathcal{S}_{h}$ of the boundary element $h\in H^\omega_{\rho,p}(D')$ contains $y$ , and

$\begin{equation*} \operatorname{cap}\bigl((\{y\}, F_0); L^1_p(V_i,D';\omega)\bigr)=0, \end{equation*} \notag$

where

$V_i$ is the connected component associated with

$\mathcal{S}_{h}$ at

$y$ , then

$f(z)\to \mathcal{S}_{\widetilde f(h)}$ as

$z\to y$ with

$z\in V_i\cap D'$ in the topology of the extended space

$\mathbb R^n$ ;

(5) if the domain $D'$ is locally $\mu$ -connected at a boundary point $y\in\partial D'$ , the support $\mathcal{S}_{h}$ of the boundary element $h\in H^\omega_{\rho,p}(D')$ contains $y$ and

$\begin{equation*} \operatorname{cap}\bigl((\{y\}, F_0); L^1_p(V_i,D';\omega)\bigr)=0, \end{equation*} \notag$

where

$V_i$ is the connected component associated with

$\mathcal{S}_{h}$ at

$y$ and

$\mathcal{S}_{\widetilde f(h)}=\{x\}\in \partial D$ , then the mapping

$f \colon D'\to D$ extends by continuity to

$y\in\partial D'$ and

$\begin{equation*} \lim_{z\to y,\,z\in V_i\cap D'}f(z)=x; \end{equation*} \notag$

(6) if the domain $D'$ is locally connected at $y\in\partial D'$ and

$\begin{equation*} \operatorname{cap}\bigl((\{y\}, F_0); L^1_p(D';\omega)\bigr)=0, \end{equation*} \notag$

then

$y$ lies in the support

$\mathcal{S}_{h}$ of some boundary element

$h\in H^\omega_{\rho,p}(D')$ ;

(7) if $\mathcal{S}_{\widetilde f( h)}=\{x\}\in \partial D$ , then the mapping $f \colon D'\to D$ extends by continuity to $y\in\mathcal{S}_{h}$ of the boundary element $h\in H^\omega_{\rho,p}(D')$ and

$\begin{equation*} \lim_{z\to y,\,z\in D'} f(z)=x \quad \textit{for every points}\quad y\in \mathcal{S}_{h}. \end{equation*} \notag$

Let us compare the above example with the mapping of [72], which considers a $W^{1}_{1, \mathrm{loc}}$ -homeomorphism $f\colon D' \to D$ with finite distortion, whose outer distortion function

$\begin{equation} K^{1,1}_{n,n}(y,f) = \begin{cases} \dfrac{|Df(y)|}{|{\det Df (y)}|^{1/n}} &\text{if }\det Df (y)\neq 0, \\ 0 &\text{if } \det Df (y)=0 \end{cases} \end{equation} \tag{5.3}$

belongs to

$L_{(n-1)n, \mathrm{loc}}(D')$ .

Verify that this mapping is a particular case for $s=n$ of the scale mapping considered above: $f\in W^1_{n-1, \mathrm{loc}}(D')$ with the distortion function (5.1). To this end, we have to show that the $W^{1}_{1, \mathrm{loc}}$ -homeomorphism $f\colon D' \to D$ is of class $f\in W^1_{n-1, \mathrm{loc}}(D')$ . To verify the last property, observe that $f$ induces the composition operator

$\begin{equation*} f^*\colon L^1_{n}(D)\cap \operatorname{Lip}_{\mathrm{loc}}(D) \to L^1_{n-1, \mathrm{loc}}(D') \end{equation*} \notag$

in the sense that

$u\circ f \in L^1_{n-1, \mathrm{loc}}(D')$ for every function

$u\in L^1_{n}(D)\cap \operatorname{Lip}_{\mathrm{loc}}(D)$ .

Indeed, consider a compactly embedded domain $U\Subset D'$ . Take $u\in {L}^1_n(f(U)) \cap \operatorname{Lip}_{\mathrm{loc}}(f(U))$ . The composition $u\circ f$ clearly lies in $\mathrm{ACL}(U)$ . Let us show that the derivatives of the composition are integrable. We can find the derivative of the composition as

$\begin{equation*} \frac{\partial (u\circ f)}{\partial y_i}(y)= \sum_{j=1}^n\frac{\partial u}{\partial x_j}(f(y))\, \frac{\partial f_j}{\partial y_i}(y) \end{equation*} \notag$

provided that

$f(y)$ is a point of differentiability of

$u$ and

$\partial (u\circ f)(y)/\partial y_i=0$ otherwise because in this case,

$y\in Z'$ and

$Df(y)=0$ a.e. Since the distortion function (5.3) is of class

$L_{(n-1)n}(U)$ , we have

$\begin{equation} \int_{U} |\nabla(u\circ f)(y)|^{n-1}\,dy \nonumber \end{equation} \notag$

$\begin{equation} \qquad\leqslant \int_{U\setminus (Z'\cup \Sigma')}|\nabla u(f(y))|^{n-1}\det Df(y)^{(n-1)/n} \cdot\frac{|Df(y)|^{n-1}}{\det Df(y)^{(n-1)/n}} \,dy \end{equation} \tag{5.4}$

$\begin{equation} \qquad\leqslant \biggl(\int_{U\setminus (Z'\cup \Sigma')}|\nabla u(f(y))|^n\det Df(y)\,dy\biggr)^{(n-1)/n} \nonumber \end{equation} \notag$

$\begin{equation} \qquad\qquad \times \biggl(\int_{U\setminus (Z'\cup\Sigma')} \biggl(\frac{|Df(y)|}{|{\det Df(y)}|^{1/n}}\biggr)^{(n-1)n} \,dy\biggr)^{1/n} \end{equation} \tag{5.5}$

$\begin{equation} \qquad=\|K^{1,1}_{n,n}(\,{\cdot}\,,f)\mid L_{(n-1)n}(U)\|^{n-1} \biggl(\int_{f(U)}|\nabla u(x)|^n\,dx\biggr)^{(n-1)/n}. \nonumber \end{equation} \notag$

To change from (5.4) to (5.5), we use Hölder’s inequality with the summability exponents

$n/(n-1)$ and

$n$ .

Furthermore, we note that $f(U)$ is a bounded open set, so that the coordinate function $u_j(x)\mapsto x_j$ lies in $L^1_n(f(U))$ . By (5.4), (5.5) the composition $(u_j\circ f)(y)=f_j(y)$ for $y\in D'$ is of class $f_j\in L^1_{n-1, \mathrm{loc}}(D')$ , for $j=1,\dots,n$ , while the mapping $f\colon D'\to D$ is of class $W^1_{n-1, \mathrm{loc}}(D')$ .

Therefore, the mapping of [72] satisfies all hypotheses of Example 1.13 with $s=n$ , and thus, the claim of Proposition 5.1 holds for it.

5.2. The homeomorphism of Example 1.16

Consider the mapping of Example 1.16 in the case that it is a homeomorphism. Then we have some homeomorphism $f\colon D'\to D$ of class $\mathcal{OD}(D';s,r;\theta,1)$ , where $n-1< s\leqslant r<\infty$ , with outer bounded $\theta$ -weighted $(s,r)$ -distortion, meaning that

(1) $f \in W^1_{n-1,\mathrm{loc}}(D')$ ;

(2) $f$ has finite distortion;

(3) the distortion function

$\begin{equation*} D' \ni x\mapsto K_{s,r}^{\theta,1}(x,f)= \begin{cases} \dfrac{\theta^{1/s}(x)|D f(x)|}{|{\det Df(x)}|^{1/r}} &\text{if } \det Df(x)\ne0, \\ 0 &\text{otherwise} \end{cases} \end{equation*} \notag$

is of class

$L_{\rho}(D')$ , where

$\rho$ is found from the condition

$1/\rho = 1/s-1/r$ and

$\rho = \infty$ for

$s=r$ .

Proposition 5.2. On assuming that $\omega(x)=\theta^{-(n-1)/(s-(n-1))}(x)$ is locally integrable, the homeomorphism $f\colon D'\to D$ of class $\mathcal{OD}(D';s,r;\theta,1)$ , where $n\leqslant s \leqslant r < n+ 1/(n+2)$ , belongs to the family $\mathcal Q_{p,q}(D',\omega;D)$ , where $q=r/(r-(n-1))$ and $p=s/(s-(n-1))$ with $n-1<q\leqslant p\leqslant n$ . Furthermore, the factors in the right-hand side of (1.8) are equal to $K_p=\|K_{r,r}^{\theta,1}(\,{\cdot}\,,f)\mid L_{\infty}(D')\|^{n-1}$ for $q=p$ and

$\begin{equation*} \Psi_{p,q}\bigl(Q(x,R)\setminus \overline{Q(x,r)}\bigr)^{1/\sigma}= \bigl\|K_{s,r}^{\theta,1}(\,{\cdot}\,,f)\bigm| L_{\rho}(Q(x,R)\setminus \overline{Q(x,r)})\bigr\|^{n-1}\quad \textit{for} \quad q<p, \end{equation*} \notag$

where

$1/\sigma=1/q-1/p=(n-1)/\varrho$ .

Therefore, Theorems 2.22 and 2.37 concerning boundary behaviour and their Corollaries 2.38 and 2.39 apply to the mapping $f\colon D'\to D$ . In particular, applying Corollary 2.39, we obtain the following proposition.

Proposition 5.3. Under the hypotheses of Proposition 5.2, assume that

(1) the domain $D'$ is locally connected at every point $y\in\partial D'$ and

$\begin{equation*} \operatorname{cap}\bigl((\{y\}, F_0); L^1_p(D';\omega)\bigr)=0, \end{equation*} \notag$

(2) the support $\mathcal{S}_{\widetilde f(h)}$ of the boundary element $\widetilde f(h)$ is a singleton: $\mathcal{S}_{\widetilde f( h)}=\{x\}\in \partial D$ , where $h\in H^\omega_{\rho,p}(D')$ is the boundary element containing $\{y\}$ .

Then we obtain an extension by continuity of the homeomorphism $f \colon D'\to D$ at the point $y$ of the support $\mathcal{S}_{h}$ of the boundary element $h\in H^\omega_{\rho,p}(D')$ such that

$\begin{equation*} \lim_{z\to y,\,z\in D'} f(z)=x \quad \textit{for every point}\quad y\in \mathcal{S}_{h}. \end{equation*} \notag$

A similar result is obtained in [67], Theorem 2, under stronger restrictions: $f\in W^1_{s,\mathrm{loc}}(D')$ , and so $n-1< s$ , condition (1) holds, but instead of condition (2) it is assumed that the points $x\in\partial D$ are $q$ -strongly accessible for $q=r /(r- (n- 1))$ . Recall that under this condition the support $\mathcal{S}_{h}$ of $x\in h$ is a singleton, see Proposition 4.4. Therefore, the fulfillment of the hypotheses of Theorem [67], Theorem 2, ensures that conditions (1) and (2) above hold. Then, there exists a continuous extension of the mapping $f \colon D'\to D$ to the Euclidean boundary.

Proposition 5.4. Assume the hypotheses of Proposition 5.2. If the domain $D'$ is locally connected at the boundary, while the boundary $\partial D$ is $q$ -weakly flat for $q=r /(r-(n-1))$ , then the mapping $f^{-1}$ admits a continuous extension $\widetilde{f}^{-1}\colon\overline{D} \to \overline{\mathbb{R}^{n}}$ .

Proof. Assume on the contrary that the mapping

$f^{-1}$ has no limit at some point

$x_{0} \in \partial D$ . Then there exist two distinct points

$y_1,y_2\in \partial D'$ and two sequences

$\{x_{1,k} \in D\}$ ,

$\{x_{2,k} \in D\}$ such that

$\begin{equation*} \lim_{x_{1,k}\to x_0} f^{-1}(x_{1,k}) = y_1 \neq y_2 = \lim_{x_{2,k}\to x_0} f^{-1}(x_{2,k}). \end{equation*} \notag$

Choose two balls

$B_i=B(y_i, r_i)$ , for

$i=1,2$ , satisfying

$\overline{B}_1 \cap \overline{B}_2=\varnothing$ . Since the domain

$D'$ is locally connected at the boundary, for the ball

$B_i$ there is a connected component of

$B_i \cap D'$ which includes

$U_i=B(y_i, \widetilde{r}_i) \cap D'$ for some

$\widetilde{r}_i \in(0, r_i)$ , for

$i=1,2$ .

Consider a positive number $h<\operatorname{dist}(B_1, B_2)$ . By the subordination principle, Property 1.2, the piecewise linear function $u$ defined as

$\begin{equation*} u(y)= \begin{cases} 1 &\text{for } y \in B(y_1, r_1) \cap D', \\ 0 & \text{for } y \in \mathbb{R}^{n} \setminus(B(y_1, r_1+h) \cap D') \end{cases} \end{equation*} \notag$

is admissible for the condenser

$E'=(F_1',F_2')$ for every continuum

$F_i'\Subset B_i \cap D'$ . Let

$P$ be a number such that

$P > C \|u \mid L^1_p (D', \omega)\|$ , where

$C$ is the constant in (1.9).

By construction, $x_{0} \in \overline{f(U_1)} \cap \overline{f(U_2)}$ . Suppose that $V$ is a neighbourhood of $x_{0}$ so small that

$\begin{equation*} f(U_i) \setminus V \neq \varnothing, \qquad i=1,2. \end{equation*} \notag$

Since

$\partial D$ is

$q$ -weakly flat, for some neighbourhood

$W \subset V$ of

$x_{0}$ and some continuum

$F_i {\subset}\, f(U_i)$ , for

$i\,{=}\,1,2$ , intersecting

$\partial V$ and

$\partial W$ , we have

$\begin{equation*} \operatorname{cap}^{1/q} ((F_1, F_2); L_q(D))\,{\geqslant}\, P. \end{equation*} \notag$

Let

$F_i'$ be such that

$F_i'=f(F_i)$ . Then the relations

$\begin{equation*} \begin{aligned} \, P &\leqslant \operatorname{cap}^{1/q}\bigl((F_1,F_2); L^1_q(D)\bigr) = \operatorname{cap}^{1/q}\bigl(f^{-1}(E'); L^1_q(D)\bigr) \\ &\leqslant C \operatorname{cap}^{1/p} (E'; L^1_p(D',\omega)) < P \end{aligned} \end{equation*} \notag$

lead to a contradiction. Proposition 5.4 is proved.

Some results similar to Propositions 5.2–5.4 were obtained in [67], Theorem 1, under stronger restrictions: $f\in W^1_{s,\mathrm{loc}}(D')$ and $s>n-1$ .



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Citation: S. K. Vodopyanov, A. O. Molchanova, “The boundary behavior of

$\mathcal Q_{p,q}$ -homeomorphisms”, Izv. Math., 87:4 (2023), 683–725

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\paper The boundary behavior of $\mathcal Q_{p,q}$-homeomorphisms

\jour Izv. Math.

\yr 2023

\vol 87

\issue 4

\pages 683--725

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This publication is cited in the following 4 articles:

S. K. Vodopyanov, S. V. Pavlov, “O granichnykh znacheniyakh v geometricheskoi teorii funktsii v oblastyakh s podvizhnymi granitsami”, Sib. matem. zhurn., 65:3 (2024), 489–516
Nataliya Ilkevych, Evgeny Sevost'yanov, Alexander Ukhlov, “On the Equicontinuity of Generalized Quasiconformal Mappings by Prime Ends”, Complex Anal. Oper. Theory, 18:4 (2024)
S. K. Vodopyanov, D. A. Sboev, “Semicontinuity under Convergence of Homeomorphisms in $L_{1,\operatorname{loc}}$ of the Operator Distortion Function”, Sib Math J, 65:4 (2024), 737
S. K. Vodopyanov, D. A. Sboev, “Polunepreryvnost operatornoi funktsii iskazheniya pri skhodimosti gomeomorfizmov v $L_{1, \mathrm{loc}}$ ”, Sib. matem. zhurn., 65:4 (2024), 605–621

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Introduction

§ 1. Classes of Qp,q\mathcal Q_{p,q}-homeomorphisms

1.1. Definitions of Sobolev spaces and the capacity of condensers

1.2. Condensers and their capacity in Sobolev spaces

1.3. A quasi-additive set function and its properties

1.4. Definition of the class of \mathcal Q_{p,q}(D',\omega;D)\mathcal Q_{p,q}(D',\omega;D)-homeomorphisms and their properties

§ 2. Behaviour of mappings with respect to the capacity metric

2.1. Capacity metric functions in domains for the homeomorphisms of class \mathcal Q_{p,q}(D',\omega;D)\mathcal Q_{p,q}(D',\omega;D) for n-1<q\leqslant p\leqslant nn-1<q\leqslant p\leqslant n

2.2. Capacity metric and completion of the domain

2.3. Capacity boundary. Boundary correspondence of mappings

2.4. Support of a boundary element

2.5. Continuous extension of mappings of class \mathcal{Q}_{p, q}(D',\omega;D)\mathcal{Q}_{p, q}(D',\omega;D) to the Euclidean boundary

§ 3. Moduli of curve families and homeomorphisms of class \mathcal Q_{p,q}(D',\omega)\mathcal Q_{p,q}(D',\omega)

§ 4. Geometry the boundary

§ 5. Applications

5.1. The homeomorphism of Example 1.13

5.2. The homeomorphism of Example 1.16

Bibliography

§ 1. Classes of $\mathcal Q_{p,q}$ -homeomorphisms

1.4. Definition of the class of $\mathcal Q_{p,q}(D',\omega;D)$ -homeomorphisms and their properties

2.1. Capacity metric functions in domains for the homeomorphisms of class $\mathcal Q_{p,q}(D',\omega;D)$ for $n-1<q\leqslant p\leqslant n$

2.5. Continuous extension of mappings of class $\mathcal{Q}_{p, q}(D',\omega;D)$ to the Euclidean boundary

§ 3. Moduli of curve families and homeomorphisms of class $\mathcal Q_{p,q}(D',\omega)$