On the Linear Dilatation of the Map**s Satisfying an Inverse Poletsky Modular Inequality in Metric Spaces

Abstract

We study the linear dilatation of the map**s satisfying an inverse Poletsky inequality in metric spaces. We also show that under certain conditions such map**s 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 map**s 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 map** $f:D\subset {\mathbb {R}}^n\rightarrow {\mathbb {R}}^n$ (see the monographs [41, 42, 54] for more information about quasiregular map**s). Several generalizations of quasiregular map**s were developed in the last 35 years. One of the most important is the class of map**s of finite distortion (see the monographs [26, 29]) for which a generalized Poletsky modular inequality is established in [33]. A map** $f:D\rightarrow {\mathbb {R}}^n$ 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 map**s $f:D\rightarrow {\mathbb {R}}^n$ 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 map**s distinguished by moduli inequalities has been intensively studied in the last 20 years. Such an approach was proposed by Martio, first on map**s 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 map**s 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 map**s 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 map**s (see also [7, 9, 43, 47]).

Quasiconformal and quasiregular map**s between metric measure spaces X and Y were also studied in [2, 6, 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 map** 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 map** $f:X\rightarrow Y$ 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,

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

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

Also, a large class of map**s 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 map** 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 map**, $G\subset \subset D$ 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 map**s 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 map**s. 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 map** 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 map** 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 map** 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 map**s, a class with important applications to mathematical models of non-linear elasticity (see [30]). Such map**s 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 map** 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 map**s 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 map**s of finite distortion. Our result (1.3) is the first one of this type given for a large class of map**s of finite distortion. Some other properties of the class of map**s of finite distortion from Theorem 1.9 simply result because an inverse Poletsky modular inequality of type (1.3) holds for such map**s. We will attend these matters in some future paper.

2 Preliminaries

2.1 Topological preliminaries

     A map** $f:D\rightarrow Y$ 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 map** 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 map** 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 map** 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 map** $f:D\rightarrow {\mathbb {R}}^n$ 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 map** $f:D\subset X\rightarrow Y$ 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 map**.

The proof of Lemma 2.4 follows the ones from [10, 47] given for map**s 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 map**. 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 map**.

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 map**s (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 map**s, 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 map**s 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, 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 map** $g=f|(U{\setminus } f^{-1}(f(\partial U)):U{\setminus } f^{-1}(f(\partial U))\rightarrow f(U){\setminus } f(\partial U))$ 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 map** 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 map**

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

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 map** 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 map**. 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 map** 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 map** 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.