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This article is cited in 2 scientific papers (total in 2 papers) Coincidence of set functions in quasiconformal analysis S. K. Vodopyanov Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
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     § 1. IntroductionIt is known that quasiconformal mappings have several equivalent descriptions (for instance, see [1] and [2]): a metric one, an analytic one, a geometric description in the language of moduli (see [2]), a geometric description in the language of capacities (see [3]), a functional one (see [4]). Associated with each of these descriptions is a set function defined on open subsets of the domain of definition of the mapping. The main result of our paper is that all these set functions coincide. This result is apparently new even for classical quasiconformal mappings (of course, apart from conformal ones). More precisely, given a homeomorphism φ:D→D′ of domains D,D′⊂Rn, n⩾2, assume that it induces (by the change of variables formula φ∗(f)=f∘φ) a bounded composition operator1 with parameters n−1<q⩽p<∞ for n⩾3, or 1⩽q⩽p<∞ for n=2 and with weight function ω∈L1,loc(D′).Then the following hold. I. For q<p the set functions2
are equal; for each open set W⊂D′.II. For q=p the two quantities,
are equal: For classical quasiconformal mappings (when q=p=n and ω≡1) the last equality says that the norm of the composition operator coincides withProblems in quasiconformal analysis stated more than 50 years ago has led to the concept of mappings characterized by a controllable change of the capacity of the image of a condenser in terms of the weighted capacity of the condenser in the source space: for instance, see [5]. At the same time authors from Finland, Ukraine, Israel and other countries investigated maps (see [6]) whose definition, in place of capacity, involved another geometric characteristic, the modulus of a family of curves. Note that for arbitrary weights there are no two-sided estimates connecting weighted capacity with weighted modulus (with the exception of weights in the Muckenhoupt class \mathcal A_n; see [7]–[9]). So both approaches were investigated independently, particularly since the formal impression was that by using the modulus-based approach (see [6]) one can treat a wider class of maps than in the framework of the capacity-based approach. The central result of our paper (cf. [10]) is that, in fact, whether we use the capacity-based characteristic or the modulus-based characteristic, we obtain the same classes of maps. A striking feature of this result is that we do not obtain it by comparing a weighted capacity and a weighted modulus, but we use a functional analytic description of mappings characterized by a controllable change of the capacity in the target space in terms of the capacity in the source space, which was developed in 2020 (see [11]–[14]). In § 3 we define two other set functions (one is based on bounds for capacities and the other on bounds for moduli) and show that, depending on their parameters, they either coincide with those in (1.1) or with those in (1.2). This paper can be viewed s a natural part of the cycle [11]–[14], and also of the cycle [15]–[19], where the history of the problem was exposed and a comprehensive bibliography was presented. Before those papers [4], [20] and [21] were published, where methods from the theory of Sobolev function spaces (see [22] and [23]) and geometric function theory (see [1]–[3] and [24]–[27]) were synthesized. Some results obtained in that cycle of papers found applications to nonlinear elasticity theory: see [28]. § 2. Classes of \mathcal Q_{q,p}-homeomorphismsHere and below D and D' are domains (connected open sets) in \mathbb{R}^n. 2.1. The definitions of Sobolev spaces and of condenser capacityRecall that a function u\colon D\to\mathbb R belongs to the Sobolev class L^1_{p}(D) if u is locally integrable in D (that is, u\in L_1(U) for each compactly embedded subdomain U\Subset D), has generalized derivatives {\partial u}/{dx_j}\in L_{1,\mathrm{loc}}(D) for each j=1,\dots,n and has a 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
A map \varphi=(\varphi_1,\dots, \varphi_n) belongs to the Sobolev class W^1_{p,\mathrm{loc}}(D; \mathbb{R}^n) if {\varphi_j(x) \in L_{p,\mathrm{loc}}(D)} and the generalized derivatives {\partial\varphi_j}/{dx_i} belong to L_{p,\mathrm{loc}}(D) for all {j,i=1,\dots,n}. A map \varphi\colon D\to \mathbb R^n in the Sobolev class W^1_{1,\mathrm{loc}}(D;\mathbb{R}^n) is called a map with bounded distortion if3
\begin{equation*}
D\varphi(x)=0 \quad\text{a.e. on } Z=\{x\in D\colon \det D\varphi (x)=0\}.
\end{equation*}
\notag
Here and throughout, D\varphi (x)=({\partial\varphi_j}/{\partial x_i}(x)) is the Jacobian matrix of \varphi at x\in D, |D\varphi (x)| is the Euclidean operator norm of this matrix and \det D\varphi (x) is its determinant (Jacobian). A locally integrable function \omega\colon D'\!\to\!\mathbb R is called a weight function if {0\!<\!\omega(y)\!<\!\infty} for almost all y\in D'. Recall that u\colon D'\to\mathbb R belongs to the weighted Sobolev class L^1_{p}(D';\omega), p\in[1,\infty), if u is locally integrable in D' and has generalized derivatives {\partial u}/{\partial y_j} in D' that belong to L_{p}(D';\omega) for all j=1,\dots,n. The seminorm of u\in L^1_{p}(D';\omega) is the quantity
\begin{equation*}
\|u\mid L^1_{p}(D';\omega)\|=\biggl(\int_{D'}|\nabla u|^p(y)\omega(y)\,dy\biggr)^{1/p}.
\end{equation*}
\notag
For \omega\equiv 1 we simply write L^1_{p}(D') instead of L^1_{p}(D';1). In what follows we let \operatorname{Lip}_{\mathrm{loc}}(D') denote the space of locally Lipschitz functions on the domain D'. Clearly, \operatorname{Lip}_{\mathrm{loc}}(D')=W^1_{\infty,\mathrm{loc}}(D')\cap C(D'). Recall that a homeomorphism \varphi \colon D \to D' between domains D and D' in \mathbb{R}^n induces a 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
which acts by the formula D\ni x\mapsto(\varphi^*u)(x)=u(\varphi(x)), provided that for some constant K_{q,p}<\infty we have
\begin{equation*}
\begin{gathered} \, \|\varphi^*u\mid L^1_q(D)\|\leqslant K_{q,p}\|u\mid L^1_p(D';\omega)\| \\ \text{for each } u\in L^1_p(D')\cap \operatorname{Lip}_{\mathrm{loc}}(D'). \end{gathered}
\end{equation*}
\notag
Definition 1. A condenser in a domain D\subset \mathbb{R}^n is a pair E=(F_1,F_0) of connected compact sets (continua) in D: F_1,F_0\subset D. If F lies in a connected compactly embedded subset U\Subset D, then we denote the condenser E=(F,\partial U) by E=(F,U). A condenser E=(F,\partial U) is called a ring condenser if U\Subset D' is a connected open set and F\subset U is a continuum such that \mathbb R^n\setminus F is a connected open set and the complement \overline{\mathbb R^n}\setminus (U\setminus F) has two connected components, F and \overline{\mathbb R^n}\setminus U (here \overline{\mathbb R^n}=\mathbb R^n\cup\{\infty\} is a one-point compactification of \mathbb R^n). A ring condenser E=(F,\partial U) in \mathbb R^n is said to be spherical (cubic) if U is a ball4 B(x,R)=\{y\in\mathbb R^n\colon |y-x|_2< R\} (a cube Q(x,R)=\{y\in\mathbb R^n\colon |y-x|_\infty< R\}) and F\subset U is the closure F=\{y\in\mathbb R^n\colon |y-x|_2\leqslant r\} of a ball B(x,r) (the closure F=\{y\in\mathbb R^n\colon |y-x|_\infty\leqslant r\} of a cube Q(x,r)), where r\in (0,R). A continuous function u\colon D\to\mathbb R in the class W^1_{1,\mathrm{loc}}(D) is said to be admissible for a condenser E=(F_1,F_0)\subset D if u\equiv 1 on F_1 and u \equiv 0 on F_0. We denote the set of functions admissible for E=(F_1,F_0) by \mathcal A(E). We define the capacity of the condenser E=(F_1,F_0) in the space L^1_q(D), {q\in[1,\infty)}, by
\begin{equation*}
\operatorname{cap}(E; L^1_q(D))=\inf_{u\in \mathcal A(E)}\|u\mid L^1_{q}(D)\|^q;
\end{equation*}
\notag
here we take the lower bound over all functions admissible for E = (F_1,F_0) \subset D that belong to L^1_{q}(D). The weighted capacity of a condenser E=(F_1,F_0)\subset D' in L^1_p(D';\omega) is defined similarly, by
\begin{equation*}
\operatorname{cap}(E; L^1_p(D';\omega))=\inf_{u\in \mathcal A(E)\cap\operatorname{Lip}_{\mathrm{loc}}(D')}\|u\mid L^1_{p}(D';\omega)\|^p,
\end{equation*}
\notag
where the lower bound is taken over all functions admissible for E=(F_1,F_0) that belong to \operatorname{Lip}_{\mathrm{loc}}(D')\cap L^1_{p}(D';\omega). For details on the properties of weighted capacities (for a certain special class of weight functions), see. [23], Ch. 2. The following principle is a direct consequence of the definition of capacity, Subordination principle. Let E'=(F'_1,F'_0) and E=(F_1,F_0) be two condensers in a domain D' such that the plates of the first condenser lie in the plates of the second: F'_1\subset F_1 and F'_0\subset F_0. Then
\begin{equation*}
\operatorname{cap}(E'; L^1_p(D';\omega))\leqslant\operatorname{cap}(E; L^1_p(D';\omega)).
\end{equation*}
\notag
A similar property also holds for the q-capacity of a condenser in a domain D. To prove the subordination principle it is sufficient to observe that
\begin{equation*}
\mathcal A(E')\cap \operatorname{Lip}_{\mathrm{loc}}(D')\cap L^1_{p}(D';\omega)\subset \mathcal A(E)\cap \operatorname{Lip}_{\mathrm{loc}}(D')\cap L^1_{p}(D';\omega).
\end{equation*}
\notag
The definition of the class of \mathcal Q_{q,p}-homeomorphisms presented in § 2.4 below is based on an estimate for the capacity of the image in D' of a cubic condenser in terms of the weighted capacity of the original condenser. 2.2. A quasiadditive set function and its propertiesLet {\mathcal O}(D) be a system of open subsets of D with the following properties:
The choice of the ball or a cube in this definition depends on the choice of the system of elementary sets used for the differentiation of set functions (see Proposition 1 below). Definition 2. The map \Phi\colon {\mathcal O}(D)\to[0,\infty] is called a quasiadditive set function if 1) for each point x\in D there exists \delta, 0<\delta<\mathrm{dist}(x, \partial D), such that (if D=\mathbb R^n, then 0\leqslant\Phi(D(x,{\delta}))<\infty must hold for all \delta\in(0, \delta(x)), where \delta(x) is a positive number which can depend on x); we can replace balls by cubes in this condition;2) for each finite disjoint system \{U_i\in{\mathcal O}(D), i=1,\dots,l\} of open sets such that \bigcup_{i=1}^lU_i\subset U, where U\in{\mathcal O}(D), we have If for each finite system \{U_i\in{\mathcal O}(D)\} of mutually disjoint open sets we have
\begin{equation}
\sum_{i=1}^{n}\Phi(U_i)=\Phi\biggl(\bigcup_{i=1}^{n} U_i \biggr),
\end{equation}
\tag{2.1}
then the quasiadditive set function with the corresponding property is said to be finitely additive, and if (2.1) holds for each countable system \{U_i\in{\mathcal O}(D)\} of mutually disjoint open sets, then it is said to be countably additive. A function \Phi is monotone if \Phi(U_1)\leqslant \Phi(U_2) for U_1\subset U_2 \subset D, U_1,U_2\in{\mathcal O}(D). It is obvious that all quasiadditive set functions are monotone. A quasiadditive set function \Phi\colon {\mathcal O}(D)\to[0,\infty] is called a bounded quasiadditive set function if \sup_{U\in {\mathcal O}(D)}\Phi(U)<\infty. A quasiadditive set function \Phi as defined above is known to be differentiable in the following sense. Proposition 1 (see [30]–[32]). Let \Phi be a quasiadditive set function defined on some system {\mathcal O}(D') of open subsets of D'. Then 1) for almost all y\in D' the finite derivative 5
\begin{equation*}
\Phi'(y)= \lim_{\delta\to 0,\, y\in B_\delta}\frac{\Phi(B_\delta)}{|B_{\delta}|}
\end{equation*}
\notag
exists; 2) for each open set U\in \mathcal O(D') 2.3. The formula for a change of variableWe present, in the form we require, the change of variable formula from [13] and [14], which is a modification of the formula for a change of variable in a Lebesgue integral from [33] and [34]. Let \varphi\colon \Omega\to \mathbb R^n be a map and let E \subseteq D. Then the function \mathcal N (y, \varphi, E)\colon \mathbb R^n \to\mathbb {N} \cup \{0,\infty \} defined by
\begin{equation*}
\mathbb {R}^n\ni y\mapsto \mathcal N (y, \varphi, E) = \begin{cases} 0 \quad&\text{if $\varphi^{-1}(y) \cap E$ is empty,}\\ \infty \quad&\text{if $\varphi^{-1}(y) \cap E$ is infinite,}\\ \#(\varphi^{-1}(y) \cap E) \quad&\text{otherwise}, \end{cases}
\end{equation*}
\notag
is called the Banach indicatrix of \varphi. Here the symbol \#(\varphi^{-1}(y) \cap E) stands for the number of points in the preimage \varphi^{-1}(y) \cap E of y. Proposition 2. Let \varphi\colon \Omega\to \mathbb R^n be a map in the Sobolev class W^1_{1,\mathrm{loc}}(\Omega) (or in the class \operatorname{ACL}(\Omega)). Then 1) there exists a Borel set \Sigma\subset \Omega of zero measure such that \varphi\colon \Omega\setminus\Sigma\to\mathbb R^n has the Luzin \mathcal N-property; 2) the functions
\begin{equation*}
\Omega\setminus\Sigma\ni x\mapsto (u\circ \varphi)(x) |{\det D\varphi (x)}|\quad\textit{and} \quad \mathbb R^n \ni y\mapsto u(y)\mathcal N(y,\varphi, \Omega\setminus \Sigma)
\end{equation*}
\notag
are measurable if so is u\colon \mathbb R^n \to\mathbb R; 3) if A\subset \Omega\setminus \Sigma is a measurable set, then the following area formula holds:
\begin{equation*}
\int_{A} |{\det D\varphi (x)}|\, dx=\int_{\mathbb{R}^{n}} \mathcal N (y, \varphi, A) \, dy;
\end{equation*}
\notag
4) if a measurable function h is nonnegative, then the integrands in (2.2) are measurable and the following formula for a change of variable in the Lebesgue integral holds:
\begin{equation}
\int_{\Omega\setminus \Sigma} u(\varphi(x))|{\det D\varphi (x)}|\,dx= \int_{\mathbb{R}^{n}}\sum_{x \in \varphi^{-1}(y)\setminus \Sigma} u(x)\, dy;
\end{equation}
\tag{2.2}
5) if one of the functions is integrable, then the other is too and2.4. The definition of the class of \mathcal Q_{q,p}(D',\omega;D)-homeomorphisms, and their propertiesAs the system of open sets \mathcal O_c(D') on which we define a quasiadditive set function \Psi we take the minimal system of open subsets of D' (see Definition 2) that contains
In the next definition and in Theorem 1, as a bounded quasiadditive set function we take a map \Phi\colon \mathcal O_c(D')\to[0,\infty). Definition 3 (see [11] and [14]). 1) Let D and D' be two domains in \mathbb R^n, {n\geqslant2}. We say that a homeomorphism f\colon D'\to D belongs to the class Q\mathcal{RQ}_{q,p}(D',\omega;D), where {1< q\leqslant p<\infty} for n\geqslant3, or 1\leqslant q\leqslant p<\infty for n=2 and \omega\in L_{1,\mathrm{loc}}(D') is a weight function, if there exists or such that for each cubic condenser E=(\overline{Q(x,r)}, Q(x,R)), 0<r<R, lying in D' whose image f(E)=(f(\overline{Q(x,r)}), f(Q(x,R)) lies in D we have
\begin{equation}
\begin{cases} \operatorname{cap}^{1/p}(f(E); L^1_p(D)) \leqslant K_p\operatorname{cap}^{1/p}(E; L^1_p(D';\omega)) &\text{for }q=p, \\ \operatorname{cap}^{1/q}(f(E); L^1_q(D)) \\ \ \ \leqslant \Psi_{q,p}(Q(x,R)\setminus \overline{Q(x,r)})^{1/\sigma} \operatorname{cap}^{1/p}(E; L^1_p(D';\omega)) &\text{for }q<p, \end{cases}
\end{equation}
\tag{2.4}
where 1/\sigma=1/q-1/p. 2) Let D and D' be domains in \mathbb R^n, n\geqslant2. We say that a homeomorphism is in the class \mathcal{Q}_{q,p}(D',\omega;D), where 1< q\leqslant p<\infty for n\geqslant3, or 1\leqslant q\leqslant p<\infty for n=2 and \omega\in L_{1,\mathrm{loc}}(D') is a weight function, if there exists (\widetilde{\rm{a}}) a constant \widetilde K_p>0 in the case q=p, or (\widetilde{\rm{b}}) a bounded quasiadditive function \widetilde\Psi_{q,p} defined on the system \mathcal O(D') of all open subsets of D' in the case q<p such that for each condenser E=(F_1,F_0) lying in D' and its image f(E)=(f(F_1), f(F_0)) lying in D we have
\begin{equation}
\begin{aligned} \, \notag &\operatorname{cap}^{1/q}(f(E); L^1_q(D)) \\ &\qquad\leqslant \begin{cases} \widetilde K_p \operatorname{cap}^{1/p}(E; L^1_p(D';\omega)) &\text{for }1<q=p<\infty, \\ \widetilde\Psi_{q,p}(D'\setminus(F_0\cup F_1))^{1/\sigma} \operatorname{cap}^{1/p}(E; L^1_p(D';\omega)) &\text{for }1<q<p<\infty, \end{cases} \end{aligned}
\end{equation}
\tag{2.5}
where {1}/{\sigma}={1}/{q}-{1}/{p}. 3) If (2.5) holds for all ring condensers E=(F,\partial U) in D', then we say that the mapping f belongs to the class \mathcal{RQ}_{q,p}(D',\omega;D). The following inclusions are obvious: Definition 4. Let \varphi\colon D\to D' be a homeomorphism between domains D, D'\subset \mathbb R^n that induces a 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
where \omega\colon D'\to (0,\infty) is a locally integrable weight function. Fix an arbitrary open set W\subset D' and consider the restriction of \varphi^* to the subspace6
\begin{equation}
\mathcal R(W)={L}^1_p(W;\omega) \cap \mathring{\operatorname{Lip}}_{\mathrm{loc}}(W)\subset{L}^1_p(D';\omega) \cap \operatorname{Lip}_{\mathrm{loc}}(D').
\end{equation}
\tag{2.6}
(Here \mathring{\operatorname{Lip}}_{\mathrm{loc}}(W)\subset \operatorname{Lip}_{\mathrm{loc}}(D') is a subspace of the space of locally Lipschitz functions on D' that vanish identically outside W.) It is obvious that the norm of the restriction \varphi_W\colon {L}^1_p(W;\omega) \cap \mathring{\operatorname{Lip}}_{\mathrm{loc}}(W)\to L^1_q(D) can depend on W:
\begin{equation}
\|\varphi^*_W\|=\sup_{\substack{u\in \mathcal R(W)\\ u\ne0}} \frac{\|\varphi_W^*u\mid L^1_q(D)\|}{\|u\mid L^1_p(W;\omega)\|}, \qquad1\leqslant q \leqslant p<\infty
\end{equation}
\tag{2.7}
(here we assume that the denominator in (2.7) is distinct from zero). For 1\leqslant q<p<\infty we define a set function by assigning to W\subset D' the quantity In the following theorem we describe analytically those mappings whose inverse maps belong to Q\mathcal{RQ}_{q,p}(D',\omega;D). Theorem 1 (see [11]–[14]). A homeomorphism f\colon D' \to D belongs to the class Q\mathcal{RQ}_{q,p}(D',\omega;D), where 1<q\leqslant p<\infty for n\geqslant 3, or 1\leqslant q\leqslant p<\infty for n=2, if and only if the inverse homeomorphism \varphi=f^{-1}\colon D\to D' has one of the following properties 1)–4). 1) The composition operator \varphi^*\colon {L}^1_p(D';\omega) \cap \operatorname{Lip}_{\mathrm{loc}}(D')\to L^1_q(D), 1\!<\!q\!\leqslant\!p\!<\!\infty, is bounded. 2) For any condenser E=(F_1,F_0) in D' with inverse image \varphi^{-1}(E)=(\varphi^{-1}(F_1), \varphi^{-1}(F_0)) in D,
\begin{equation}
\operatorname{cap}^{1/q}(\varphi^{-1}(E); L^1_q(D)) \leqslant \begin{cases} \widetilde K_p \operatorname{cap}^{1/p}(E; L^1_p(D';\omega)), \\ \widetilde\Psi_{q,p}(D'\setminus(F_0\cup F_1))^{1/\sigma} \operatorname{cap}^{1/p}(E; L^1_p(D';\omega)), \end{cases}
\end{equation}
\tag{2.9}
where the ranges of the parameters q and p are as indicated in Definition 3, {{1}/{\sigma}={1}/{q}- {1}/{p}}, and \widetilde\Psi_{q,p} is a bounded quasiadditive set function defined on open subsets of D'. 3) For each ring condenser E=(F,U) in D' with inverse image \varphi^{-1}(E)=(\varphi^{-1}(F),\varphi^{-1}(U)) in D
\begin{equation}
\begin{aligned} \, \notag &\operatorname{cap}^{1/q}(\varphi^{-1}(E); L^1_q(D)) \\ &\qquad\leqslant \begin{cases} K_p \operatorname{cap}^{1/p}(E; L^1_p(D';\omega)), &1<q=p<\infty, \\ \Psi(U\setminus F)^{1/\sigma} \operatorname{cap}^{1/p}(E; L^1_p(D';\omega)), &1<q<p<\infty, \end{cases} \end{aligned}
\end{equation}
\tag{2.10}
where K_p is a constant and \Psi is a bounded quasiadditive set function defined on some system7 \mathcal O(D') of open subsets of D'. 4) For the homeomorphism \varphi \colon D \to D' the following hold: \qquad(a) it belongs to the Sobolev class W^1_{q, \operatorname{loc}}(D); \qquad(b) it has a finite distortion: D\varphi(x)=0 almost everywhere on the set Z={\{x\in D \mid \det D\varphi(x)=0\}}; \qquad(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{for }\det D\varphi (x)\neq 0, \\ 0 &\textit{for }\det D\varphi (x) = 0, \end{cases}
\end{equation}
\tag{2.11}
belongs to L_{\sigma}(D), where {1}/{\sigma}={1}/{q}-{1}/{p} for 1\leqslant q<p<\infty and \sigma=\infty for {q=p}. Moreover, \varphi\in W^1_{q, \operatorname{loc}}(D) and
\begin{equation}
\begin{aligned} \, &\begin{cases} 2^{-n/p}\biggl(\dfrac{3n}2\biggr)^{-1}\|K^{1,\omega}_{p,p}(\,\cdot\,)\mid L_\sigma(\varphi^{-1}(W))\|, \\ 2^{-n/q}\biggl(\dfrac{3n}2\biggr)^{-1}\|K^{1,\omega}_{q,p}(\,\cdot\,)\mid L_\sigma(\varphi^{-1}(W))\| \end{cases}\notag \\ &\quad\leqslant\|\varphi_W^*\|\leqslant\|K^{1,\omega}_{q,p}(\,\cdot\,)\mid L_\sigma(\varphi^{-1}(W))\| \notag \\ &\quad\leqslant\begin{cases} 7^{n/p} nK_p, &1< q= p<\infty, \\ 7^{n/q} n\Psi_{q,p}(W)^{1/\sigma}, &1< q< p<\infty, \end{cases} \quad\textit{for } \varphi\in Q\mathcal{RQ}_{q,p}(D',\omega;D), \notag \\ &\quad\leqslant\begin{cases} {3n} 2^{(n-p)/p}\widetilde K_p, &q=p, \\ {3n} 2^{(n-q)/q}\widetilde\Psi(W)^{1/\sigma}, &q<p, \end{cases} \quad\textit{for } \varphi\in \mathcal{Q}_{q,p}(D',\omega;D)\qquad \end{aligned}
\end{equation}
\tag{2.12}
for each open set W\in \mathcal O(D'). (The quantity \|\varphi_W^*\| was defined above in (2.7).) 5) Properties 1)–4) also hold for n=2 and 1=q \leqslant p<\infty. The proof of Theorem 1 was presented in [11], Theorem 1, [12], [13] and [14], Theorem 1. Here are some relevant comments to these references. That conditions 1) and 4) in Theorem 1 are necessary was shown in [13], Theorem 18, under the assumption that f\in Q\mathcal{RQ}_{q,p}(D',\omega;D), where 1<q\leqslant p<\infty for n\geqslant 3 and 1\leqslant q\leqslant p<\infty for n=2. The implication 1) \Rightarrow 2) is proved just as the implication 1) \Rightarrow 2) in Theorem 1 from [14]. To prove that condition 1) is sufficient, it was shown in [14], Theorem 1 (in [13], Theorem 18) that condition 2) also holds for 1<q\leqslant p<\infty and n\geqslant 3 (for 1\leqslant q\leqslant p<\infty and n=2, respectively). If a homeomorphism \varphi\colon D\to D' satisfies condition 2), then it also clearly satisfies condition 1). If condition 3) holds, then the inverse homeomorphism f=\varphi^{-1}\colon D'\to D belongs to Q\mathcal{RQ}_{q,p}(D',\omega;D). It was shown in [14], Theorem 1 (in [13], Theorem 18, respectively) that when condition 4) holds, so does condition 1), and therefore f=\varphi^{-1}\in Q\mathcal{RQ}_{q,p}(D',\omega;D) by the above. The second line of inequality (2.12) was proved8 in [13], while the first and third lines were proved in [14]. Corollary 1. Let f\colon D'\to D be a homeomorphism in the class Q\mathcal{RQ}_{q,p}(D',\omega;D), where 1<q\leqslant p<\infty for n\geqslant 3, or 1\leqslant q\leqslant p<\infty for n=2. Then {f\in\mathcal{Q}_{q,p}(D',\omega;D)}, that is, (2.9) holds for each condenser E=(F_1,F_0) in D' with image f(E)=(f(F_1),f(F_0)) in D. Furthermore, the least constant \widetilde K_p for q=p (the quantity
\begin{equation*}
\sup_E\frac{\operatorname{cap}^{1/q}(f(E); L^1_p(D))}{\operatorname{cap}^{1/p}(E; L^1_p(D';\omega))},
\end{equation*}
\notag
where for q<p the upper bound in taken over all condensers E=(F_1,F_0) in D' such that the denominator is distinct from zero), just as in Theorem 1, is estimated in terms of the least constant K_p (the quantity \Psi_{q,p}(D')^{1/\sigma} in (2.4)) times a multiplicative constant depending only on n and p (on n, p and q, respectively). The differentiability properties of maps in the classes \mathcal Q_{q,p}(D',\omega;D) were established in [12] and [13], Theorem 2. The homeomorphisms \varphi\colon D \to D' in Theorem 1 have the following properties:
We can extract two classes of Q_{q,p}-homeomorphisms from Theorem 1. Example 1 (see [11] and [14]). If the 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), where 1<q\leqslant p<\infty for n\geqslant 3, or 1\leqslant q\leqslant p<\infty for n=2, then the inverse homeomorphism f=\varphi^{-1}\colon D'\to D belongs to the class Q\mathcal{RQ}_{q,p}(D',\omega;D). Example 2 (see [11] and [14]). Consider a homeomorphism \varphi \colon D \to D' with the following properties:
Then the inverse homeomorphism f=\varphi^{-1}\colon D'\to D belongs to the class Q\mathcal{RQ}_{q,p}(D',\omega;D). In addition to Examples 1 and 2, the reader can find in [13] further examples of classes of maps in the family \mathcal Q_{q,p}(D',\omega;D). Example 3 (see [13], Example 3). Let \varphi\colon D\to D' be a homeomorphism in the Sobolev class W^1_{p,\mathrm{loc}}(D), where 1<p<\infty for n\geqslant3, or 1\leqslant p<\infty for n=2, which has a finite distortion. Then the inverse homeomorphism f=\varphi^{-1}\colon D'\to D belongs to the class \mathcal Q_{p,p}(D',\omega;D) with constant K_p=1 and weight function Example 4 (see [13], Example 4). Let n-1< s<\infty, and let f \colon D' \to D be a homeomorphism between open domains D', D\subset \mathbb{R}^n, n\geqslant 2, such that
Then the inverse homeomorphism \varphi=f^{-1}\colon D\!\to\! D' has the following properties: while the direct homeomorphism f\colon D'\to D
Example 5 (see [13], Example 5). Let n-1< s<\infty, and let f \colon D' \to D be a homeomorphism between open domains D', D\subset \mathbb{R}^n, n\geqslant 2, such that
Then the inverse homeomorphism \varphi=f^{-1}\colon D\!\to\! D' has the following properties: while the direct homeomorphism f\colon D'\to DExample 6 (see [16], Definition 11 and Theorem 34). A homeomorphism f\colon D'\to D is called a homeomorphism with bounded inner \theta-weighted (s,r)-distortion (belongs to the class \mathcal{ID}(D;s,r;\theta,1)), where n-1< s\leqslant r<\infty, if
Set \mathcal K^{\theta,1}_{s,r}(f;D')=\|\mathcal K_{s,r}^{\theta,1}(\,\cdot\,,f)\mid L_{\varrho}(D')\|. If n-1< s\leqslant r<\infty and the function \omega(x)=\theta^{-(n-1)/(s-(n-1))}(x) is locally integrable, then f\colon D'\to D belongs to the family where q=r/(r-(n-1)) and p=s/(s-(n-1)), 1<q\leqslant p<\infty. Furthermore, the coefficients on the right-hand sides of relations in (2.4) are K_p={\|\mathcal K_{r,r}^{\theta,1}(\,\cdot\,,f)\,{\mid}\, L_{\infty}(\Omega)\|} for q=p and
\begin{equation*}
\Psi_{q,p}(Q(x,R)\setminus \overline{Q(x,r)})^{1/\sigma}= \bigl\|\mathcal K_{s,r}^{\theta,1}(\,\cdot\,,f)\mid L_{\varrho}(Q(x,R)\setminus \overline{Q(x,r)})\bigr\| \quad\text{for } q<p,
\end{equation*}
\notag
where 1/\sigma=1/q-1/p={1}/{\varrho}. Example 7 (see [17], Definition 3 and Theorem 19). A homeomorphism f\colon D'\to D belongs to the class \mathcal{OD}(D';s,r;\theta,1), where n-1< s\leqslant r<\infty (is called a map with bounded outer \theta-weighted (s,r)-distortion) if
Then, provided that n-1< s\leqslant r<\infty and the function \omega(x)=\theta^{-(n-1)/(s-(n-1))}(x) is locally integrable, the homeomorphism f\colon D'\!\to\! D belongs to where q=r/(r-(n-1)) and p=s/(s-(n-1)), 1<q\leqslant p<\infty. Moreover, the coefficients on the right-hand sides of (2.4) are equal to K_p=\|K_{r,r}^{\theta,1}(\,\cdot\,,f) | L_{\infty}(D')\|^{n-1} for q= p and
\begin{equation*}
\Psi_{q,p}(Q(x,R)\setminus \overline{Q(x,r)})^{1/\sigma}= \bigl\|K_{s,r}^{\theta,1}(\,\cdot\,,f)\mid L_{\rho}(Q(x,R)\setminus \overline{Q(x,r)})\bigr\|^{n-1} \quad\text{for } q<p,
\end{equation*}
\notag
where 1/\sigma=1/q-1/p=(n-1)/{\varrho}. It was proved in [17], Theorem 8, that for n-1< s\leqslant r<\infty we have
\begin{equation*}
\mathcal{OD}(\Omega;s,r;\theta,1)\subset\mathcal{ID}(\Omega;s,r;\theta,1).
\end{equation*}
\notag
Moreover, for each homeomorphism f\colon D'\to D in the class \mathcal{OD}(D';s,r;\theta,1), where n-1< s\leqslant r<\infty, we have
\begin{equation*}
\bigl\|\mathcal K_{s,r}^{\theta,1}(\,\cdot\,,f)\mid L_{\sigma}(D')\bigr\| \leqslant\bigl\|K_{s,r}^{\theta,1}(\,\cdot\,,f)\mid L_\rho(D')\bigr\|^{n-1},
\end{equation*}
\notag
where \rho and \sigma were defined in Examples 6 and 7. 2.5. The moduli of families of curves and homeomorphisms in the class \mathcal Q_{q,p}(D',\omega)Let D' is a domain in \mathbb{R}^{n}, n \geqslant 2, and \omega\colon D'\to (0, \infty) be a weight function in L_{1,\mathrm{loc}}. Also let \Gamma be an arbitrary family of locally rectifiable, continuous curves (briefly, paths) \gamma\colon [a,b]\to D'. Recall that, given a family of curves \Gamma в D' and a real number p\geqslant 1, the (weighted) p-modulus of the family \Gamma is defined by
\begin{equation*}
\operatorname{mod}_{p}(\Gamma)=\inf_\rho \int_{D'} \rho^{p}\, dx \qquad \biggl( \operatorname{mod}_{p}(\Gamma)=\inf_\rho \int_{D'} \rho^{p}\omega(x)\, dx\biggr),
\end{equation*}
\notag
where the infimum is taken over all nonnegative Borel functions \rho\colon D' \to [0, \infty] such that for all paths \gamma \in \Gamma. Recall that, for a rectifiable curve \gamma\colon [a,b]\to D', the integral in (2.18) is defined as the quantity where l(\gamma) is the length of the curve \gamma\colon [a,b]\to D' and \widetilde{\gamma}\colon [0,l(\gamma)]\to D' is the natural parametrization of this curve, that is, the unique continuous map such that \gamma=\widetilde{\gamma}\,{\circ}\, S_{\gamma}, where S_{\gamma}\colon [a,b]\to[0,l(\gamma)] is the length function, whose value at t\in [a,b] is defined by S_{\gamma}(t)=l(\gamma\vert_{[a,t]}). If \gamma is a locally rectifiable curve, then we set here the supremum is taken over all rectifiable subcurves \gamma'\colon [a', b'] \to D' of \gamma, where [a', b']\subset(a,b) and \gamma'= \gamma_{[a', b']}. Functions \rho satisfying (2.18) are called admissible functions or metrics for the family \Gamma. Theorem 2 (inequalities between moduli for maps in the class \mathcal{OD}(D,D';q,p;1,\omega); see [10]). Given a homeomorphism \varphi\colon D\to D' between two domains D,D'\subset\mathbb R^n, n\geqslant2, and a locally integrable weight function \omega\colon D'\to (0,\infty), assume that \varphi\colon {D \to D'} belongs to the family
\begin{equation*}
\mathcal{OD}(D,D'; q,p; 1,\omega), \quad \textit{where}\ \begin{cases} n-1< q \leqslant p<\infty &\textit{for } n>2, \\ 1\leqslant q \leqslant p<\infty &\textit{for }n=2. \end{cases}
\end{equation*}
\notag
Then for the same values of the parameters q\leqslant p the following inequality holds for each family \Gamma of paths in D:
\begin{equation}
(\operatorname{mod}_q(\Gamma))^{1/q}\leqslant K_{q,p}(\operatorname{mod}^\omega_p(\varphi \Gamma))^{1/p},
\end{equation}
\tag{2.19}
where K_{q,p}=\|K^{1,\omega}_{q,p}(\,\cdot\,,\varphi)\mid L_\sigma(D)\|, K^{1,\omega}_{q,p}(\,\cdot\,,\varphi) is the distortion function (2.11), and {1}/{\sigma}={1}/{q}-{1}/{p} for q<p<\infty, while \sigma=\infty for q=p. The claim also holds for 1=q \leqslant p<\infty and n=2. As a corollary to Theorem 2 (see [14], Lemma 2.3, and [13], Theorem 18), we obtain the following. Theorem 3 (modulus-based description of maps in the classes Q\mathcal{RQ}_{q,p}(D',\omega); see [10]). Fix a homeomorphism f\colon D'\to D between domains D',D\subset \mathbb R^n, n\geqslant2, and a locally integrable weight function \omega\colon D'\to (0,\infty). 1) Assume that f\colon D'\to D belongs to the family
\begin{equation*}
Q\mathcal{RQ}_{q,p}(D',\omega), \qquad n-1< q \leqslant p<\infty.
\end{equation*}
\notag
Then for each family \Gamma of paths in D'
\begin{equation}
(\operatorname{mod}_q(f\Gamma))^{1/q}\leqslant \begin{cases} K_{p,p} (\operatorname{mod}^\omega_p(\Gamma))^{1/p} &\textit{for } n-1<q=p<\infty, \\ K_{q,p}(\operatorname{mod}^\omega_p(\Gamma))^{1/p} &\textit{for } n-1<q<p<\infty, \end{cases}
\end{equation}
\tag{2.20}
where K_{q,p} in the constant from (2.19), and also
\begin{equation}
(\operatorname{mod}_q(f\Gamma))^{1/q}\leqslant \begin{cases} K_{p,p}(E) (\operatorname{mod}^\omega_p(\Gamma))^{1/p} &\textit{for } n-1<q=p<\infty, \\ K_{q,p}(E)(\operatorname{mod}^\omega_p(\Gamma))^{1/p} &\textit{for } n-1<q<p<\infty, \end{cases}
\end{equation}
\tag{2.21}
for the family \Gamma of all curves10 in the condenser E=(F,U), where
\begin{equation}
K_{q,p}(E)=\bigl\|K^{1,\omega}_{q,p}(\,\cdot\,,\varphi)\mid L_\sigma(f(U\setminus F))\bigr\|,
\end{equation}
\tag{2.22}
K^{1,\omega}_{q,p}(\,\cdot\,,\varphi) is the distortion function (2.11), and {1}/{\sigma}={1}/{q}-{1}/{p} for n-1< q<p< \infty, or \sigma=\infty for q=p. 2) Assume that f\colon D'\to D satisfies the relations
\begin{equation}
\begin{aligned} \, &(\operatorname{mod}_q(f\Gamma))^{1/q}\notag \\ &\quad\leqslant \begin{cases} K_{p,p} (\operatorname{mod}^\omega_p(\Gamma))^{1/p} &\textit{for } n-1<q=p<\infty, \\ \Psi_{q,p}(Q(x,R)\setminus\overline{Q(x,r)})^{1/\sigma}(\operatorname{mod}^\omega_p(\Gamma))^{1/p} &\textit{for } n-1<q<p<\infty, \end{cases} \end{aligned}
\end{equation}
\tag{2.23}
involving the constant K_{p,p} for 1< q=p<\infty and the bounded quasiadditive function \Psi_{q,p} for 1< q<p<\infty, for all cubic condensers (\overline{Q(x,r)}, Q(x,R)) in D', r\in (0,R), whose plates are concentric cubes, and for the family \Gamma of all curves \gamma\colon [a,b]\to D' in the condenser (\overline{Q(x,r)},Q(x,R)) such that \gamma(a)\in\overline{Q(x,r)} and \gamma(b)\in\partial{Q(x,R)}. Then \qquad (a) the homeomorphism f belongs to the class \mathcal Q_{q,p}(D',\omega) , n-1< q \leqslant p<\infty; \qquad (b) relations (2.20) hold for f. The least constants K_{q,p} in (2.20), the quantities11
\begin{equation}
\sup_\Gamma\frac{(\operatorname{mod}_q(f\Gamma))^{1/q}}{(\operatorname{mod}^\omega_p(\Gamma))^{1/p}}\quad\textit{and} \quad \sup_E\frac{(\operatorname{mod}_q(f\Gamma))^{1/q}}{(\operatorname{mod}^\omega_p(\Gamma))^{1/p}},
\end{equation}
\tag{2.24}
corresponding to all possible choices of the family of curves in (2.20) and all possible condensers in (2.21), have an upper estimate in terms of the quantity \|K^{1,\omega}_{q,p}(\,\cdot\,,\varphi)\,|\, L_\sigma(D)\| in (2.11), with coefficients depending only on q, p and n (the denominators of the expressions in (2.24) are distinct from zero). The above claims also hold for n=2 and 1\leqslant q\leqslant p<\infty. Remark 2. In the proof of Theorem 3 presented in [10] it was shown that the homeomorphism f\colon D'\to D satisfies the estimates
\begin{equation}
\begin{aligned} \, \notag &\bigl(\operatorname{cap}(f(\overline{Q(x,r)}),f(Q(x,R));L^1_q(D))\bigr)^{1/q} =(\operatorname{mod}_q(f\Gamma))^{1/q} \\ \notag &\qquad\leqslant\begin{cases} K_{p,p} (\operatorname{mod}^\omega_p(\Gamma))^{1/p}, & n-1<q=p<\infty, \\ \Psi_{q,p}(Q(x,R)\setminus\overline{Q(x,r)})^{1/\sigma}(\operatorname{mod}^\omega_p(\Gamma))^{1/p}, &n-1<q<p<\infty, \end{cases} \\ &\qquad\leqslant\begin{cases} K_{p,p} \bigl({\operatorname{cap}}(\overline{Q(x,r)},Q(x,R); L^1_p(D';\omega))^{1/p}, \\ \Psi_{q,p}(Q(x,R)\setminus\overline{Q(x,r)})^{1/\sigma} \bigl({\operatorname{cap}}\big(\overline{Q(x,r)},Q(x,R); L^1_p(D';\omega)\bigr)^{1/p} \end{cases} \end{aligned}
\end{equation}
\tag{2.25}
provided that (2.23) holds for f. From Theorems 3 and 1 we can make the following striking conclusion. Corollary 2. Homeomorphisms f\colon D'\to D in the family \mathcal Q_{q,p}(D',\omega), where {n-1}< q \leqslant p<\infty or 1=q \leqslant p<\infty for n=2, have two equivalent descriptions, (2.4) in the language of capacities and (2.20) in the language of moduli. Remark 3. For q=p=n (n - 1<q=p<n) we can deduce from (2.25) that the class \mathcal Q_{n,n}(D',\omega;D) (\mathcal Q_{p,p}(D',\omega;D), respectively) contains (see [14], § 4.4) the class of so-called Q-homeomorphisms ((p, Q)-homeomorphisms)12, which are defined in terms of a controllable change of the modulus of a family of curves; see [6] (and [39], respectively). From Theorem 3 above we can conclude that the class \mathcal Q_{n,n}(D',\omega) coincides in fact with the family of Q-homeomorphisms from [6], § 4.1. Let D' and D be two domains in \mathbb{R}^{n}, n \geqslant 2, and let Q\colon D' \to [1, \infty) be a function in L_{1,\mathrm{loc}}. Recall that a homeomorphism f\colon D' \to D is a Q-homeomorphism (see [6], § 4.1) if
\begin{equation}
\operatorname{mod}_n(f \Gamma) \leqslant \int_{D'} Q(x) \cdot \rho^{n}(x)\,dx
\end{equation}
\tag{2.26}
for each family \Gamma of paths in D' and any function \rho admissible for \Gamma. By Theorem 3 the homeomorphisms satisfying (2.26) coincide with the homeomorphisms f\colon {D'\to D} in the class \mathcal Q_{n,n}(D',\omega) for \omega=Q. Some properties of homeomorphisms in \mathcal Q_{q,p}(D',\omega) were studied in [40] (for n-1<q<p=n, \Psi_{q,n}(U) in place of \Psi_{q,n}(U\setminus F) and \omega\equiv1), [6], [41]–[45] (for {q=p=n} and \omega=Q), [46], [47] (for 1<q=p<n and \omega=Q) and other papers. In all these works, apart from [40], the distortion of the geometry of condensers was described using the language of the moduli of families of curves. In some cases such a characterisation is more restrictive in its essential capabilities than a description using the language of capacities. § 3. New set functionsIt this section we define two new set functions; the first is defined in terms of ratios of capacities and the second in terms of ratios of moduli. 3.1. A set function based on capacitiesLet \varphi\colon D\to D' be a homomorphism such that the composition operator \varphi^*\colon {L}^1_p(D';\omega) \cap \operatorname{Lip}_{\mathrm{loc}}(D')\to L^1_q(D), {1< q < p<\infty}, is bounded. For the pair of numbers q, p, 1<q<p<\infty, consider the set function N_c(W) defined on connected open sets W\subset D' by
\begin{equation}
W\mapsto N_c(W)=\sup_{E=(F,U)}\frac{\operatorname{cap}^{\sigma/q}(\varphi^{-1}(E); L^1_q(D))}{\operatorname{cap}^{\sigma/p}(E; L^1_p(D';\omega))},
\end{equation}
\tag{3.1}
where the supremum is taken over all condensers E=(F,U) such that U\subset W and the denominator in (3.1) is distinct from zero. By (2.10) the denominator in (3.1) is always nonzero for n-1<q if n\geqslant3 and for 1\leqslant q if n\geqslant2, provided that the plate F of the condenser E=(F,U) contains a continuum: for instance, see inequality (3.2) in [14]. In this case, for the condenser E=(F, U) in D we have
\begin{equation*}
\operatorname{cap}_q^{n-1}(E; L^1_q(U)) \geqslant c\,\frac{(\operatorname{diam} F)^{q}}{|U|^{q-(n-1)}}>0,
\end{equation*}
\notag
where c is a constant depending only on n and q. Note the following properties of the function N_c(W) for connected open sets W\subset D', which are easy to verify:
Consider the variation (N_c, U) of the above set function D'\supset W\mapsto N_c(W). As is known (for instance, see [30]), the variation V(N_c, U) of N_c with respect to an open set U\subset D' is defined by where the supremum is taken over all possible finite systems of disjoint connected open sets W_{1}, \dots, W_{m} lying in U. The variation V(N_c, U) is a quasiadditive function of the open set U \subset D'. It is obvious thatNote that V(N_c, U), U\subset D', is a bounded quasiadditive function of the open set U \subset D': In fact, as \Phi is finitely additive, we obtain
\begin{equation}
V(N_c, U)=\sup \sum_{k=1}^{m} N_c(W_{k})\leqslant \sup \sum_{k=1}^{m} \Phi(W_{k}) = \Phi\biggl(\bigcup_{k=1}^{m}W_{k}\biggr)\leqslant\Phi(U).
\end{equation}
\tag{3.3}
(Here W_{1},\dots,W_{m} are the disjoint connected open sets from (3.2).) Proposition 3. For each ring condenser E=(F,U) in D' relation (2.10) holds for 1<q<p<\infty and the quasiadditive function V(N_c, U\setminus F) in place of \Phi(U\setminus F). Proof. Set W=U\setminus F. For the condenser E=(F,U) in W\subset D' we have
\begin{equation}
\begin{aligned} \, \notag \frac{\operatorname{cap}^{1/q}(\varphi^{-1}(E); L^1_q(D))}{\operatorname{cap}^{1/p}(E; L^1_p(D';\omega))} &=\sup_{u\in \mathcal A(E;L^1_p(D';\omega))} \frac{\displaystyle\biggl(\int_{\varphi^{-1}(U\setminus F)} |\nabla v_e (x)|^q\,dx\biggr)^{1/q}} {\displaystyle\biggl(\int_{U\setminus F} |\nabla u(y)|^p\omega(y)\,dy\biggr)^{1/p}} \\ &=\sup_{u\in \mathcal A(E;L^1_p(D';\omega))} \frac{\displaystyle\biggl(\int_{\varphi^{-1}(U\setminus F)} |\nabla \widetilde v_e(x)|^q\,dx\biggr)^{1/q}} {\displaystyle\biggl(\int_{U\setminus F} |\nabla\widetilde u(y)|^p\omega(y)\,dy\biggr)^{1/p}}, \end{aligned}
\end{equation}
\tag{3.4}
where \mathcal A(E;\operatorname{Lip}_{\mathrm{loc}}(D')) is the class of functions admissible for E=(F,U) in D', v_e is the extremal function for the capacity \operatorname{cap}(\varphi^{-1}(E); L^1_q(D)), and \widetilde u (\widetilde v_e) is defined in terms of the function u (of v_e, respectively), by the formula In verifying the last equality in (3.4) note that the equalities |\nabla \widetilde v_e(x)|=2|\nabla v_e(x)| and |\nabla \widetilde u(x)|=2|\nabla u(x)| hold for almost all x\in U.
Note that13
\begin{equation}
\int_{U\setminus F} |\nabla\widetilde u(y)|^p\omega(y)\,dy \geqslant\operatorname{cap}\biggl(\biggl(u^{-1}\biggl(\frac12\biggr),U\setminus F\biggr); L^1_p(D';\omega)\biggr)
\end{equation}
\tag{3.6}
and
\begin{equation}
\int_{\varphi^{-1}(U\setminus F)} |\nabla \widetilde v_e(x)|^q\,dx =\operatorname{cap}\biggl(\biggl(\varphi^{-1}\biggl(u^{-1}\biggl(\frac12\biggr)\biggr), \varphi^{-1}(U\setminus F)\biggr); L^1_q(D)\biggr).
\end{equation}
\tag{3.7}
Inequality (3.6) holds because \widetilde u is admissible for the capacity \operatorname{cap}((u^{-1}(1/2), {U\,{\setminus}\, F}); L^1_q(D)). Inequality (3.7) holds because otherwise, starting from the extremal function for the capacity \operatorname{cap}((\varphi^{-1}(u^{-1}(1/2)),\varphi^{-1}(U\setminus F)); L^1_q(D)) we can construct (by inverting the operation in (3.5)) an admissible function for \operatorname{cap}(\varphi^{-1}(E); L^1_q(D)) and thus arrive at a contradiction with the fact that v_e is an extremal function. Next, substituting the smaller value from (3.6) into the denominator in (3.4) and the greater value from (3.7) into the numerator, using inequalities 3) and 4) we obtain an upper estimate for the left-hand side on (3.4):
\begin{equation}
\begin{aligned} \, \notag &\frac{\operatorname{cap}^{\sigma/q}(\varphi^{-1}(E); L^1_q(D))}{\operatorname{cap}^{\sigma/p}(E; L^1_p(D';\omega))} \\ \notag &\qquad\leqslant\sup_{u\in \mathcal A(E;L^1_p(D';\omega))} \frac{\operatorname{cap}^{\sigma/q}\bigl((\varphi^{-1}(u^{-1}(1/2)),\varphi^{-1}(U\setminus F)); L^1_q(D)\bigr)}{\operatorname{cap}^{\sigma/p}\bigl((u^{-1}(1/2),U\setminus F); L^1_p(D';\omega)\bigr)} \\ &\qquad\leqslant N_c(U\setminus F)\leqslant V(N_c, U\setminus F)\leqslant\|\varphi_{U\setminus F}^*\|^\sigma\leqslant \bigl\|K^{1,\omega}_{q,p}(\,\cdot\,)\mid L_\sigma(\varphi^{-1}(U\setminus F))\bigr\|^\sigma. \end{aligned}
\end{equation}
\tag{3.8}
From (3.8) we obtain
\begin{equation*}
N_c(W)=\sup_{E=(F,U)}\frac{\operatorname{cap}^{\sigma/q}(\varphi^{-1}(E); L^1_q(D))}{\operatorname{cap}^{\sigma/p}(E; L^1_p(D';\omega))}= N_c(U\setminus F)\leqslant V(N_c, U\setminus F)\leqslant\|\varphi_{U\setminus F}^*\|^\sigma,
\end{equation*}
\notag
where the upper bound is taken over all condensers E=(F,U) such that U\subset W.
Proposition 3 is proved. Remark 4. Since (2.10) holds for 1<q<p<\infty and the bounded quasiadditive function V(N_c, U\setminus F) in place of \Phi(U\setminus F), we obtain the following inequalities (see the second line in (2.12) and inequality 4) above):
\begin{equation}
\|\varphi_{W}^*\|\leqslant\bigl\|K^{1,\omega}_{q,p}(\,\cdot\,)\mid L_\sigma(\varphi^{-1}(W))\bigr\| \leqslant n7^{n/q} V(N_c, W)^{1/\sigma}\leqslant n7^{n/q}\|\varphi_{W}^*\|
\end{equation}
\tag{3.9}
for each open set W\subset D'. 3.2. A set function based on moduliIn this subsection our considerations rely on the following result, which is a direct consequence of Theorem 3. Proposition 4 (modulus-based description of maps in the classes \mathcal{RQ}_{q,p}(D',\omega); see [10]). Fix a homeomorphism f\colon D'\to D between domains D',D\subset \mathbb R^n, n\geqslant2, and a locally integrable weight function \omega\colon D'\to (0,\infty). 1) Assume that f\colon D'\to D belongs to the class
\begin{equation*}
\mathcal{RQ}_{q,p}(D',\omega), \qquad n-1< q \leqslant p<\infty.
\end{equation*}
\notag
Then for each family \Gamma of curves 14 in an arbitrary condenser E=(F,U)\subset D',
\begin{equation}
(\operatorname{mod}_q(f\Gamma))^{1/q}\leqslant \begin{cases} K_{p,p}(E) (\operatorname{mod}^\omega_p(\Gamma))^{1/p} &\textit{for } n-1<q=p<\infty, \\ K_{q,p}(E)(\operatorname{mod}^\omega_p(\Gamma))^{1/p}&\textit{for } n-1<q<p<\infty, \end{cases}
\end{equation}
\tag{3.10}
where
\begin{equation}
K_{q,p}(E)=\bigl\|K^{1,\omega}_{q,p}(\,\cdot\,,\varphi)\mid L_\sigma(f(U\setminus F))\bigr\|,
\end{equation}
\tag{3.11}
K^{1,\omega}_{q,p}(\,\cdot\,,\varphi) is the distortion function (2.11), and {1}/{\sigma}={1}/{q}-{1}/{p} for n-1< q<p< \infty, or \sigma=\infty for q=p. 2) Assume that f\colon D'\to D satisfies the relations
\begin{equation}
\begin{aligned} \, &(\operatorname{mod}_q(f\Gamma))^{1/q}\notag \\ &\qquad\leqslant \begin{cases} K_{p,p} (\operatorname{mod}^\omega_p(\Gamma))^{1/p} &\textit{for } n-1<q=p<\infty, \\ \Psi_{q,p}(Q(x,R)\setminus\overline{Q(x,r)})^{1/\sigma}(\operatorname{mod}^\omega_p(\Gamma))^{1/p} &\textit{for } n-1<q<p<\infty, \end{cases} \end{aligned}
\end{equation}
\tag{3.12}
involving the constant K_{p,p} for n-1< q=p<\infty and the bounded quasiadditive function \Psi_{q,p} for n-1< q<p<\infty, for all condensers E=(F,U) and the family \Gamma of all curves \gamma\colon [a,b]\to D' in E=(F,U) such that \gamma(a)\in F and \gamma(b)\in\partial U. Then \qquad (a) f\colon D'\to D belongs to the class \mathcal{RQ}_{q,p}(D',\omega) for n-1< q \leqslant p<\infty; \qquad (b) relations (2.20) hold for f\colon D'\to D. These results also hold for n=2 and 1\leqslant q\leqslant p<\infty. Let \varphi\colon D\to D' be a homeomorphism such that the composition operator \varphi^*: {L}^1_p(D';\omega) \cap \operatorname{Lip}_{\mathrm{loc}}(D')\to L^1_q(D), n-1< q < p<\infty, is bounded. Then f=\varphi^{-1}: D'\to D belongs to \mathcal{RQ}_{q,p}(D',\omega) for n-1< q \leqslant p<\infty. For a pair of numbers q, p, n-1<q<p<\infty, consider the set function N_m(W) defined on connected open sets W\subset D' by
\begin{equation}
W\mapsto N_m(W)=\sup_{E=(F,U)}\sup_{\Gamma}\frac{(\operatorname{mod}_q(f\Gamma))^{\sigma/q}} {(\operatorname{mod}^\omega_p(\Gamma))^{\sigma/p}},
\end{equation}
\tag{3.13}
where the inner supremum is taken over the families \Gamma of curves in the condenser E=(F,U), and the outer one is taken over the condensers E=(F,U) such that U\subset W (in view of (3.10) the denominator in (3.13) is distinct from zero, provided that the plate F of E=(F,U) contains a continuum). Note the following properties of N_m(W) for connected open sets W\subset D':
Consider the variation V(N_m, U) of the above set function D'\supset W\mapsto N_m(W). As is known (for instance, see [30]), the variation V(N_m, U) of a set function N_m with respect to an open set U\subset D' is defined by where the upper bound is taken over various finite systems of disjoint connected open sets W_{1}, \dots, W_{m} lying in U. The variation V(N_m, U) is a quasiadditive function of the open set U \subset D'. It is obvious thatNote that V(N_m, U), U\subset D', is a bounded quasiadditive function of the open set U \subset D': Really, from the fact that \Phi is finitely additive we deduce
\begin{equation}
V(N_m, U)=\sup \sum_{k=1}^{m} N_m(W_{k})\leqslant \sup \sum_{k=1}^{m} K_{q,p}(W)^\sigma = K_{q,p}\biggl(\bigcup_{k=1}^{m}W_{k}\biggr)\leqslant K_{q,p}(U).
\end{equation}
\tag{3.15}
Here W_{1},\dots,W_{m} are the disjoint connected open sets from (3.14).
§ 4. Coincidence of four set functionsIn this section we establish some new properties of homeomorphisms f\colon D'\to D between domains D' and D in \mathbb R^n, n\geqslant2, that belong to the class Q\mathcal{RQ}_{q,p}(D',\omega;D). Here we prove the central result of this paper. Theorem 4 (coincidence of the set functions associated with a homeomorphism Q\mathcal{RQ}_{q,p}(D',\omega)). Let \varphi\colon D\to D' be a homeomorphism between domains {D,D'\!\subset\! \mathbb R^n}, n\geqslant2, such that the composition operator
\begin{equation*}
\varphi^*\colon {L}^1_p(D';\omega) \cap \operatorname{Lip}_{\mathrm{loc}}(D')\to L^1_q(D)
\end{equation*}
\notag
with parameters n-1< q\leqslant p<\infty for n\geqslant3, or 1\leqslant q\leqslant p<\infty for n=2 and weight function \omega\in L_{1,\mathrm{loc}}(D') is bounded. Then the following hold. I. For q<p the following set functions coincide: \qquad 1) \mathcal O(D')\ni W\mapsto \Phi(W)=\|\varphi^*_W\|^\sigma (see (2.8)); \qquad 2) \mathcal O(D')\ni W\mapsto \|K^{1,\omega}_{q,p}(\,\cdot\,)\mid L_\sigma(\varphi^{-1}(W))\|^\sigma (see (2.11)); \qquad 3) \mathcal O(D')\ni W\mapsto V(N_c, W) (see (3.2)); \qquad 4) \mathcal O(D')\ni W\mapsto V(N_m, W) (see (3.14)); so that
\begin{equation}
\|\varphi^*_W\|^\sigma=\bigl\|K^{1,\omega}_{q,p}(\,\cdot\,)\mid L_\sigma(\varphi^{-1}(W))\bigr\|^\sigma=V(N_c, W)=V(N_m, W)
\end{equation}
\tag{4.1}
for each open set W\in \mathcal O(D'). II. For q=p the following quantities coincide: \qquad 5) the norm of the composition operator \varphi^*\colon {L}^1_p(D';\omega) \cap \operatorname{Lip}_{\mathrm{loc}}(D')\to L^1_p(D) in Theorem 1; \qquad 6) \|K^{1,\omega}_{p,p}(\,\cdot\,)\mid L_\infty(D)\|; \qquad 7) the least constant K_p in (2.10); \qquad 8) the least constant K_{p,p} in (3.12); so that Fix a homeomorphism \varphi\colon D\to D' meeting the assumption of Theorem 4. Then by Theorem 1 the inverse homeomorphism f=\varphi^{-1} belongs to the class \mathcal{RQ}_{q,p}(D',\omega;D) with parameters n-1< q\leqslant p<\infty for n\geqslant3, or 1\leqslant q\leqslant p<\infty for n=2 and with weight function \omega\in L_{1,\mathrm{loc}}(D'). The following result on the properties of the inverse homeomorphism f=\varphi^{-1}\colon D'\to D was proved in [13]. Theorem 5 (see [13], Theorem 23). Let n-1<q<\infty for n\geqslant3, or 1\leqslant q<\infty for n=2. Then each homeomorphism f\colon D'\to D in the class \mathcal{RQ}_{q,p}(D',\omega;D), q\leqslant p<\infty: \qquad 1) belongs to the Sobolev class W^1_{1,\mathrm{loc}}(D'); \qquad 2) has a finite distortion; \qquad 3) is differentiable almost everywhere in D'. The proof of Theorem 4 is based on the following lemma. Lemma 1. Let f\colon D'\to D be a homeomorphism in the class \mathcal{RQ}_{q,p}(D',\omega;D) with parameters satisfying n-1< q\leqslant p<\infty for n\geqslant3, or 1\leqslant q\leqslant p<\infty for n=2 and with weight function \omega\in L_{1,\mathrm{loc}}(D'). Then the following inequality holds for almost all points a\in D' such that \det Df(a)\ne0 and \omega(a)\ne0:
\begin{equation}
\frac{|{\operatorname{adj} Df(a)}|}{|{\det Df(a)}|^{(q-1)/q}\omega(a)^{1/p}} \leqslant \begin{cases} \Psi'(a)^{1/\sigma}&\textit{for } q<p, \\ K_p &\textit{for }q=p, \end{cases}
\end{equation}
\tag{4.3}
where \Psi'(a) is the derivative of the quasiadditive set function \Psi in (2.10). Let H_{I}(a, f) denote the quantity on the left-hand side of (4.3). This quantity is well defined at points a\in D' such that \det Df(a)\ne0 and \omega(a)\ne0. At points a\in D', where \det Df(a)=0, we have \operatorname{adj} Df(a)= 0; then we set H_{I}(a, f)=0. Proof of Lemma 1. Consider the case q<p (for q=p the arguments are simpler). The mapping is differentiable almost everywhere in D'. Hence the quantity H_{I}(x, f) is well defined almost everywhere in D'. We prove that H_{I}(x, f)^\sigma\leqslant \Psi'(x) almost everywhere in D'.
By Theorem 5, f is differentiable at almost all points x\in D'. It is sufficient to verify (4.3) for almost all points x\in D' at which \det Df(x) \neq 0 (if \det Df(x)=0, then this inequality is obvious) and \omega(x)\ne0 (almost all points have this property). Let a \in D' be a point at which
By the above, conditions 1)–3) hold simultaneously at almost all points {a\!\in\! D'\setminus Z'}, where Z'=\{y\in D'\colon \det Df(a)=0\}. Now we use the following algebraic lemma. Lemma 2. Let L\colon \mathbb R^n\to\mathbb R^n be a nondegenerate linear operator. Then there exist orthonormal bases \{u_1,\dots,u_n\} and \{v_1,\dots,v_n\} in \mathbb R^n and nonnegative numbers \{\lambda_1,\dots,\lambda_n\} such that L(u_i)=\lambda_iv_i, i=1,\dots,n. The vectors \{u_1,\dots,u_n\} are eigenvectors of the operator L^*L, and \lambda_1^2,\dots,\lambda_n^2 are its eigenvalues. Moreover, |{\det L}|=\lambda_1\dotsb\lambda_n. In Lemma 2 we set L=Df(a) and arrange the semiaxes of the ellipsoid obtained as the image of the unit ball under the linear map Df(a) so that \lambda_{1} \geqslant \lambda_{2} \geqslant \dots \geqslant \lambda_{n}>0. To prove (4.3) it is sufficient to show that
\begin{equation}
\frac{|{\det Df(a)}|}{\lambda_{n}^{q}}=\frac{|{\operatorname{adj} Df(a)}|^q}{|{\det Df(a)}|^{q-1}} \leqslant \Psi_{p}'(a)^{q/\sigma}\omega(a)^{q/p},
\end{equation}
\tag{4.5}
where |{\det Df(a)}|=\lambda_{1} \dotsb \lambda_{n}, \lambda_{n}^{-1}={|{\operatorname{adj} Df(a)}|}/{|{\det Df(a)}|} and where E is the identity matrix. Let T_a\mathbb R^n (T_{f(a)}\mathbb R^n) be the tangent space to \mathbb R^n at the point a (at f(a)) with basis vectors \{u_i\} (\{v_i\}, respectively). We fix an arbitrary t\!\in\!(0,\lambda_{n}) and select r\!>\!0 so that the condenser {E_r\!=\!(F_r, U_r)}, where
\begin{equation}
F_r=\biggl\{y=\sum_{i=1}^ny_iu_i\colon y_{n}=0,\ |y_{i}| \leqslant r,\ i=1, \dots, n-1\biggr\}
\end{equation}
\tag{4.6}
and
\begin{equation}
U_r=\biggl\{y=\sum_{i=1}^ny_iu_i \colon|y_{n}|<r t ,\ |y_{i}|<r(1+ t),\ i=1, \dots, n\biggl\},
\end{equation}
\tag{4.7}
lies in D'. By the condition f \in \mathcal{RQ}_{q,p}(D',\omega;D) we have
\begin{equation}
\begin{aligned} \, \notag &\operatorname{cap}(\varphi^{-1}(E_r); L^1_q(D)) \\ &\qquad\leqslant \begin{cases} K^p_p \operatorname{cap}(E_r; L^1_p(D';\omega)) &\text{for }1<q=p<\infty, \\ \Psi(U_r\setminus F_r)^{q/\sigma} \operatorname{cap}^{q/p}(E_r; L^1_p(D';\omega)) &\text{for }1<q<p<\infty, \end{cases} \end{aligned}
\end{equation}
\tag{4.8}
for each ring condenser E_r=(F_r,U_r) in D' with inverse image \varphi^{-1}(E_r)\,{=}\,(\varphi^{-1}(F_r), \varphi^{-1}(U_r)) in D, where K_p is a constant and \Psi is a bounded quasiadditive set function from (2.10), which is defined on a system of open subsets of D' that contains the complements U_r \setminus F_r for all sufficiently small r. We estimate the capacities in (4.8). For the one on the right-hand side we use Lemma 19 in [13], which claims that
\begin{equation}
\begin{aligned} \, \notag \operatorname{cap}(E_r; L^1_p(D';\omega)) &\leqslant\frac{\omega(U_r)}{\operatorname{dist}(F_r, \partial U_r)^{p}} \\ & =\frac{\displaystyle\int_{U_r}\omega(y)\,dy}{(r t)^{p}} =\frac{\omega(a)\mathcal{H}^{n}(U_{r})+o(\mathcal{H}^{n}(U_{r}))}{(rt)^{p}}. \end{aligned}
\end{equation}
\tag{4.9}
On the other hand, by Proposition 5 in [40],
\begin{equation}
\operatorname{cap}(f(E_r); L^1_q(D)) \geqslant \frac{\bigl(\inf \mathcal{H}^{n-1}(S)\bigr)^{q}}{\mathcal{H}^{n}(f(U_r))^{q-1}},
\end{equation}
\tag{4.10}
where \mathcal{H}^{n-1}(S) is the (n-1)-dimensional Hausdorff measure of the C^{\infty}-manifold S, the boundary of an open set A containing f(F_r) and contained in f(U_r) together with its closure, and the infimum is taken over all such sets S. (The proof of (4.10) is based on the inequality \operatorname{cap}(f(E_r); L^1_1(D)) \geqslant \inf_S \mathcal{H}^{n-1}(S), established in [48].) We continue the estimate from below in (4.10) by using the differentiability properties of the map D'\ni y\mapsto f(y) at x=a. Let r>0 be sufficiently small so that in the expansion we have |o(1)|<t for y \in U_r. Then the measure of f(U_r) exhibits the asymptotic behaviour
\begin{equation*}
\mathcal{H}^{n}(f(U_r))=(|{\det Df(a)}|+o(1))\cdot\mathcal{H}^{n}(U_{r})\quad\text{as}\quad r\to0.
\end{equation*}
\notag
The projection of f(F_r) onto the subspace x_{n}=0 contains the (n-1)-dimensional ellipsoid where \operatorname{Id} is the linear operator given by the formula \operatorname{Id}(u_i)=v_i, i=1,\dots,n, on the basis vectors. Hence, in view of (4.6) and (4.7) we have
\begin{equation}
\begin{aligned} \, \notag \mathcal{H}^{n-1} (S) &\geqslant 2 \mathcal{H}^{n-1}(V_{r})=2\mathcal{H}^{n-1}(F_{r}) \prod_{i=1}^{n-1}(\lambda_{i}-o(1)) \\ &=2|{\operatorname{adj} Df(a)}|(1-o(1))\cdot\mathcal{H}^{n-1}(F_{r}), \end{aligned}
\end{equation}
\tag{4.11}
so that for the left-hand side of (4.10) we have the lower bound
\begin{equation*}
\operatorname{cap}(f(E_r); L^1_q(D)) \geqslant\frac{(2|{\operatorname{adj} Df(a)}|(1-o(1))\cdot\mathcal{H}^{n-1}(F_{r}) )^{q}}{((|{\det Df(a)}|+o(1))\cdot\mathcal{H}^{n}(U_{r}))^{q-1}} \quad\text{as } r\to0.
\end{equation*}
\notag
Substituting the estimates for capacities we have obtained into inequality (4.8) above we see that
\begin{equation*}
\frac{(2 |{\operatorname{adj} Df(a)}|\cdot \mathcal{H}^{n-1}(F_{r}))^{q}\cdot (1-o(1))} {((|{\det Df(a)}|+o(1))\cdot\mathcal{H}^{n}(U_{r})))^{q-1}}\leqslant \Psi(U_r)^{q/\sigma} \frac{((\omega(a)+o(1))\cdot\mathcal{H}^{n}(U_{r}))^{q/p} }{(r t )^{q}}
\end{equation*}
\notag
as r\to0. Multiplying both sides by (r t )^{q} and dividing by \mathcal{H}^{n}(U_{r}) =2rt\mathcal{H}^{n-1}(F_{r})(1+t)^{n-1} yields
\begin{equation*}
\begin{aligned} \, &\frac{(|{\operatorname{adj} Df(a)}|\cdot 2rt \mathcal{H}^{n-1}(F_{r})(1+t)^{n-1} )^{q}\cdot(1-o(1))}{((|{\det Df(a)}|+o(1)) \cdot\mathcal{H}^{n}(U_{r}))^{q}} \, \frac{(|{\det Df(a)}|+o(1))\cdot\mathcal{H}^{n}(U_{r})}{(1+t)^{q(n-1)} \cdot\mathcal{H}^{n}(U_{r})} \\ &\qquad\to\frac{|{\operatorname{adj} Df(a)}|^{q}}{(1+t)^{q(n-1)}|{\det Df(a)}|^{q-1}} \leqslant \Psi'(a)^{q/\sigma} \omega(a)^{q/p} \end{aligned}
\end{equation*}
\notag
as r \to0. Since t > 0 can be arbitrary, we conclude that
\begin{equation*}
\frac{|{\operatorname{adj} Df(a)}|^q}{|{\det Df(a)}|^{q-1}} \leqslant \Psi'(a)^{q/\sigma}\omega(a)^{q/p},
\end{equation*}
\notag
which yields the required estimate (4.3). For a proof for q=p one must replace q by p and \Psi_{p}'(a)^{q/\sigma} by K_p^p in the above arguments. Lemma 1 is proved. Proof of Theorem 4. Combining (4.3) with the first line of (2.12) we obtain
\begin{equation}
\begin{aligned} \, \notag \|\varphi_W^*\|^\sigma &\leqslant\bigl\|K^{1,\omega}_{q,p}(\,\cdot\,)\mid L_\sigma(\varphi^{-1}(W))\bigr\|^\sigma =\int_{W\setminus Z'}\biggl(\frac{|{\operatorname{adj} Df(y)}|}{|{\det Df(y)}|^{(q-1)/q}\omega(y)^{1/p}}\biggr)^\sigma\,dy \\ &\leqslant\int_{W\setminus Z'}\Psi'(y)\,dy\leqslant\int_{W}\Psi'(y)\,dy\leqslant \Psi(W) \end{aligned}
\end{equation}
\tag{4.12}
for q<p and \|\varphi^*\|\leqslant \|K^{1,\omega}_{p,p}(\,\cdot\,)\mid L_\infty(D)\| \leqslant K_p for q=p.
Since we can take \Phi(W)=\|\varphi^*_W\|^\sigma as the quasiadditive set function, from (4.12) we infer
\begin{equation*}
\|\varphi_W^*\|^\sigma\leqslant\bigl\|K^{1,\omega}_{q,p}(\,\cdot\,)\mid L_\sigma(\varphi^{-1}(W))\bigr\|^\sigma\leqslant \|\varphi_W^*\|^\sigma.
\end{equation*}
\notag
In a similar way, we can take V(N_c, W) as the quasiadditive set function in (2.10). Then taking (3.3) into account, from (4.12) we obtain
\begin{equation*}
\|\varphi_W^*\|^\sigma \leqslant\bigl\|K^{1,\omega}_{q,p}(\,\cdot\,)\mid L_\sigma(\varphi^{-1}(W))\bigr\|^\sigma\leqslant V(N_c, W)\leqslant \|\varphi_W^*\|^\sigma
\end{equation*}
\notag
for q<p and \|\varphi^*\|\leqslant \|K^{1,\omega}_{p,p}(\,\cdot\,)\mid L_\infty(D)\| \leqslant K_p\leqslant\|\varphi^*\| for q=p.
This completes the proof of equalities (4.1) between the first three quasiadditive functions (of equalities (4.2) between the first three constants, respectively). To finish the proof it remains to show that the quasiadditive function V(N_m,W) coincides with any of the first three quasiadditive functions in (4.1) (the constant K_{p,p} is equal to the first three quantities in (4.2)). Note that the proof of Lemma 1 is based on inequalities (4.9) and (4.10) for capacities and on (4.8). In place of (4.8) we can now write an updated inequality (3.10) for the condenser E_r=(F_r,U_r):
\begin{equation}
(\operatorname{mod}_q(f\Gamma))^{1/q}\leqslant \begin{cases} K_{p,p}(U_r) (\operatorname{mod}^\omega_p(\Gamma))^{1/p} &\text{for } n-1<q=p<\infty, \\ V(N_m, U_r) K_{q,p}(E_r)(\operatorname{mod}^\omega_p(\Gamma))^{1/p} &\text{for } n-1<q<p<\infty, \end{cases}
\end{equation}
\tag{4.13}
for the family \Gamma of all curves15 in E_r=(F_r,U_r). In this inequality
\begin{equation}
K_{p,p}(U_r)=\bigl\|K^{1,\omega}_{q,p}(\,\cdot\,,\varphi)\mid L_\infty(f(U_r))\bigr\|
\end{equation}
\tag{4.14}
and K^{1,\omega}_{p,p}(\,\cdot\,,\varphi) is the distortion function (2.11) (for n=2 inequalities (4.13) also hold for 1\leqslant q\leqslant p< \infty).
By means of the arguments used in [10] to verify (2.25) we can also establish the following relations for the family \Gamma of all curves in E_r=(F_r,U_r)\subset D':
\begin{equation}
\begin{aligned} \, \notag &\bigl(\operatorname{cap}(f(F_r),f(U_r);L^1_q(D))\bigr)^{1/q} =(\operatorname{mod}_q(f\Gamma))^{1/q} \\ \notag &\qquad\leqslant\begin{cases} K_{p,p}(D') (\operatorname{mod}^\omega_p(\Gamma))^{1/p} &\text{for } n-1<q=p<\infty, \\ V(N_m, U_r)(\operatorname{mod}^\omega_p(\Gamma))^{1/p} &\text{for } n-1<q<p<\infty, \end{cases} \\ &\qquad\leqslant\begin{cases} K_{p,p}(D') ({\operatorname{cap}}(F_r,U_r); L^1_p(D';\omega))^{1/p}, \\ V(N_m, U_r)({\operatorname{cap}}(F_r,U_r); L^1_p(D';\omega))^{1/p} \end{cases} \end{aligned}
\end{equation}
\tag{4.15}
provided that (2.21) holds for f. Hence, applying the upper estimate (4.10) to the right-hand side of (4.15) and the lower estimate (4.9) to the left-hand side we arrive at the same relations as those obtained for (4.8) and underlying the proof of (4.3). Since in (4.15) we take V(N_m, \cdot) as the quasiadditive function, we have
\begin{equation}
\frac{|{\operatorname{adj} Df(a)}|}{|{\det Df(a)}|^{(q-1)/q}\omega(a)^{1/p}} \leqslant \begin{cases} V'(N_m,\,\cdot\,)(a)&\text{for } q<p, \\ K_{p,p}(D')&\text{for } q=p, \end{cases}
\end{equation}
\tag{4.16}
where V'(N_m,\,\cdot\,)(a) is the derivative of the quasiadditive set function V'(N_m,\,\cdot\,) at a. Hence, just as in (4.12), we obtain
\begin{equation*}
\|\varphi_W^*\|^\sigma\leqslant\int_{W\setminus Z'}V'(N_m,\,\cdot\,)(y)\,dy \leqslant\int_{W}V'(N_m,\,\cdot\,)(y)\,dy\leqslant V(N_m,W).
\end{equation*}
\notag
Taking account of inequality 4) at the end of § 3.2 and the equality of the first two quantities in (4.1), we arrive at the required result: Theorem 4 is proved. From (4.1) and (4.12) we can derive the following unexpected result. Proposition 5. Under the assumptions of Theorem 4 the set functions in (4.1) can be represented analytically as follows:
\begin{equation}
\begin{aligned} \, \|\varphi^*_W\|^\sigma &=\|K^{1,\omega}_{q,p}(\,\cdot\,)\mid L_\sigma(\varphi^{-1}(W))\|^\sigma= V(N_c, W)=V(N_m, W) \nonumber \\ &=\int_{W\setminus Z'}\biggl(\frac{|{\operatorname{adj} Df(y)}|}{|{\det Df(y)}|^{(q-1)/q}\omega(y)^{1/p}}\biggr)^\sigma\,dy, \end{aligned}
\end{equation}
\tag{4.17}
where W is an open subset of D', f = \varphi^{-1}\colon D'\to D (the analytic properties of f were stated in Theorem 5), and Z' = \{y \in D'\colon \!\det Df(y) = 0\} is the zero set of the Jacobian of f. In addition, the representation (4.17) ensures an extension of set functions to the Borel \sigma-algebra in D' such that the extended functions are absolutely continuous. Remark 5. The interesting part of results in § 4 is related to the case when one considers the quantities (3.1) and (3.2) in the context of a homeomorphism \varphi\colon {D\to D'} inducing a bounded composition operator
\begin{equation*}
\varphi^*\colon {L}^1_p(D';\omega) \cap \operatorname{Lip}_{\mathrm{loc}}(D')\to L^1_q(D),
\end{equation*}
\notag
where n-1< q < p<\infty for n\geqslant3, or 1\leqslant q <p<\infty for n=2. Note that the quantities (3.1) and (3.2) can be defined for any homeomorphism \varphi\colon D\to D', without assuming that \varphi^* is bounded. Of course, such a definition is consistent only when the denominator in (3.1) is distinct from zero and the quantities (3.1) and (3.2) are bounded. For our arguments in this paper to be valid it is sufficient to assume that condensers of two types have positive capacities in L^1_p(D';\omega), namely, the spherical condensers E=(B(y_0,r), B(y_0,R))\subset D' , 0<r<R<\infty, and the condensers E_r=(F_r, U_r)\subset D' (see (4.6) and (4.7)), so that \operatorname{cap}(E; L^1_p(D';\omega))>0 and \operatorname{cap}(E_r; L^1_p(D';\omega))>0. This always holds if \omega\equiv 1. For an arbitrary weight we indicate sufficient conditions ensuring that the capacities of these condensers are positive for p>1. We can obtain such conditions from an estimate for an arbitrary function u that is admissible for the condenser E=(B(y_0,r), B(y_0,R)) \subset D' in L^1_p(D';\omega):
\begin{equation}
\begin{aligned} \, \notag 0 &<\operatorname{cap}(E; L^1_1(D')) \\ &\leqslant \int_{B(y_0,R)}|\nabla u(y)|\,dy \leqslant \int_{B(y_0,R)}|\nabla u(y)|\omega^{1/p}(y)\omega^{-1/p}(y)\,dy \notag \\ &\leqslant\biggl( \int_{B(y_0,R)}|\nabla u(y)|^p\omega(y)\,dy\biggr)^{1/p}\biggl( \int_{B(y_0,R)}\omega^{-1/(p-1)}(y)\,dy\biggr)^{(p-1)/p}. \end{aligned}
\end{equation}
\tag{4.18}
The left-hand side of (4.18) is a consequence of estimate (4.10) for q=1. Hence, if the integral is finite, then \operatorname{cap}(E; L^1_p(D';\omega))>0 for each ball B(y_0,R)\subset D'. The last condition always holds for p>1, provided that \omega belongs to the Muckenhoupt class A_p. Note that condition (4.19) also ensures that the capacity \operatorname{cap}(E_r; L^1_p(D';\omega)) is positive. Based on the above, we make the following conclusion. Let the weight function \omega satisfy (4.19) and \varphi\colon D\to D' be a homeomorphism. If the quantity (3.1), where the condenser capacity in the denominator is distinct from zero, defines a bounded quasiadditive function (3.2), then condition 2) in Theorem 1 holds for \varphi\colon D\to D', so the conclusions of Theorem 1 hold for this homeomorphism. We can also make similar conclusion for the quantities (3.13) and (3.14). In fact, if \rho is an admissible metric for the family of curves \Gamma in the condenser {E=(B(y_0,r), B(y_0,R))\subset D'} that have their endpoints at the boundary spheres S(y_0,r) and S(y_0,R), then
\begin{equation}
\begin{aligned} \, \notag 0&<\operatorname{cap}(E; L^1_1(D'))=\operatorname{mod}_1(\Gamma) \\ \notag &\leqslant\int_{B(y_0,R)}\rho(y)\,dy\leqslant \int_{B(y_0,R)}\rho(y)\omega^{1/p}(y)\omega^{-1/p}(y)\,dy \\ &\leqslant\biggl( \int_{B(y_0,R)}\rho(y)^p\omega(y)\,dy\biggr)^{1/p}\biggl( \int_{B(y_0,R)}\omega^{-1/(p-1)}(y)\,dy\biggr)^{(p-1)/p}. \end{aligned}
\end{equation}
\tag{4.20}
Hence, if (4.19) holds, then the modulus \operatorname{mod}^\omega_p(\Gamma) is positive. The equality of a capacity and a modulus in the first line of (4.20) was proved in [7]. AcknowledgementI am grateful to the referee for their useful comments.
Citation in format AMSBIB
\Bibitem{Vod22}
This publication is cited in the following 2 articles:
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