On the Linear Dilatation of the Mappings Satisfying an Inverse Poletsky Modular Inequality in Metric Spaces

Abstract

We study the linear dilatation of the mappings satisfying an inverse Poletsky inequality in metric spaces. We also show that under certain conditions such mappings are quasiregular.

1 Introduction

Throughout this paper X and Y will be metric measure spaces endowed with Borel measures $\mu$ and $\nu$ such that $0<\mu (B)<\infty$ for every ball B in X and $0<\nu (B)<\infty$ for every ball B in Y. Moreover X and Y will be assumed to be locally compact and locally pathwise connected. Also $D\subset X$ will be an open set.

Let $D\subset X$. We denote by A(D) the set of all non-constant path families in D and if $\Gamma \in A(D)$ we denote by $F(\Gamma )$, the collection of Borel functions $\rho :D\rightarrow [0,\infty ]$ such that $\int _\gamma \rho ds\ge 1$ for every locally rectifiable $\gamma \in \Gamma$.

If $p>1$ and $\omega :D\rightarrow [0,\infty ]$ is $\mu$-measurable and finite $\mu$-almost everywhere we define the p-modulus of weight $\omega$ by

$$\begin{aligned} M_\omega ^p(\Gamma )=\inf _{\rho \in F(\Gamma )}\int _X\omega (x)\rho (x)^pd\mu \hspace{0.1cm}\text {if}\hspace{0.1cm} \Gamma \in A(D). \end{aligned}$$

If $F(\Gamma )=\phi$ we set $M_\omega ^p(\Gamma )=0$. If $\omega =1$ we put

$$\begin{aligned} M_p(\Gamma )=\inf _{\rho \in F(\Gamma )}\int _{X}\rho (x)^pd\mu \hspace{0.1cm}\text {if} \hspace{0.1cm}\Gamma \in A(D). \end{aligned}$$

One of the basic tools in studying quasiregular mappings is the modular inequality of Poletsky

$$\begin{aligned} M_n(f(\Gamma ))\le KM_n(\Gamma )\quad \text {for every }\Gamma \in A(D) \end{aligned}$$

valid for a K-quasiregular mapping $f:D\subset {\mathbb {R}}^n\rightarrow {\mathbb {R}}^n$ (see the monographs [41, 42, 54] for more information about quasiregular mappings). Several generalizations of quasiregular mappings were developed in the last 35 years. One of the most important is the class of mappings of finite distortion (see the monographs [26, 29]) for which a generalized Poletsky modular inequality is established in [33]. A mapping $f:D\rightarrow {\mathbb {R}}^n$ is of finite distortion if $n\ge 2$, $D\subset {\mathbb {R}}^n$ is open, $f\in W_{loc}^{1,1}(D,{\mathbb {R}}^n)$, $J_f\in L_{loc}^1(D,{\mathbb {R}}^n)$ and there exists $K:D\rightarrow [0,\infty ]$ measurable and finite a.e. such that $|f'(x)|^n\le K(x)J_f(x)$ a.e. and we set the outer dilatation

$$\begin{aligned} K_0(x,f)= {\left\{ \begin{array}{ll} \frac{|f'(x)|^n}{J_f(x)}, &{}\text {if }J_f(x)\ne 0 \\ 1, &{}\text {if }J_f(x)=0. \end{array}\right. } \end{aligned}$$

If $p\ge 1$ and $D\subset {\mathbb {R}}^n$ is open, we let $W_{loc}^{1,p}(D,{\mathbb {R}}^n)$ be the Sobolev space of all mappings $f:D\rightarrow {\mathbb {R}}^n$ such that f and its first distributional derivatives are $L^p(D,{\mathbb {R}}^n)$-integrable. For the definition of Sobolev spaces in the metric setting see [25]. The classes of mappings distinguished by moduli inequalities has been intensively studied in the last 20 years. Such an approach was proposed by Martio, first on mappings between open sets in ${\mathbb {R}}^n$ in [7, 9, 11, 16,17,18,19,20, 36,37,38, 43, 45,46,47] and then on metric measure spaces in [12, 13, 44, 49, 50]. Such mappings satisfy a generalized Poletsky modular inequality of type

$$\begin{aligned} M_p(f(\Gamma ))\le M_\omega ^q(\Gamma )\quad \text {for every }\Gamma \in A(D). \end{aligned}$$

(1.1)

It must be mentioned that in these classes of mappings we can give analogues of Liouville, Montel, Picard type theorems, boundary extension results, equicontinuity results and estimates of the modulus of continuity and we recommend to the reader the monograph [38] for further information about this class of mappings (see also [7, 9, 43, 47]).

Quasiconformal and quasiregular mappings between metric measure spaces X and Y were also studied in [2, 6, 21,22,23,24,25, 28, 31, 32, 39, 40, 56].

Let $f:D\rightarrow Y$ be continuous, open and discrete and let

$$\begin{aligned} L(x,f,r)&=\sup _{y\in S(x,r)}d(f(x),f(y)) \quad \text {for}\quad x\in D\hbox { and }r>0\hbox { such that }{\overline{B}}(x,r)\subset D, \\ l(x,f,r)&=\inf _{y\in S(x,r)}d(f(x),f(y)) \quad \text {for}\quad x\in D\hbox { and }r>0\hbox { such that }{\overline{B}}(x,r)\subset D, \\ H(x,f)&=\limsup _{r\rightarrow 0}\frac{L(x,f,r)}{l(x,f,r)} \end{aligned}$$

the linear dilatation of f at a point $x\in D$.

We denote by d the distance on X and Y and by

$$\begin{aligned} B(x,r)&=\{y\in X|d(x,y)<r\},\\ {\overline{B}}(x,r)&=\{y\in X|d(x,y)\le r\},\\ S(x,r)&=\{y\in X|d(x,y)=r\}. \end{aligned}$$

We say that f is metrically K-quasiregular if $H(\cdot ,f)$ is bounded from above by K in D. If f is also injective, we say that f is metrically K-quasiconformal.

Let $f:D\rightarrow Y$ be continuous and open and let $\tilde{l}(x,f,r)=\sup \{s>0| B(f(x),s)\subset f(B(x,r))\}$ if $x\in D$, $r>0$ and ${\overline{B}}(x,r) \subset D$.

We propose in this case a suitable linear dilatation

$$\begin{aligned} \tilde{H}(x,f)=\limsup _{r\rightarrow 0}\frac{L(x,f,r)}{\tilde{l}(x,f,r)}\hspace{0.1cm}\quad \text {if}\hspace{0.1cm}x\in D. \end{aligned}$$

We denote by $B_f=\{x\in D|f$ is not a local homeomorphism at $x\}$ the branch set of f and we denote by $\mu _n$ the Lebesgue measure on ${\mathbb {R}}^n$.

If $f:D\rightarrow Y$ is a mapping and $x\in D$, we define

$$\begin{aligned} L(x,f)=\limsup _{y\rightarrow x}\frac{d(f(x),f(y))}{d(x,y)}. \end{aligned}$$

As in the classical case, a continuous, open and discrete mapping $f:X\rightarrow Y$ between n-Loewner spaces is metrically quasiregular if and only if it is geometrically quasiregular, i.e.

$$\begin{aligned} M_n(\Gamma )\le K\int _YN(y,f,G)\rho ^n(y)d\nu (y) \end{aligned}$$

(1.2)

for every open set $G\subset \subset D$, every $\Gamma \in A(D)$ and every $\rho \in F(f(\Gamma ))$ (see [6, 21]). Here N(y, f, G) is the cardinality of $f^{-1}(y)\cap G$.

Partially inspired by property (1.2), several authors have studied mappings satisfying a generalized inverse Poletsky modular inequality:

$$\begin{aligned} M_n(\Gamma )\le \gamma _G(M_{\omega _G}^p(f(\Gamma ))), \end{aligned}$$

(1.3)

where $n,p>1$ are constants, $f:D\rightarrow Y$ is continuous, open and discrete, $\omega _G:Y\rightarrow [0,\infty ]$ is $\nu$-measurable and finite $\nu$-almost everywhere,$\gamma _G:(0,\infty )\rightarrow (0,\infty )$ is increasing and $\lim _{t\rightarrow 0}\gamma _G(t)=0$ for every $G\subset \subset D$ (see [8, 51, 52]).

The following example shows that there are mappings satisfying (1.3) which are not quasiregular.

Example 1.1

Let $F\in C^{\infty }((0,1)^3,{\mathbb {R}}^3)$ be given by $F(x,y,z)=(x,f(y,z))$ where f is analytic non-constant and such that $J_f(y,z)=0$ if and only if $y=z=0$. Then

$$\begin{aligned} K_0((x,y,z),F)\ge \frac{|F^\prime (x,y,z)(1,00)|^3}{|J_f(y,z)|}\ge \frac{1}{|J_f(y,z)|}\rightarrow \infty \end{aligned}$$

if $(x,y,z)\rightarrow (0,0,0)$ and hence F is not quasiregular.

We see that F is open, discrete, $\mu _3(B_F)=0$, $\mu _3(F(B_F))=0$, F satisfies condition $(N^{-1})$ and we use Lemma 2.5 to see that there exists $\omega \in L^1({\mathbb {R}}^3)$ such that $M_3(\Gamma )\le M_\omega ^3(F(\Gamma ))$ for every $\Gamma \in A((0,1)^3)$.

An example of a mapping $f:{\mathbb {R}}^n\rightarrow {\mathbb {R}}^n$ which is not open and satisfies a modular inequality of type (1.3) is given in [8].

Also, a large class of mappings of finite distortion satisfy an inverse Poletsky modular inequality of type (1.3). Indeed, let $n\ge 2$, $D\subset {\mathbb {R}}^n$ be open, let $f:D\rightarrow {\mathbb {R}}^n$ be a mapping of finite distortion such that $f\in W_{loc}^{1,n}(D,{\mathbb {R}}^n)$ and $K_0(\cdot ,f)\in L_{loc}^p(D)$, where $p>n-1$ if $n\ge 3$, $p=1$ if $n=2$. Then f is continuous, open, discrete, is a.e. differentiable, $\mu _n(B_f)=0$, $\mu _n(f(B_f))=0$ and f satisfies condition $(N^{-1})$ (see [26]). Let $G\subset \subset D$. Then

$$\begin{aligned} \int _G L(x,f)^nd\mu =\int _G |f'(x)|^n d\mu <\infty \end{aligned}$$

and we use again Lemma 2.5 to see that there exists $\omega _G\in L^1({\mathbb {R}}^n)$ such that

$$\begin{aligned} M_n(\Gamma )\le M_{\omega _G}^n(f(\Gamma ))\quad \text {for every }\Gamma \in A(G). \end{aligned}$$

(1.4)

In fact, using Hölder’s inequality, we see that relation (1.4) holds with n replaced by some $n-1<q<n$.

If $f:D\rightarrow Y$ is a mapping, $G\subset \subset D$ is open, $0<\epsilon <\delta$ and $y\in f(G)$, we set $\Gamma _{G,f}(y,\epsilon ,\delta )=\{\gamma :[0,1]\rightarrow G$ path $|f(\gamma (0))\in S(y,\epsilon ),f\circ \gamma (1)\in S(y,\delta )$ and $f(\gamma (0,1))\subset B(y,\delta ){\setminus } {\overline{B}}(y,\epsilon )\}$.

Also, if $D\subset X$ is a set, $E,F\subset {\overline{D}}$, we define $\Delta (E,F,D)=\{\gamma :[0,1]\rightarrow {\overline{D}}$ path $|\gamma (0)\in E, \gamma (1)\in F$ and $\gamma ((0,1))\subset D\}$ and if $y\in Y$, $0<\epsilon <\delta$, we set $\Gamma _{y,\epsilon ,\delta }=\Delta ({\overline{B}}(y,\epsilon ), S(y,\delta ),C_{y,\epsilon ,\delta })$, where $C_{y,\epsilon ,\delta }=B(y,\delta ){\setminus }{\overline{B}}(y,\epsilon )$.

The aim of this paper is to study the linear dilatation $H(\cdot ,f)$ of the mappings satisfying the inverse Poletsky modular inequality (1.3). We show that in some very general cases we have $H(x,f)<\infty$ a.e. in this class of mappings. We prove:

Theorem 1.2

Let $n,p>1$ and consider an n-Loewner space X with Loewner function $\Phi$ which is $C_2$-upper n-regular and a c-LLC and $C_1$-upper regular space Y. Let $f:D\rightarrow Y$ be continuous and open such that (1.3) holds for every $G\subset \subset D$ and $x\in G\subset \subset D$. If f is discrete, then $H(x,f)<C_x$ or if f is light, then ${\tilde{H}}(x,f)<C_x$ for every $x\in D$. In both cases, the constant $C_x$ depends on $(\omega _G)_{f(x)}, c,C_1,\Phi$ and p. More precisely,

$$\begin{aligned} C_x=\max \left\{ c^2,\frac{c^2}{e}\exp \left( \exp \left( \frac{C(\omega _G)_{f(x)}}{\epsilon }\right) ^{1/p}\right) \right\} \end{aligned}$$

where $\epsilon >0$ is chosen such that $\gamma _G(t)<\Phi (1)/2$ for every $0<t\le \epsilon$ and $C=C_1e^p\sum _{k=1}^\infty (1/k^p)$.

Remark 1.3

If $\gamma _G:(0,\infty )\rightarrow (0,\infty )$ is a homeomorphism, in Theorem 1.2 we just take the constant $\epsilon =\gamma _G^{-1} (\Phi (1)/2)$.

Remark 1.4

We use the notations from Theorem 1.2 and we additionally suppose that there exists $\omega \in L^1(Y)$ such that $\omega |G=\omega _G$ for every $G\subset \subset D$. Then $\omega _y<\infty$ for $\nu$ a.e. $y\in Y$ and if f also satisfies condition $(N^{-1})$, we see that $H(x,f)<\infty$ a.e. if f is discrete on D and that ${\tilde{H}}(x,f)<\infty$ a.e. if f is a light mapping on D.

Remark 1.5

We use the notations from Theorem 1.2 and we additionally suppose that there exists $M>0$ and $G\subset \subset D$ such that $\omega _G(y)\le M$ for every $y\in G$. We see from Theorem 1.2 that $H(x,f)<H(G)<\infty$ for every $x\in G$ and hence f is metrically quasiregular on G. If the relation

$$\begin{aligned} M_n(\Gamma )\le \gamma _G(M_p(f(\Gamma )) \end{aligned}$$

(1.5)

is satisfied for every $G\subset \subset D$ and every $\Gamma \in A(G)$, then $H(x,f)<H<\infty$ for every $x\in D$ and hence f is metrically quasiregular on D.

Suppose now that $D\subset {\mathbb {R}}^n$ is open and relation (1.4) is satisfied for some homeomorphism $f:D\rightarrow f(D)\subset {\mathbb {R}}^n$. We see that $p=n$ and $\gamma (t)=Kt$ for every $t>0$, simply because f is quasiconformal and for such a mapping the relation

$$\begin{aligned} M_n(\Gamma )\le KM_n(f(\Gamma ))\quad \text {for every }\Gamma \in A(D) \end{aligned}$$

(1.6)

is satisfied. This is surprising, since the function $\gamma$ and the constant $p>1$ are arbitrary in relation (1.4) and this still implies the quasiconformality of the mapping f. Also, in the particular case (1.4) of Theorem 1.2 we extend on very general metric spaces our results from [4, 5] given on Euclidean spaces.

We also prove:

Theorem 1.6

Let X be a Riemannian n-manifold which is n-Loewner, let $p>1$ and Y be $c-LLC$ and upper p-regular and let $f:D\rightarrow Y$ be continuous and open and satisfying condition $(N^{-1})$. Let $\omega \in L^1(Y)$ be such that for every $G\subset \subset D$ there exists $\gamma _G:(0,\infty )\rightarrow (0,\infty )$ increasing with $\lim _{t\rightarrow 0}\gamma _G(t)=0$ and such that $M_n(\Gamma )\le \gamma _G(M_\omega ^p(\Gamma ))$ for every $\Gamma \in A(G)$. Then ${\tilde{H}}(x,f)<\infty$ a.e. If f is discrete on D, then $H(x,f)<\infty$ a.e.

Remark 1.7

If the weight $\omega$ from Theorem 1.2 is such that $\omega _y<\infty$ for every $y\in Y$, then ${\tilde{H}}(x,f)<\infty$ for every $x\in D$ and if f is discrete, then $H(x,f)<\infty$ for every $x\in D$. It is interesting that if f is only open, continuous and relation (1.3) is satisfied, we find that the dilatation ${\tilde{H}}(\cdot ,f)$ if finite on D.

Remark 1.8

Lemma 2.5 may be applied to the class of bi-conformal mappings, a class with important applications to mathematical models of non-linear elasticity (see [30]). Such mappings are homeomorphisms $f:X\rightarrow Y$ between two domains X, Y in ${\mathbb {R}}^n$ such that $f\in W_{loc}^{1,n} (X,Y)$, $f^{-1}\in W_{loc}^{1,n}(Y,X)$ and the energy $E_{XY}(f)=\int _X|f'(x)|^ndx+\int _Y|(f^{-1})'(y)|^ndy<\infty$. Since $f^{-1}$ satisfies condition (N) and f is a.e. differentiable, we see that $L(x,f)=|f'(x)|$ a.e. and hence $\int _XL(x,f)^ndx<\infty$. We apply Lemma 2.5 and we see that if f is of bi-conformal energy, then $H(x,f)<\infty$ a.e.

Theorem 1.9

Let $n\ge 2$, $D\subset {\mathbb {R}}^n$ be open, let $f:D\rightarrow {\mathbb {R}}^n$ be a mapping of finite distortion such that $f\in W_{loc}^{1,n}(D,{\mathbb {R}}^n)$ and $K_0(\cdot ,f)\in L_{loc}^p(D)$, where $p>n-1$ if $n\ge 3$ and $p=1$ if $n=2$. Then $H(x,f)<\infty$ a.e.

Note that while a Poletsky modular inequality is known for some very general classes of mappings of finite distortion (see [7, 33]), an inverse Poletsky modular inequality with a very general weight $\omega$ is not yet known in the class of mappings of finite distortion. Our result (1.3) is the first one of this type given for a large class of mappings of finite distortion. Some other properties of the class of mappings of finite distortion from Theorem 1.9 simply result because an inverse Poletsky modular inequality of type (1.3) holds for such mappings. We will attend these matters in some future paper.

2 Preliminaries

2.1 Topological preliminaries

     A mapping $f:D\rightarrow Y$ is open if it carries open sets into open sets, it is discrete if either $f^{-1}(y)=\phi$ or $f^{-1}(y)$ is a discrete subset of D for every $y\in Y$ and we say that f is a light mapping if $\dim f^{-1}(y)=0$ for every $y\in Y$. If $\dim f^{-1}(y)=0$, then for every compact connected subset $C\subset D$ such that $f(x)=y$ for every $x\in C$, we have that Card $C=1$ (see [27, p. 15]).

Let $f:X\rightarrow Y$ be continuous, open and light and $D\subset X$ open. We say that D is normal if ${\overline{D}}$ is compact and $\partial f(D)=f(\partial D)$. We see from page 186 in [55] (see also [35, Lem. 2.7]) that if $D\subset X$ is a normal domain, $p:[0,1]\rightarrow f(D)$ is a path and $x\in D$ is such that $f(x)=p(0)$, then there exists a path $q:[0,1]\rightarrow D$ such that $q(0)=x$ and $f\circ q=p$. We also say that the mapping f lifts the path p from x and the path q is a lifting of p from the point x. We also see that such a mapping has a fundamental system $(U_n)_{n\in {\mathbb {N}}}$ of neighbourhoods which are normal around each point $x\in X$.

It is interesting that even if $n\ge 3$ and $D\subset {\mathbb {R}}^n$ is a domain, a result of Wilson [57] shows that there exists a continuous, open and light mapping $f:D\rightarrow {\mathbb {R}}^n$ which is not discrete.

We say that $f:X\rightarrow Y$ is proper if $f^{-1}(K)$ is compact in X for every compact set $K\subset Y$. If $x\in X$ and $0<a<b$, we set $C_{x,a,b}= B(x,b){\setminus }{\overline{B}}(x,a)$ and if $\gamma :[a,b]\rightarrow X$ is a path, we set $|\gamma |=Im\gamma$. If $\Gamma _1,\Gamma _2\in A(X)$, we say that $\Gamma _1>\Gamma _2$ if every path $\gamma _1\in \Gamma _1$ has a subpath $\gamma _2\in \Gamma _2$ and if $\Gamma _1>\Gamma _2$, then $M_\omega ^p(\Gamma _1)\le M_\omega ^p(\Gamma _2)$. We say that the mapping $f:D\subset X\rightarrow Y$ satisfies condition $(N^{-1})$ if $\mu (f^{-1}(E))=0$ whenever $\nu (E)=0$.

We denote $G\subset \subset D$ if $G\subset D$ is open, ${\overline{G}}\subset D$ and ${\overline{G}}$ is compact.

We say that $\mathcal{{U}}$ is a fundamental system of neighbourhoods of a point x if for every $V\in \mathcal{{V}}(x)$ there exists $U\in \mathcal{{V}}(x)$, $U\in \mathcal{{U}}$ such that ${\overline{U}}\subset V$.

2.2 Geometric prerequisites

A metric space X is Ahlfors p-regular if there exists a constant $C\ge 1$ such that $r^p/C\le \mu (B(x,r))\le Cr^p$ for every $x\in X$ and every $r>0$ and we say that X is upper p-regular if only the second inequality holds.

We say that $(X,\mu )$ is a Loewner n-space if X has Hausdorff dimension n and there exists a function $\Phi :(0,\infty )\rightarrow (0,\infty )$ such that $M_n(\Delta (E,F,X))\ge \Phi (t)$ for every non-degenerate continua E and F in X with $d(E,F)/\min \{d(E),d(F)\}\ge t$ for $t\in (0,\infty )$. Here d(E) is the diameter of E and d(E, F) is the distance between E and F.

A metric space X is called linearly locally connected of constant $c\ge 1$ (c-LLC) if there exists $c\ge 1$ such that any two points in B(x, r) can be joined by a path in B(x, cr) and any two points in $X{{\setminus }}{\overline{B}}(x,r)$ can be joined by a path in $X{\setminus }{\overline{B}}(x,r/c)$ for every ball B(x, r) in X.

If $\omega :Y\rightarrow [0,\infty ]$ is $\nu$ measurable and finite a.e. and A is a Borel subset of Y, we put

and if $y\in Y$, we set

Lemma 2.1

Let (M, g), $(M',g')$ be smooth Riemannian n-manifolds having volume elements $d\nu _g$, respectively $d\nu _{g'}$, let $D\subset M$, $D'\subset M'$ be domains and let $f:D\rightarrow D'$ be a $C^1$ diffeomorphism. Then, if $p>1$ and $G\subset \subset D$ is a domain, there exists a constant $K\ge 1$ such that $M_p(\Gamma )\le KM_p(f(\Gamma ))$ for every $\Gamma \in A(G)$.

Proof

Let $G\subset \subset D$ be a domain, $\Gamma \in A(G)$, $\rho \in F(f(\Gamma ))$ and let $L>0$, $0<M<\infty$ be such that

$$\begin{aligned} L(x,f)\le M\quad \text {for every }x\in G \end{aligned}$$

and

$$\begin{aligned} J_f(x)=\lim _{r\rightarrow 0}\frac{\nu _{g^\prime }(f(B(x,r)))}{\nu _g(B(x,r))}\hspace{0.1cm}\quad \text {for}\quad \mu \hbox {-almost }x\in G. \end{aligned}$$

Let $\gamma \in \Gamma$. We see from [12, Lem. 2.3] that

$$\begin{aligned} 1\le \int _{f\circ \gamma }\rho ds\le \int _{\gamma } \rho \left( f(x)\right) L(x,f)ds\le \int _\gamma M\rho \circ f ds \end{aligned}$$

and hence $M\rho \circ f\in F(\Gamma )$.

Using the change of variable formulae we have

$$\begin{aligned} M_p(\Gamma )\le M^p\int _G(\rho \circ f)^pd\nu _g\le M^pL\int _G\rho \left( f(x)\right) ^pJ_f(x)d\nu _g\le M^pL\int _{f(G)}\rho ^p(y)d\nu _{g^\prime }. \end{aligned}$$

Lemma 2.2

Let (M, g) be a smooth n-Riemannian manifold, $n-1<p\le n$ and $x\in M$. Then there exists $r_x>0$ and a constant $C>0$ such that $M_p(\Delta (E,F,C_{x,a,b}))\ge C$ for all sets $E,F\subset C_{x,a,b}\subset M$ such that $S(x,t)\cap E\ne \phi$, $S(x,t)\cap F\ne \phi$ for every $0<a<t<b<r_x$ and the constant C depends on n, p, a, b and x.

Proof

We see from [34, Lem. 5.10, Cor. 6.11] that there exists an open set $U\subset {\mathbb {R}}^n$ such that $0\in U$ and local coordinates $\Phi :U\rightarrow M$ such that every geodesic sphere centered at x and of radius r in the Riemannian metric from M can be associated with the sphere centered at 0 and of radius r in the Euclidean metric from U. Let $r_0>0$ be such that ${\overline{B}}(x,r_0)\subset \Phi (U)$ and let $0<a<b<r_0$ and sets $E,F\subset C_{x,a,b}\subset M$ be such that $S(x,t)\cap E\ne \phi$, $S(x,t)\cap F\ne \phi$ for every $a<t<b$. Then $E_1=\Phi ^{-1}(E)$, $F_1=\Phi ^{-1}(F)$ and $C_{0,a,b}$ are subsets of ${\mathbb {R}}^n$ and using the fact that $E_1\cap S(0,t)\ne \phi$, $F_1\cap S(0,t)\ne \phi$ for every $a<t<b$ and [3, Thm. 3] or [53, Thm. 10.2], we find a constant $C>0$ such that

$$\begin{aligned} M_p(\Delta (E_1,F_1,C_{0,a,b}))\ge C, \quad \text {and}\quad C= {\left\{ \begin{array}{ll} \frac{C(n,p)}{n-p}\left( b^{n-p}-a^{n-p}\right) &{}\text {if }n-1<p<n,\\ C(n)\ln \frac{b}{a}&{}\text {if }p=n. \end{array}\right. } \end{aligned}$$

We see from Lemma 2.1 that there exists a constant $K>1$ depending only on p and x such that $M_p(\Gamma )\le KM_p(\Phi (\Gamma ))$ for every $\Gamma \in A(\Phi ^{-1}(B(x,r_0)))$. Then

$$\begin{aligned} 0<C<M_p(\Delta (E_1,F_1,C_{0,a,b})) \le KM_p(\Phi (\Delta (E_1,F_1,C_{0,a,b})))\le KM_p(\Delta (E,F,C_{x,a,b})) \end{aligned}$$

and hence $M_p(\Delta (E,F,C_{x,a,b}))\ge C/K>0$.

Lemma 2.3

Let $p>1$, X a metric measure space, $x\in X$ such that

$$\begin{aligned} \limsup _{r\rightarrow 0}\frac{\mu (B(x,r))}{r^p}<C<\infty \end{aligned}$$

and let $\omega :X\rightarrow [0,\infty ]$ be $\mu$ measurable and finite a.e. such that $\omega _x<\infty$. We recall that . Then there exists $r_x>0$ such that

$$\begin{aligned} M_{\omega }^p \left( \Gamma _{x,a,b}\right) \le \frac{Ce^p\omega _x\sum _{k=1}^\infty \frac{1}{k^p}}{\left( \ln \ln \left( \frac{be}{a}\right) \right) ^p} \end{aligned}$$

for every $0<a<b<be<r_x$.

Proof

Let $r_x>0$ be such that

$$\begin{aligned} \sup _{0<r<r_x}\frac{\mu (B(x,r))}{r^p}\le C \end{aligned}$$

and

Let $0<a<b<be<r_x$ and let $B_k = (B(x,be^{-k})$ and $\Delta _k=B_k{\setminus }{\overline{B}}_{k+1}$ for $k\in {\mathbb {N}}$. Let $\rho :C_{x,a,b}\rightarrow [0,\infty ]$ be defined by

$$\begin{aligned} \rho (z)=\frac{1}{d(x,z)\ln \left( \frac{be}{d(x,z)}\right) } \end{aligned}$$

if $z\in C_{x,a,b}$. Let $\gamma \in \Gamma _{x,a,b}$. We see from [12, Lem. 2.4] that

$$\begin{aligned} \int _\gamma \rho ds\ge \int _a^b\frac{dt}{t\ln \left( \frac{be}{t}\right) }=\ln \ln \!\left( \frac{be}{a}\right) \end{aligned}$$

and hence

$$\begin{aligned} \frac{\rho }{\ln \ln \left( \frac{be}{a}\right) }\in F(\Gamma _{x,a,b}). \end{aligned}$$

We also see that if $z\in \Delta _k$, then

$$\begin{aligned} \frac{1}{d(x,z)^p}\le \frac{1}{(be^{-(k+1)})^p}= \frac{e^p}{(be^{-k})^p}\le \frac{Ce^p}{\mu (B_k)} \end{aligned}$$

and

$$\begin{aligned} \frac{be}{d(x,z)}\ge \frac{be}{be^{-k}}=e^{k+1} \end{aligned}$$

and then

$$\begin{aligned} \frac{1}{\left( \ln \left( \frac{be}{d(x,z)}\right) \right) ^p}\le \frac{1}{(k+1)^p} \end{aligned}$$

. We have

Lemma 2.4

Let X be a Riemannian n-manifold, $n-1<p\le n$, $q>1$, Y a metric measure space, $D\subset X$ a domain and let $f:D\rightarrow Y$ be continuous and non-constant such that for every open set $G\subset \subset D$ and every $y\in f(G)$ there exists an increasing function $\gamma _{G,y}: (0,\infty )\rightarrow (0,\infty )$ with $\lim _{t\rightarrow 0}\gamma _{G,y}(t)=0$ and a function $\omega _{G,y}:Y\rightarrow [0,\infty ]$ $\nu$ measurable and finite $\nu$ a.e. depending on G and y such that

$$\begin{aligned} M_p(\Gamma _{G,f}(y,\epsilon ,\delta ))\le \gamma _{G,y}(M_{\omega _{G,y}}^q \left( f\left( \Gamma _{G,f}(y,\epsilon ,\delta )\right) \right) \quad \text {for every } 0<\epsilon <\delta \end{aligned}$$

(2.1)

and

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}M_{\omega _{G,y}}^q \left( \Gamma _{y,\epsilon ,\delta }\right) =0\quad \text {for every fixed } 0<\delta . \end{aligned}$$

(2.2)

Then f is a light mapping.

The proof of Lemma 2.4 follows the ones from [10, 47] given for mappings between Euclidean spaces. The crucial step is the use of Lemma 2.2 instead of the similar one valid in ${\mathbb {R}}^n$ and we give the whole proof for the sake of completeness.

Proof of Lemma 2.4

Suppose that f is not a light mapping. Then there exists $y\in f(G)$ and a component C of $f^{-1}(y)$ with Card $C>1$ and since f is not constant on D we see that $C\ne D$. Let $x\in D\cap \partial C$ and $\rho >0$ such that ${\overline{B}}(x,\rho )\subset D$ and $S(x,t)\cap C\ne \phi$ for every $0<t<\rho$. Since C is closed and $x\in \partial C\cap D$, we can find a ball $B_0=B(x_0,\rho _0)$ small enough such that ${\overline{B}}_0$ is connected and ${\overline{B}}_0\subset B(x,\rho ){{\setminus }} C$. Let $d=d(x,x_0)$ and

$$\begin{aligned} a=\inf _{z\in {\overline{B}}_0}d(x,z)\quad \text {and}\quad b=\sup _{z\in {\overline{B}}_0}d(x,z). \end{aligned}$$

Since Int $S(x,d)=\phi$, we see that $0<a<b<\rho$ and since

$$\begin{aligned} {\overline{B}}_0\cap S(x,a)\ne \phi ,{\overline{B}}_0\cap S(x,b)\ne \phi \end{aligned}$$

and ${\overline{B}}_0$ is connected, we see that ${\overline{B}}_0\cap S(x,t)\ne \phi$ for every $0<a<t<b<\rho$. Let $\Gamma =\Delta (C,B_0,C_{x,a,b})$. We see from Lemma 2.2 that there exists a constant $C=C(n,p,a,b,x)>0$ such that $M_p(\Gamma )>C$ and let $0<\delta <d(y,f({\overline{B}}_0))$ and $0<\epsilon <\delta$. If $B=B(x,\rho )$, we have $\Gamma >\Gamma _{B,f}(y,\epsilon ,\delta )$ and $f(\Gamma _{B,f} (y,\epsilon ,\delta ))\subset \Gamma _{y,\epsilon ,\delta }$ and then

$$\begin{aligned} 0<C<M_p (\Gamma )&\le M_p \left( \Gamma _{B,f} (y,\epsilon ,\delta )\right) \le \gamma _{B,y} (M_{\omega _{B,y}}^q \left( f\left( \Gamma _{B,f} (y,\epsilon ,\delta )\right) \right) \\&\le \gamma _{B,y}\left( M_{\omega _{B,y}}^q \left( \Gamma _{y,\epsilon ,\delta }\right) \right) \rightarrow 0\quad \text {as } \epsilon \rightarrow 0. \end{aligned}$$

We reached a contradiction and therefore proved that f is a light mapping.

The following result is a slight modification of [8, Lem. 2] and we give its proof for the sake of completeness.

Lemma 2.5

Let $n\ge 2$, $q>1$, $D\subset {\mathbb {R}}^n$ a domain, let $f\in W^{1,n-1}_{loc}(D,{\mathbb {R}}^n)$ be continuous, open, discrete satisfying the condition $(N^{-1})$ such that $\mu _n(B_f)=0$, $\mu _n(f(B_f))=0$ and such that $\int _DL(x,f)^qdx<\infty$. Suppose also that its Jacobian $J_f$ is strictly positive almost everywhere on the set where $|\nabla f|$ does not vanish. Then there exists $\omega \in L^1({\mathbb {R}}^n)$ such that $M_q(\Gamma )\le M_\omega ^q(f(\Gamma ))$ for every $\Gamma \in A(D)$.

Proof

Let $x\in D{{\setminus }} B_f$ and $U\in \mathcal{{V}}(x)$, $V\in \mathcal{{V}}\left( f(x)\right)$ be such that ${\overline{U}}\subset D{{\setminus }} B_f$ and $f|U:U\rightarrow V$ is a homeomorphism. We see from [14] that $(f|U)^{-1}\in W_{loc}^{1,1}(V,U)$.

Let $D_k\subset \subset D{\setminus } B_f$ be such that ${\overline{D}}_k\subset D_{k+1}$ for every $k\in {\mathbb {N}}$ and $D{\setminus } B_f=\bigcup _{k=1}^\infty D_{k}$. Let $y\in f(D_k){\setminus } f(B_f)$. Then $f^{-1}(y)\cap {\overline{D}}_k$ is a finite set and we find $V\in \mathcal{{V}}(y)$ and open sets $Q_k\subset \subset D{\setminus } B_f$ such that $f^{-1}(V)\cap {\overline{D}}_k\subset \bigcup _{i=1}^m Q_i\subset D_{k+1}$ and $f|Q_i:Q_i\rightarrow V$ is a homeomorphism such that $(f|Q_i)^{-1}\in W_{loc}^{1,1}(V,Q_i)$ for $i=1,\ldots ,m$ and $k\in {\mathbb {N}}$.

Let $k\in {\mathbb {N}}$ be fixed. Using Vitali’s covering theorem, we find $H_k\subset {\mathbb {R}}^n$ with $\mu _n(H_k)=0$, disjoint balls $V_i$ such that $f(D_k){\setminus } H_k=\bigcup _{i=1}^\infty V_i$, disjoint sets $Q_{ij}$ such that $f|Q_{ij}:Q_{ij}\rightarrow V_i$ is a homeomorphism and such that if $g_{ij}$ is its inverse, then $g_{ij}\in W_{loc}^{1,1}(V_i,Q_{ij})$ for $i\in {\mathbb {N}}$, $j=1,\ldots ,j(i)$ and $D_k{\setminus } f^{-1}(H_k)\subset \bigcup _{i=1}^\infty \bigcup _{j=1}^{j(i)}Q_{ij}$. Since f satisfies condition $(N^{-1})$, we see that $\mu _n(f^{-1}(H_k))=0$. Let

$$\begin{aligned}&\omega _k:{\mathbb {R}}^n\rightarrow [0,\infty ], \quad \omega _k(y)\\&\quad = {\left\{ \begin{array}{ll} \sum _{i=1}^\infty \chi _{V_i}(y)\sum _{j=1}^{j(i)}J_{g_{ij}}(y)L(g_{ij}(y),f)^q&{} \text {if }y\in f(D_k){\setminus } H_k, \\ \omega _k(y)=0 &{}\text {otherwise}. \end{array}\right. } \end{aligned}$$

Using the fact that $g_{ij}\in W_{loc}^{1,1}(V_i,Q_{ij})$ and $g_{ij}$ satisfies condition (N) and the change of variable formulae for Sobolev mappings (see [29, p. 107]), we have

$$\begin{aligned} \int _{f(D_k)}\omega _k(y)d\mu _n&=\sum _{i=1}^\infty \int _{V_i}\omega _k(y)d\mu _n=\sum _{i=1}^\infty \int _{V_i} \sum _{j=1}^{j(i)}J_{g_{ij}}(y)L(g_{ij}(y),f)^qd\mu _n\\ {}&=\sum _{i=1}^\infty \sum _{j=1}^{j(i)}\int _{Q_{ij}}L(z,f)^qd\mu _n\le \int _{D_k}L(z,f)^qd\mu _n. \end{aligned}$$

We proved that

$$\begin{aligned} \int _{f(D_k)}\omega _k(y)d\mu _n\le \int _{D_k}L(z,f)^qd\mu _n. \end{aligned}$$

(2.3)

If $\Gamma \in A(D)$ is such that $F(\Gamma )=\phi$, then $M_q(\Gamma )=0$ and the theorem is proved and we can suppose that $F(\Gamma )\ne \phi$. Let $\Gamma \in A(D)$ and $\Delta =\{\gamma \in \Gamma |\gamma$ is locally rectifiable and $f\circ \gamma ^0$ is absolutely continuous}. Using Fuglede’s theorem from [53, p. 95], we see that $M_q(\Gamma )=M_q(\Delta )$. Here, if $\gamma :[a,b]\rightarrow {\mathbb {R}}^n$ is rectifiable, the normalisation of $\gamma$ is the path $\gamma ^0:[a,l(\gamma )]\rightarrow {\mathbb {R}}^n$ such that $\gamma =\gamma ^0\circ s_\gamma$, where $s_\gamma$ is the length function of $\gamma$.

Let $\eta \in F(f(\Gamma ))$ and let $\rho :D\rightarrow [0,\infty ]$ be defined by $\rho (z)=\eta (f(z))L(z,f)$ if $z\in D$, $\rho (z)=0$ otherwise.

We see from [53, p. 12] that $\rho \in F(\Delta )$ and using the change of variable formulae for Sobolev mappings, we have

$$\begin{aligned} \int _{D_k}\rho \left( z\right) ^qd\mu _n&=\int _{D_k}\eta \left( f\left( z\right) \right) ^qL\left( z,f\right) ^qd\mu _n \le \sum _{i=1}^\infty \sum _{j=1}^{j\left( i\right) }\int _{Q_{ij}} \eta \left( f\left( z\right) \right) ^qL\left( z,f\right) ^qd\mu _n\\&=\sum _{i=1}^\infty \sum _{j=1}^{j\left( i\right) }\int _{V_i}\eta \left( f\left( g_{ij}\left( y\right) \right) \right) ^qL\left( g_{ij}\left( y\right) ,f\right) ^q J_{g_{ij}}\left( y\right) d\mu _n \\&=\sum _{i=1}^\infty \int _{V_i}\eta \left( y\right) ^q\chi _{V_i}\left( y\right) \sum _{j=1}^{j\left( i\right) }L\left( g_{ij}\left( y\right) ,f\right) ^q J_{g_{ij}}\left( y\right) d\mu _n\\&=\int _{V}\eta \left( y\right) ^q \omega _k\left( y\right) d\mu _n. \end{aligned}$$

We proved that

$$\begin{aligned} \int _{D_k}\rho (z)^qd\mu _n\le \int _{{\mathbb {R}}^n}\eta (y)^q\omega _k(y)d\mu _n. \end{aligned}$$

(2.4)

Let $\omega :{\mathbb {R}}^n\rightarrow [0,\infty ]$, $\omega (y)=\sum _{x\in f^{-1}(y)}J_{g_x}(y)L(g_x(y),f)^q$ if $y\in f(D){\setminus } f(B_f)$, $\omega (y)=0$ otherwise.

Here, if $x\in f^{-1}(y)$, we denote by $g_x$ a local inverse of f such that $g_x(y)=x$. We see that $\omega _k=\omega |D_k$ and $\omega _k\nearrow \omega$ $\mu _n$-almost everywhere.

We now use the fact that $\mu _n(B_f)=0$, $\mu _n(f(B_f))=0$ and letting $k\rightarrow \infty$ in (2.3) and (2.4) we find that

$$\begin{aligned} \int _{f(D)}\omega (y)d\mu _n<\infty \end{aligned}$$

and

$$\begin{aligned} M_q(\Gamma )=M_q(\Delta )\le \int _D\rho (z)^qd\mu _n\le \int _{{\mathbb {R}}^n}\eta (y)^q\omega (y)d\mu _n. \end{aligned}$$

Since $\eta \in F(f(\Gamma ))$ was arbitrarily chosen, we have proven that

$$\begin{aligned} M_q(\Gamma )\le M_\omega ^q(f(\Gamma ))\quad \text {for every }\Gamma \in A(D). \end{aligned}$$

(2.5)

Remark 2.6

Let us take in Lemma 2.5 the Borel measure $\nu _\omega$ given by $\nu _\omega (A)=\int _A\omega (x)d\mu _n$ for every Borel set $A\subset D$. Then $\nu _\omega \le \mu _n$ and we have the equality

$$\begin{aligned} \int _A\omega (y)\eta (y)^qd\mu _n=\int _A\eta (y)^qd\nu _\omega \end{aligned}$$

for every Borel set $A\subset {\mathbb {R}}^n$ and every Borel function $\eta :{\overline{{\mathbb {R}}^n}}\rightarrow [0,\infty ]$. Using (2.5), we see that

$$\begin{aligned} M_q(\Gamma )\le M_q(f(\Gamma ))\quad \text {for every }\Gamma \in A(D). \end{aligned}$$

(2.6)

Here, if $f:D\rightarrow {\mathbb {R}}^n$ is as in Lemma 2.5, we take on D the Lebesgue measure $\mu _n$ and we take on ${\mathbb {R}}^n$ the measure $\nu _\omega$. We see in this way that if $q=n$ in Lemma 2.5, we have from Theorem 1.2 that f is metrically quasiregular, taking the measure $\mu _n$ on D and the measure $\nu _\omega$ on ${\mathbb {R}}^n$.

As a special case, the mappings of finite distortion $f:D\rightarrow {\mathbb {R}}^n$ from Theorem 1.9 are quasiregular for some special metric $\nu _\omega$ on ${\mathbb {R}}^n$.

Using the equivalence between metric quasiconformality and geometric quasiconformality established in [6, 21], we see that the mappings of finite distortion from Theorem 1.9 also satisfy the equality

$$\begin{aligned} M_n(\Gamma )\le K\int _{{\mathbb {R}}^n}N(y,f,G)d\nu _\omega \end{aligned}$$

(2.7)

for some $K\ge 1$, every open set $G\subset \subset D$, every $\Gamma \in A(D)$ and every $\rho \in F(f(\Gamma ))$.

Remark 2.6 was pointed out to me by the referee.

3 Proof of the results

Proof of Theorem 1.2

Let $G\subset \subset D$ be open and $x\in G$ with $(\omega _G)_{f(x)}<\infty$. Let $U\in \mathcal{{V}}(x)$ be such that ${\overline{U}}\subset G$, $f(x)\not \in f(\partial U)$ and if f is discrete at x we take U such that ${\overline{U}}\cap f^{-1}(f(x))=\{x\}$ and let $\rho '=d(f(x),f(\partial U))$. Then $f^{-1}(f(\partial U))$ is compact in U and the mapping $g=f|(U{\setminus } f^{-1}(f(\partial U)):U{\setminus } f^{-1}(f(\partial U))\rightarrow f(U){\setminus } f(\partial U))$ is continuous, open, light and proper and using some results of Whyburn (see [55, p. 186]) we see that g lifts the paths. Using the openness of the mapping f we see that $\partial f(U)\subset f(\partial U)$. Let $0<\rho '_1<\rho '/c$ and let us show that $B(f(x),\rho _1')\subset f(U)$. Indeed, let $y\in B(f(x),\rho _1')$. Since Y is c-LLC, we find Q connected such that $f(x)\in Q$, $y\in Q$ and $Q\subset B(f(x),c\rho _1')$. Since $B(f(x),c\rho _1')\cap f(\partial U)=\phi$ we find that $B(f(x),c\rho _1')\cap \partial f(U)=\phi$. Now $Q=(Q\cap f(U))\cup (Q\cap \complement {\overline{f(U)}})$ and since Q is connected we find that $Q\subset f(U)$. Since $y\in B(f(x),\rho _1')$ was arbitrarily chosen, we obtain that $B(f(x),\rho _1')\subset f(U){\setminus } f(\partial U)$.

We recall from [24, Lem. 3.14] that whenever $0<2s<R$ and $y\in X$, then the upper n-regularity of X implies that

$$\begin{aligned} (3.1)\hspace{1cm}M_n(\Gamma _{x,s,R})\le \frac{B(n,C_2)}{\left( \ln \left( \frac{R}{s}\right) \right) ^{n-1}}. \end{aligned}$$

Here B depends only on n and the upper n-regularity constant $C_2$. We let $U\in \mathcal{{V}}(x)$, $\rho _1^\prime >0$ as above and $\rho _1=d \left( x,f^{-1}\left( S \left( f(x),\rho _1^\prime \right) \right) \right) >0$.

We fix $k\in {\mathbb {N}}$ such that

$$\begin{aligned} \frac{B(n,C_2)}{(\ln k)^{n-1}}<\frac{\Phi (1)}{2}, \end{aligned}$$

for the constant in (3.1).

We first consider the case when f is discrete and we denote $L_r=L(x,f,r)$ and $l_r=l(x,f,r)$ for $r>0$.

Let $r_{f(x)}>0$ denote the radius from Lemma 2.3 applied to the point $f(x)\in Y$. There exists $r_x>0$ such that $cL_r<\min \{r_{f(x)},\rho _1^\prime \}, {\overline{B}}(x,2kr)\subset U{\setminus } f^{-1}(f(\partial U))$ and

$$\begin{aligned} f({\overline{B}}(x,2kr))\subset B(f(x),\rho _1^\prime )\subset f(U){\setminus } f(\partial U) \end{aligned}$$

for every $0<r<r_x$.

Now we suppose that there exists $r_m<r_x$ with $r_m\rightarrow 0$ and

$$\begin{aligned} (3.2)\hspace{1cm}\frac{L_{r_m}}{l_{r_m}}>C_X. \end{aligned}$$

We derive a contradiction from (3.2). To this end, let $r=r_m$ for some $m\in {\mathbb {N}}$ and let $y\in B(f(x),\rho ^\prime ){\setminus }{\overline{B}} (f(x),\rho _1^\prime )$.

Since Y is c-LLC, we find a path $p:[0,1]\rightarrow Y{\setminus } B(f(x),L_r/c)$ such that $p(0)=f(a_r)$, $p(1)=y$ and $d(f(x),f(a_r))=L_r$ with $a_r\in S(x,r)$. Let $0\le t_1$ be such that $p_1=p|[0,t_1]\subset {\overline{B}}(f(x), \rho _1')$ and $p_1(t_1)\in S(f(x),\rho _1')$. Since the mapping

$$\begin{aligned} g:U{\setminus } f^{-1}\left( f(\partial U)\right) \rightarrow f(U){\setminus } f(\partial U) \end{aligned}$$

lifts the paths, we can find a path $q_1:[0,t_1]\rightarrow U{{\setminus }} f^{-1}(f(\partial U))$ such that $q_1(0)=a_r$ and $f\circ q_1=p_1$. Then $q_1(t_1)\in {\overline{U}}\cap f^{-1}(S(f(x),\rho _1'))$ and hence $d(x,q_1(t_1))\ge \rho _1\ge 2kr$. We can find now a subpath $q_2$ of $q_1$ such that $a_r\in |q_2|$, $|q_2|\subset {\overline{B}}(x,2r)$ and $|q_2|\cap S(x,2r)\ne \phi$. Let $F=|q_2|$ and we see that $d(F)\ge r$.

Let $b_r\in S(x,r)$ such that $l_r=d(f(x),f(b_r))$. Since Y is c-LLC, we find a path

$$\begin{aligned} p_2:[0,1]\rightarrow B(f(x),cl_r), \end{aligned}$$

where

$$\begin{aligned} B(f(x),cl_r)\subset B\!\left( f(x),\frac{L_r}{c}\right) \subset B(f(x),\rho _1^\prime )\subset f(U){\setminus } f(\partial U) \end{aligned}$$

such that $p_2(0)=f(b_r)$ and $p_2(1)=f(x)$. We used, here the fact that $L_r/l_r\ge c^2$. Since the mapping g lifts the paths, we can find a path $q:[0,1]\rightarrow U{{\setminus }} f^{-1}(f(\partial U))$ such that $q(0)=b_r$ and $f\circ q=p_2$. Then $f(q(1))=p_2(1)=f(x)$ and $q(1)\in {\overline{U}}\cap f^{-1}(f(x))$ and hence $q(1)=x$. We now find a subpath $q_3$ of q such that $|q_3|\subset {\overline{B}}(x,r)$, $x\in |q_3|$ and $|q_3|\cap S(x,r)\ne \phi$ and let $E=|q_3|$. We see that $d(E)\ge r$ and we have $d(E,F)/\min \{d(E),d(F)\}\le r/r=1$. Let $\Gamma =\Delta (E,F,B(x,2kr))$ and $\Gamma _1=\Delta (E,F,X){{\setminus }} \Gamma$.

We see that $\Gamma \in A(G)$, that $\Gamma _1>\Gamma _{x,2r,2kr}$ and hence $M_n(\Gamma _1)\le M_n(\Gamma _{x,2r,2kr})<\Phi (1)/2$. Since X is n-Loewner, we see that

$$\begin{aligned} M_n(\Gamma )+M_n(\Gamma _1)\ge M_n(\Delta (E,F,X))\ge \Phi (1) \end{aligned}$$

and hence

$$\begin{aligned} (3.3)\hspace{1cm}\frac{\Phi (1)}{2}<M_n(\Gamma ). \end{aligned}$$

We see that $f(\Gamma )>\Gamma _{x,cl_r,L_r/c}$ and using Lemma 2.3 we have

$$\begin{aligned} \frac{\Phi (1)}{2}\le & {} M_n(\Gamma )\le \gamma _G\bigg (M_{\omega _G}^p(f(\Gamma )\bigg )\le \gamma _G\bigg (M_{\omega _G}^p\bigg (\Gamma _{x,cl_r,L_r/c}\bigg )\bigg )\\\le & {} \gamma _G \left( \frac{C(\omega _G)_{f(x)}}{\bigg (\ln \ln \bigg (\frac{eL_r}{c^2l_r}\bigg )\bigg )^p}\right) . \end{aligned}$$

Let

$$\begin{aligned} t=\frac{C(\omega _G)_{f(x)}}{\bigg (\ln \ln \bigg (\frac{eL_r}{c^2l_r}\bigg )\bigg )^p}. \end{aligned}$$

We have

$$\begin{aligned} \frac{c^2}{e}\exp \!\left( \exp \!\left( \left( \frac{C(\omega _G)_{f(x)}}{\epsilon }\right) ^{1/p}\right) \right) \le \frac{L_r}{l_r} \end{aligned}$$

and hence $0\le t\le \epsilon$. This implies that

$$\begin{aligned} \frac{\Phi (1)}{2}\le \gamma _G(t)<\frac{\Phi (1)}{2} \end{aligned}$$

and we reached a contradiction. We therefore proved that $H(x,f)\le C_x$.

Suppose now that f is a light mapping. Let $G\subset \subset D$ be open and $x\in G$ be such that $(\omega _G)_{f(x)}<\infty$, let $U\in \mathcal{{V}}(x)$ be such that ${\overline{U}}\subset G$, $f(x)\not \in f(\partial U)$ and let $g=f|U{{\setminus }} f^{-1}(f(\partial U)):U{{\setminus }} f^{-1}(f(\partial U))\rightarrow U{{\setminus }} f(\partial U)$. Let $\rho ^\prime =d(f(x),f(\partial U))$. We take $\rho _1^\prime$ and $\rho _1$ as before and let $L_r=L(x,f,r)$ and $\tilde{l}_r= \tilde{l}(x,f,r)$ for $r>0$. Let $k\in {\mathbb {N}}$, $r_{f(x)}$ and $r_x$ as before. Now suppose that there exists $r_m<r_x$, $r_m\rightarrow 0$ such that

$$\begin{aligned} (3.4)\hspace{1cm} \frac{L_{r_m}}{\tilde{l}_{r_m}}>C_x\quad \text {for every }m\in {\mathbb {N}}. \end{aligned}$$

We derive a contradiction from (3.4). To this end, let $r=r_m$ for some $m\in {\mathbb {N}}$ and let $\lambda >1$ be such that $\lambda cL_r<\rho _1^\prime$ and $\lambda c^2<L_r/\tilde{l}_r$. Let $y\in B(f(x),\lambda \tilde{l}_r){{\setminus }} f(B(x,r))$.

Since Y is c-LLC, we can find a path $p_3:[0,1]\rightarrow B(f(x),\lambda c\tilde{l}_r)\subset B(f(x),\rho _1^\prime )$ such that $p_3(0)=f(x)$ and $p_3(1)=y$. Since the mapping g lifts the paths, we can find a path $q:[0,1]\rightarrow U{{\setminus }} f^{-1}(f(\partial U))$ such that $q(0)=x$ and $f\circ q=p_3$ and $f(q(1))=p_3(1)=y\not \in f(B(x,r))$. This implies that $q(1)\not \in B(x,r)$ and let $q_4$ be a subpath of $q_3$ such that $q_4(0)=x$, $|q_4|\subset {\overline{B}}(x,r)$ and $|q_4|\cap S(x,r)\ne \phi$ and let $E=|q_4|$. Then $d(E)\ge r$.

As before, we find a path $q_2$ such that $|q_2|\subset {\overline{B}}(x,2r)$, $|q_2|\cap S(x,r)\ne \phi$, $|q_2|\cap S(x,2r)\ne \phi$, we take $F=|q_2|$ and we see that $d(E)\ge r$, $d(F)\ge r$ and $d(E,F)/\min \{d(E),d(F)\}\le 1$. Let $\Gamma =\Delta (E,F,B(x,2kr)$. As before, we see that

$$\begin{aligned} \frac{\Phi (1)}{2}\le M_n(\Gamma )\le \gamma _G\left( \frac{C(\omega _G)_{f(x)}}{\left( \ln \ln \left( \frac{eL_r}{c^2{\tilde{l}}_r}\right) \right) ^p}\right) \end{aligned}$$

and arguing as before we reached a contradiction. We therefore proved that $\tilde{H}(x,f)\le C_x$.

Proof of Theorem 1.6

We see from Lemma 2.4 that f is a light mapping and we apply Theorem 1.2.

Proof of Theorem 1.9

We have seen before that there exists $\omega \in L^1({\mathbb {R}}^n)$ such that $M_n(\Gamma )\le M_\omega ^n(f(\Gamma ))$ for every $\Gamma \in A(D)$ and we apply Theorem 1.6.