Let X be a metric measure space and let \({\overline{\Gamma }}(X)\) denote the set of Lipschitz curves \([0,1]\rightarrow X\). Recall that p-almost every curve in X admits a Lipschitz reparametrization. In this section we construct essential pull-back distances by Sobolev maps. The key notion here is the essential infimum of a functional over a path family.
Definition 5.1
Let \(p\ge 1\), \(F:{\overline{\Gamma }}(X)\rightarrow [-\infty ,\infty ]\) be a function and \(\Gamma \subset {\overline{\Gamma }}(X)\) a path family. Define the p-essential infimum of F over \(\Gamma \) by
$$\begin{aligned} \underset{\Gamma }{{\text {essinf}}_{p}}F=\underset{\gamma \in \Gamma }{{\text {essinf}}_{p}}F(\gamma ):=\sup _{{\text {Mod}}_{p}(\Gamma _0)=0}\inf \{ F(\gamma ):\ \gamma \in \Gamma \setminus \Gamma _0 \} \end{aligned}$$
with the usual convention \(\inf \varnothing =\infty \).
It is clear that the supremum may be taken over curve families \(\Gamma _\rho \) for non-negative Borel functions \(\rho \in L^p(X)\), where
$$\begin{aligned} \Gamma _\rho :=\left\{ \gamma : \int _\gamma \rho =\infty \right\} . \end{aligned}$$
Remark 5.2
We have the following alternative expression for \({\text {essinf}}_p\):
$$\begin{aligned} \underset{\Gamma }{{\text {essinf}}_{p}}F&=\max \{ \lambda>0:\ {\text {Mod}}_p(\Gamma \cap \Gamma _F(\lambda ))=0 \}\\&=\inf \{ \lambda>0:\ {\text {Mod}}_p(\Gamma \cap \Gamma _F(\lambda ))>0 \}. \end{aligned}$$
Here
$$\begin{aligned} \Gamma _F(\lambda ):=\{\gamma \in {\overline{\Gamma }}(X): F(\gamma )<\lambda \}. \end{aligned}$$
Let Y be a complete metric space, \(p\ge 1\), and \(u\in N^{1,p}_{loc}(X;Y)\). Given a path family \(\Gamma \) in X we set
$$\begin{aligned} ess\ell _{u,p}(\Gamma ):=\underset{\gamma \in \Gamma }{{\text {essinf}}_p}\ell (u\circ \gamma ). \end{aligned}$$
Definition 5.3
Let \(u\in N^{1,p}_{loc}(X;Y)\). Define the essential pull-back distance
\(\displaystyle d'_{u,p}:X\times X\rightarrow [0,\infty ]\) by
$$\begin{aligned} d'_{u,p}(x,y):=\lim _{\delta \rightarrow 0}ess\ell _{u,p}(\Gamma ({\bar{B}}(x,\delta ),{\bar{B}}(y,\delta )), \quad x,y\in X. \end{aligned}$$
In general, the essential pull-back distance may assume both values 0 and \(\infty \) for distinct points, and it need not satisfy the triangle inequality. We denote by \(d_{u,p}\) the maximal pseudometric not greater than \(d_{u,p}'\), given by
$$\begin{aligned} d_{u,p}(x,y)=\inf \left\{ \sum _{i=1}^nd_{u,p}'(x_i,x_{i-1}):\ x_0,\ldots ,x_n\in X,\ x_0=x,\ x_n=y \right\} ,\quad x,y\in X. \end{aligned}$$
The maximal pseudometric below \(d_{u,p}'\) may also fail to be a finite valued. We give two situations that guarantee finiteness and nice properties of the arising metric space. Firstly, in Sect. 5.2 we consider the simple case \(u=\mathrm {id}:X\rightarrow X\) and use it to prove Theorem 1.4. Secondly, in Proposition 5.12 we prove that, when X supports a Poincaré inequality and \(u\in N^{1,p}_{loc}(X;Y)\) for large enough p, the distance in Definition 5.3 is a finite valued pseudometric.
For both we need the notion of regular curves, whose properties we study next.
Regular curves
Throughout this subsection \((X,d,\mu )\) is a metric measure space. We define a metric on \({\overline{\Gamma }}(X)\) by setting
$$\begin{aligned} d_\infty (\alpha ,\beta ):=\sup _{0\le t\le 1}d(\alpha (t),\beta (t)) \end{aligned}$$
for any two Lipschitz curves \(\alpha ,\beta \). By a simple application of the Arzela-Ascoli theorem it follows that \(({\overline{\Gamma }}(X),d_\infty )\) is separable.
Definition 5.4
A curve \(\gamma \in {\overline{\Gamma }} (X)\) is called (u, p)-regular if \(u\circ \gamma \) is absolutely continuous and
$$\begin{aligned} ess\ell _{u,p} B(\gamma ,\delta )\le \ell (u\circ \gamma ) \end{aligned}$$
for every \(\delta >0\).
Proposition 5.5
p-almost every Lipschitz curve is (u, p)-regular.
For the proof we will denote \(\Gamma _{u,p}(\lambda )=\{\gamma : \ell (u\circ \gamma )<\lambda \}\).
Proof of Proposition 5.5
Denote by \(\Gamma _0\) the set of curves in \(\overline{\Gamma }(X)\) which are not (u, p)-regular. We may write \(\Gamma _0=\Gamma _1\cup \Gamma _2\), where
$$\begin{aligned} \Gamma _1=\{ \gamma : u\circ \gamma \text { not absolutely continuous} \},\quad \Gamma _2=\Gamma _0\setminus \Gamma _1. \end{aligned}$$
By the fact that \(u\in N^{1,p}_{loc}(X;Y)\) we have \({\text {Mod}}_p(\Gamma _1)=0\). It remains to show that \({\text {Mod}}_p(\Gamma _2)=0\). For any \(\delta >0\) and \(\gamma \in {\overline{\Gamma }}(X)\), set
$$\begin{aligned} \varepsilon (\delta ,\gamma )&:=ess\ell _{u,p}B(\gamma ,\delta )-\ell (u\circ \gamma )\\ \delta (\gamma )&:=\sup \{ \delta>0: \varepsilon (\delta ,\gamma )>0 \}. \end{aligned}$$
Note that \(\delta (\gamma )>0\) if and only if \(\gamma \in \Gamma _2\). Using Remark 5.2 we make the following observation which holds for \(\delta ,\varepsilon >0\):
$$\begin{aligned} {\text {Mod}}_p(B(\gamma ,\delta )\cap \Gamma _{u,p}(\ell (u\circ \gamma )+\varepsilon ))=0\text { implies }\delta \le \delta (\gamma )\text { and }\varepsilon \le \varepsilon (\delta ,\gamma ). \end{aligned}$$
(5.1)
Moreover, for any \(\gamma \in \Gamma _2\) and \(\delta >0\), we have
$$\begin{aligned} {\text {Mod}}_p(B(\gamma ,\delta )\cap \Gamma _{u,p}(\ell (u\circ \gamma )+\varepsilon (\delta ,\gamma )))=0. \end{aligned}$$
(5.2)
Let \(\{\gamma _i \}_{i\in \mathbb {N}}\subset \Gamma _2\) be a countable dense set. For each \(i,k\in \mathbb {N}\) and rational \(r>0\) let \(\gamma _{i,k,r}\in B(\gamma _i,r)\cap \Gamma _2\) satisfy
$$\begin{aligned} \ell (u\circ \gamma _{i,k,r})<\inf \{ \ell (u\circ \beta ):\ \beta \in B(\gamma _i,r)\cap \Gamma _2 \}+1/k. \end{aligned}$$
By (5.2) it suffices to prove that
$$\begin{aligned} \Gamma _2\subset \bigcup _{i,k\in \mathbb {N}}\bigcup _{r,\delta \in \mathbb {Q}_+}B(\gamma _{i,k,r},\delta )\cap \Gamma _{u,p} (\ell (u\circ \gamma _{i,k,r})+\varepsilon (\delta ,\gamma _{i,k,r})). \end{aligned}$$
For any \(\gamma \in \Gamma _2\) let \(i\in \mathbb {N}\) be such that \(d_{\infty }(\gamma ,\gamma _i)<\delta (\gamma )/8\). Choose rational numbers \(r,\delta \in \mathbb {Q}_+\) such that \(d_\infty (\gamma _i,\gamma )<r<\delta (\gamma )/8\) and \(\delta (\gamma )/4<\delta <\delta (\gamma )/2\), and a natural number \(k\in \mathbb {N}\) so that \(1/k<\varepsilon (2\delta ,\gamma )\).
We will show that
$$\begin{aligned} \gamma \in B(\gamma _{i,k,r},\delta )\cap \Gamma _{u,p}(\ell (u\circ \gamma _{i,k,r})+\varepsilon (\delta ,\gamma _{i,k,r})). \end{aligned}$$
(5.3)
Indeed, since \(\gamma _{i,k,r}\in B(\gamma _i,r)\cap \Gamma _2\), the triangle inequality yields
$$\begin{aligned} d_{\infty }(\gamma ,\gamma _{i,k,r})\le d_{\infty }(\gamma ,\gamma _{i})+d_{\infty }(\gamma _i,\gamma _{i,k,r})<2r<\delta (\gamma )/4<\delta . \end{aligned}$$
In particular
$$\begin{aligned} \gamma \in B(\gamma _{i,k,r},\delta ) \end{aligned}$$
(5.4)
and also
$$\begin{aligned} B(\gamma _{i,k,r},\delta )\subset B(\gamma ,2\delta ) \end{aligned}$$
(5.5)
To bound the length of \(u\circ \gamma \), observe that
$$\begin{aligned} \ell (u\circ \gamma _{i,k,r})&<\inf \{ \ell (u\circ \beta ):\ \beta \in B(\gamma _i,r)\cap \Gamma _2 \}+1/k<\ell (u\circ \gamma )+1/k\\&<\ell (u\circ \gamma )+\varepsilon (2\delta ,\gamma ) \end{aligned}$$
Setting \(\varepsilon :=\ell (u\circ \gamma )+1/k-\ell (u\circ \gamma _{i,k,r})>0\) it follows that
$$\begin{aligned} \Gamma _{u,p}(\ell (u\circ \gamma _{i,k,r})+\varepsilon )\subset \Gamma _{u,p}(\ell (u\circ \gamma )+\varepsilon (2\delta ,\gamma )) \end{aligned}$$
(5.6)
Combining (5.5) and (5.6) with (5.2) we obtain
$$\begin{aligned} {\text {Mod}}_p(B(\gamma _{i,k,r},\delta )\cap \Gamma _{u,p}(\ell (u\circ \gamma _{i,k,r})+\varepsilon ))=0, \end{aligned}$$
which, by (5.1) yields \(\varepsilon \le \varepsilon (\delta ,\gamma _{i,k,r})\). Thus
$$\begin{aligned} \ell (u\circ \gamma )<\ell (u\circ \gamma )+1/k=\ell (u\circ \gamma _{i,k,r})+\varepsilon \le \ell (u\circ \gamma _{i,k,r})+\varepsilon (\delta ,\gamma _{i,k,r}) \end{aligned}$$
which, together with (5.4), implies (5.3). This completes the proof. \(\square \)
For the next two results we assume that \(u\in N^{1,p}_{loc}(X;Y)\) is continuous. We record the following straightforward consequence of the definition of \(d_{u,p}'\) and the continuity of u as a lemma.
Lemma 5.6
For each \(x,y\in X\) we have
$$\begin{aligned} d(u(x),u(y))\le d'_{u,p}(x,y) \end{aligned}$$
For the next proposition, we denote by \(\ell _{u,p}(\gamma )\) the length of a curve \(\gamma \) with respect to the pseudometric \(d_{u,p}\).
Proposition 5.7
If \(\gamma :[a,b]\rightarrow X\) is (u, p)-regular and \(a\le t\le s\le b\) then \(\gamma |_{[t,s]}\) is (u, p)-regular and
$$\begin{aligned} \ell _{u,p}(\gamma )=\ell (u\circ \gamma ). \end{aligned}$$
Proof
Denote \(\eta :=\gamma |_{[s,t]}\). Let
$$\begin{aligned} \Gamma _n:=\left\{ c|_{[s,t]}\ :\ c \in B\left( \gamma ,\frac{1}{n}\right) \cap \Gamma _{u,p}\left( \ell (u\circ \gamma )+\frac{1}{n}\right) \right\} . \end{aligned}$$
Since \(\gamma \) is (u, p)-regular we have
$$\begin{aligned} {\text {Mod}}_p(\Gamma _n)\ge {\text {Mod}}_p\left( B\left( \gamma ,\frac{1}{n}\right) \cap \Gamma _{u,p}\left( \ell (u\circ \gamma )+\frac{1}{n}\right) \right) >0. \end{aligned}$$
We claim that for every \(\varepsilon >0\) there exists \(n_0\in \mathbb {N}\) so that
$$\begin{aligned} \Gamma _n\subset B\left( \eta ,\frac{1}{n}\right) \cap \Gamma _{u,p}(\ell (u\circ \eta )+\varepsilon ) \end{aligned}$$
(5.7)
for all \(n\ge n_0\). Indeed, otherwise there exists \(\varepsilon _0>0\) and a sequence
$$\begin{aligned} \gamma _{n_k}\in B\left( \gamma ,\frac{1}{n_k}\right) \cap \Gamma _{u,p}\left( \ell (u\circ \gamma )+\frac{1}{n_k}\right) \end{aligned}$$
so that
$$\begin{aligned} \ell (u\circ \gamma _{n_k}|_{[s,t]})\ge \ell (u\circ \eta )+\varepsilon _0. \end{aligned}$$
Thus
$$\begin{aligned} \ell (u \circ \gamma )+\frac{1}{n_k}&\ge \ell (u \circ \gamma _{n_k})=\ell \left( u \circ \gamma _{n_k}|_{[s,t]}\right) +\ell \left( u \circ \gamma _{n_k}|_{[s,t]^c}\right) \\&\ge \ell \left( u \circ \eta \right) +\varepsilon _0+\ell \left( u \circ \gamma _{n_k}|_{[s,t]^c}\right) \end{aligned}$$
yielding
$$\begin{aligned} \ell \left( u \circ \gamma _{|[s,t]^c}\right) +\frac{1}{n_k}\ge \ell \left( u \circ \gamma _{n_k}|_{[s,t]^c}\right) +\varepsilon _0. \end{aligned}$$
(5.8)
By taking \(\liminf _{k\rightarrow \infty }\) in (5.8) we obtain
$$\begin{aligned} \ell \left( u \circ \gamma _{|[s,t]^c}\right) \ge \ell \left( u \circ \gamma _{|[s,t]^c}\right) +\varepsilon _0, \end{aligned}$$
which is a contradiction.
Thus (5.7) holds true. If \(\varepsilon ,\delta >0\), let \(n\in \mathbb {N}\) be such that (5.7) holds and \(\delta >1/n\). We have
$$\begin{aligned} 0<{\text {Mod}}_p\Gamma _n\le {\text {Mod}}_p(B(\eta ,\delta )\cap \Gamma _{u,p}(\ell (u\circ \eta )+\varepsilon )), \end{aligned}$$
implying \(ess\ell _{u,p}B(\eta ,\delta )\le \ell (u\circ \eta )+\varepsilon \). Since \(\varepsilon >0\) is arbitrary, \(\eta \) is (u, p)-regular.
To prove the equality in the claim note that, since \(\gamma \) is (u, p)-regular we have
$$\begin{aligned} d_{u,p}(\gamma (t),\gamma (s))\le d_{u,p}'(\gamma (t),\gamma (s))\le \lim _{\delta \rightarrow 0}ess\ell _{u,p}B(\gamma |_{[s,t]},\delta )\le \ell (u\circ \gamma |_{[s,t]}) \end{aligned}$$
for any \(s\le t\). It follows that
$$\begin{aligned} \ell _{u,p}(\gamma )\le \ell (u\circ \gamma ). \end{aligned}$$
On the other hand Lemma 5.6 implies that
$$\begin{aligned} d(u(\gamma (t)),u(\gamma (s)))\le d_{u,p}(\gamma (t),\gamma (s)) \end{aligned}$$
whenever \(s\le t\), from which the opposite inequality readily follows. \(\square \)
The Sobolev-to-Lipschitz property in thick quasiconvex spaces
In this subsection let \(p\in [1,\infty ]\) and \(X=(X,d,\mu )\) be p-thick quasiconvex with constant \(C\ge 1\). Consider the map \(u=\mathrm {id}\in N^{1,p}_{loc}(X;X)\). We denote by \(d_p\) the pseudometric \(d_{u,p}\) associated to u.
Lemma 5.8
\(d_p\) is a metric on X, and satisfies \(d\le d_p\le Cd\).
Proof
Let \(x,y\in X\) be distinct and \(\delta >0\). Since
$$\begin{aligned} {\text {Mod}}_p\Gamma (B(x,\delta ),B(y,\delta );C)>0 \end{aligned}$$
it follows that
$$\begin{aligned} ess\ell _{\mathrm {id},p}\Gamma (B(x,\delta ),B(y,\delta ))\le C(d(x,y)+2\delta ), \end{aligned}$$
implying \(d_{\mathrm {id},p}'(x,y)\le Cd(x,y)\). Thus \(d_p\le Cd\) and in particular \(d_p\) is finite-valued. Lemma 5.6 implies that \(d(x,y)\le d_p(x,y)\) for every \(x,y\in X\). These estimates together prove the claim. \(\square \)
We are now ready to prove Theorem 1.4. For the proof, we denote by \(X_p\) the space \((X,d_p,\mu )\) and by \(B_p(x,r)\) balls in X with respect to the metric \(d_p\); \(\ell _p\) and \({\text {Mod}}_{X_p,p}\) refer to the length of curves and p-modulus taken with respect to \(X_p\).
Proof of Theorem 1.4
By Lemma 5.8 it follows that \(X_p\) is an infinitesimally doubling metric measure space and \({\text {Mod}}_{X_p,p}\Gamma =0\) if and only if \({\text {Mod}}_p\Gamma =0\). In particular if \(\Gamma _*\) denotes the set of curves \(\gamma \) in X such that \(\ell (\gamma )\ne \ell _p(\gamma )\) then by Propositions 5.5 and 5.7
$$\begin{aligned} {\text {Mod}}_{X_p,p}\Gamma _*= {\text {Mod}}_p\Gamma _*=0. \end{aligned}$$
(5.9)
From (5.9) and the definition of modulus it follows that \({\text {Mod}}_{X_p,p}\Gamma = {\text {Mod}}_p\Gamma \) for every family of curves \(\Gamma \) in X.
By (5.9) and Proposition 3.2 every \(f\in N^{1,p}(X_p)\) with \(g_f\le 1\) has a C-Lipschitz representative. If \(x,y\in X\) are distinct and \(\delta , \varepsilon >0\) note that the curve family
$$\begin{aligned} \Gamma _1=\{ \gamma \in \Gamma (B(x,\delta ),B(y,\delta )): \ell (\gamma )\le ess\ell _{\mathrm {id},p}\Gamma (B(x,\delta ),B(y,\delta ))+\varepsilon \} \end{aligned}$$
satisfies
$$\begin{aligned} {\text {Mod}}_p\Gamma _1>0 \end{aligned}$$
by Remark 5.2 and the fact that \(\varepsilon >0\). Note also that
$$\begin{aligned} \ell (\gamma )\le ess\ell _{\mathrm {id},p}\Gamma (B(x,\delta ),B(y,\delta ))+\varepsilon \le d_{\mathrm {id},p}'(\gamma (1),\gamma (0))+\varepsilon \end{aligned}$$
for \(\gamma \in \Gamma _1\). Let \(f\in N^{1,p}(X_p)\) satisfy \(g_f\le 1\) almost everywhere. Let \({\bar{f}}\) be the Lipschitz representative of f, and \(\Gamma _0\) a curve family with \({\text {Mod}}_{p}\Gamma _0=0\) and
$$\begin{aligned} |f(\gamma (1)-f(\gamma (0))|\le \ell _p(\gamma )=\ell (\gamma ) \end{aligned}$$
whenever \(\gamma \notin \Gamma _0\). We have
$$\begin{aligned} {\text {Mod}}_p(\Gamma _1\setminus \Gamma _0)>0, \end{aligned}$$
so that there exists \(\gamma _\delta \in \Gamma _1\setminus \Gamma _0\). We obtain
$$\begin{aligned} |{\bar{f}}(\gamma _\delta (1))-{\bar{f}}(\gamma _\delta (0))|\le \ell _p(\gamma _\delta )\le d_{\mathrm {id},p}'(\gamma _\delta (1),\gamma _\delta (0))+\varepsilon . \end{aligned}$$
Letting \(\delta \rightarrow 0\) yields \(|{\bar{f}}(x)-{\bar{f}}(y)|\le d_{\mathrm {id},p}'(x,y)+\varepsilon \). Since \(x,y\in X\) and \(\varepsilon >0\) are arbitrary it follows that \({\bar{f}}\) is 1-Lipschitz with respect to \(d_p\). In particular \(X_p\) has the p-Sobolev-to-Lipschitz property and hence Theorem 1.7 implies it is p-thick geodesic. We prove the minimality in Proposition 5.9. \(\square \)
The metric \(d_p\) is the minimal metric above d which has the p-Sobolev-to-Lipschitz property. Proposition 5.9 provides a more general statement, from which the minimality discussed in the introduction immediately follows.
Proposition 5.9
Let Y be a p-thick geodesic metric measure space and \(f:Y\rightarrow X\) be a volume preserving 1-Lipschitz map. Then the map
$$\begin{aligned} f_p:=f:Y\rightarrow (X,d_p) \end{aligned}$$
is volume preserving and 1-Lipschitz.
Proof
It suffices to show that
$$\begin{aligned} d_{\mathrm {id},p}'(f(x),f(y))\le d(x,y),\quad x,y\in Y. \end{aligned}$$
For any \(A>1\), the curve family
$$\begin{aligned} \Gamma :=\{ \gamma \in \Gamma (B(x,\delta ),B(y,\delta )): \ell (\gamma )\le Ad(\gamma (1),\gamma (0)) \} \end{aligned}$$
has positive p-modulus in Y. Since f is volume preserving and 1-Lipschitz we have
$$\begin{aligned} 0<{\text {Mod}}_{Y,p}\Gamma \le {\text {Mod}}_{X,p}f\Gamma . \end{aligned}$$
If \(\gamma \in \Gamma \) then \(f\circ \gamma \in \Gamma (B(f(x),\delta ),B(f(y),\delta ))\) and, moreover
$$\begin{aligned} \ell (f\circ \gamma )\le \ell (\gamma )\le Ad(\gamma (1),\gamma (0))\le Ad(x,y)+2A\delta . \end{aligned}$$
It follows that \(ess\ell _{\mathrm {id},p}\Gamma (B(f(x),\delta ),B(f(y),\delta ))\le Ad(x,y)+2A\delta \), and thus
$$\begin{aligned} d_{\mathrm {id},p}'(f(x),f(y))\le Ad(x,y)+2A\delta . \end{aligned}$$
Since \(A>1\) and \(\delta >0\) are arbitrary, the claim follows. \(\square \)
We have the following immediate corollary.
Corollary 5.10
Assume X is infinitesimally doubling and p-thick geodesic. Then \(d=d_p\).
Before considering essential pull-back metrics by non-trivial maps, we prove Proposition 1.5. The proof is based on the fact that spaces with Poincaré inequality are thick quasiconvex, and on the independence of the minimal weak upper gradient on the exponent.
Proposition 5.11
([12]) Let \(p\in [1,\infty ]\) and X be a doubling metric measure space satisfying a p-Poincaré inequality. There is a constant \(C\ge 1\) depending only on the data of X so that X is p-thick quasiconvex with constant C.
The inverse implication in Proposition 5.11 only holds if \(p=\infty \), see [11, 12].
Proof of Proposition 1.5
Assume X is a doubling metric measure space supporting a p-Poincaré inequality, and \(q\ge p\ge 1\). Since \({\text {Mod}}_q\Gamma =0\) implies \({\text {Mod}}_p\Gamma =0\), see [2, Proposition 2.45], we have
$$\begin{aligned} d\le d_q\le d_p\le Cd \end{aligned}$$
for some constant C depending only on p and the data of X.
Fix \(x_0\in X\) and consider the function
$$\begin{aligned} f:X\rightarrow \mathbb {R},\quad x\mapsto d_p(x_0,x). \end{aligned}$$
Then f is Lipschitz and, by Propositions 5.5 and 5.7, it has 1 as a p-weak upper gradient. Since X is doubling and supports a p-Poincaré inequality, [2, Corollary A.8] implies that the minimal q-weak and p-weak upper gradients of f agree almost everywhere, and thus 1 is a q-weak upper gradient of f, i.e.
$$\begin{aligned} |f(\gamma (1))-f(\gamma (0))|\le \ell (\gamma )\le \ell _q(\gamma ) \end{aligned}$$
for q-almost every curve. The space \(X_q\) has the q-Sobolev-to-Lipschitz property, see Theorem 1.7, and the 1-Lipschitz representative of f agrees with f everywhere, since f is continuous. By this and the definition of \(d_q\) we obtain
$$\begin{aligned} d_p(x_0,x)\le d_q(x_0,x),\quad x\in X. \end{aligned}$$
Since \(x_0\in X\) is arbitrary the equality \(d_p=d_q\) follows.
For the remaining equality, note that \(d\le d_\infty \le d_{ess}\le Cd\). Indeed, X supports an \(\infty \)-Poincaré inequality, whence [11, Theorem 3.1] implies the rightmost estimate with a constant C depending only on the data of the \(\infty \)-Poincaré inequality. As above, the function
$$\begin{aligned} g:X\rightarrow \mathbb {R},\quad x\mapsto d_{ess}(x_0,x) \end{aligned}$$
satisfies
$$\begin{aligned} |g(\gamma (1))-g(\gamma (0))|\le \ell (\gamma )\le \ell _\infty (\gamma ) \end{aligned}$$
for \(\infty \)-almost every \(\gamma \), from which the inequality \(d_{ess}\le d_\infty \) follows. \(\square \)
Note that in the proof of Proposition 5.11 we use that in p-Poincaré spaces the q-weak upper gradient does not depend on \(q\ge p\). In general such an equality is not true, see [9], and we do not know whether we can weaken the assumptions in Proposition 1.5 from p-Poincaré inequality to p-thick quasiconvexity.
Essential pull-back metrics by Sobolev maps
Throughout this subsection \((X,d,\mu )\) will be a doubling metric measure space satisfying (2.2) with \(Q\ge 1\) and supporting a weak (1, Q)-Poincaré inequality, and \(Y=(Y,d)\) a proper metric space.
We will use the following observation without further mention. If \(p>Q\), and \(u:X\rightarrow Y\) has a Q-weak upper gradient in \(L^p_\mathrm {loc}(X)\), then u has a representative \({\bar{u}}\in N^{1,p}_{loc}(X;Y)\) and the minimal p-weak upper gradient of \({\bar{u}}\) coincides with the minimal Q-weak upper gradient of u almost everywhere. See [2, Chapter 2.9 and Appendix A] and [22, Chapter 13.5] for more details.
The next proposition states that higher regularity of a map is enough to guarantee that the essential pull-back distance in Definition 5.3 is a finite-valued pseudometric.
Proposition 5.12
Let \(p>Q\), and suppose \(u\in N^{1,p}_{loc}(X;Y)\). The pull-back distance \(d_u:=d_{u,Q}'\) in Definition 5.3 is a pseudometric satisfying
whenever \(B\subset X\) is a ball and \(x,y\in B\), where the constant C depends only on p and the data of X.
For the proof of Proposition 5.12 we define the following auxiliary functions. Let \(x\in X\), \(\delta >0\) and \(\Gamma _0\) a curve family with \({\text {Mod}}_Q\Gamma _0=0\). Set \(f:=f_{x,\delta ,\Gamma _0}:X\rightarrow \mathbb {R}\) by
$$\begin{aligned} f(y)=\inf \{ \ell (u\circ \gamma ):\gamma \in \Gamma (\bar{B}(x,\delta ),y)\setminus \Gamma _0 \},\quad y\in X. \end{aligned}$$
When \(\rho \in L^Q(X)\) is a non-negative Borel function, whence \({\text {Mod}}_Q(\Gamma _\rho )=0\), we denote \(f_{x,\delta ,\rho }:=f_{x,\delta ,\Gamma _\rho }\).
Lemma 5.13
Let \(x,\delta \) and \(\rho \) be as above. The function \(f=f_{x,\delta ,\rho }\) is finite \(\mu \)-almost everywhere and has a representative in \(N^{1,p}_{loc}(X)\) with p-weak upper gradient \(g_u\). The continuous representative \({\bar{f}}\) of f satisfies
Proof
Let \(g\in L^p_{\mathrm {loc}}(X)\) be a genuine upper gradient of u and let \(\varepsilon >0\) be arbitrary. We fix a large ball \(B\subset X\) containing \({\bar{B}}(x,\delta )\) and note that there exists \(x_0\in {\bar{B}}(x,\delta )\) for which \(\mathcal {M}_B(g+\rho )^Q(x_0)<\infty \), since \((g+\rho )|_B\in L^Q(B)\); cf. (2.3). Arguing as in [41, Lemma 4.6] we have that
$$\begin{aligned} f(y)\le \inf \left\{ \int _\gamma (g+\rho ): \gamma \in \Gamma _{x_0y}\setminus \Gamma _\rho \right\} <\infty \end{aligned}$$
for almost every \(y\in B\).
Let \(\gamma \notin \Gamma _\rho \) be a curve such that \(f(\gamma (1)),f(\gamma (0))<\infty \) and \(\int _\gamma g<\infty \). We may assume that \(|f(\gamma (1))-f(\gamma (0))|=f(\gamma (1))-f(\gamma (0))\ge 0\). If \(\beta \) is an element of \({\bar{\Gamma }}(B(x,\delta ),\gamma (0))\setminus \Gamma _\rho \) such that \(\ell (u\circ \beta )<f(\gamma (0))+\varepsilon \) then the concatenation \(\gamma \beta \) satisfies \(\gamma \beta \in \Gamma ({\bar{B}}(x,\delta ),\gamma (1))\setminus \Gamma _\rho \). Thus
$$\begin{aligned} |f(\gamma (1))-f(\gamma (0))|\le \ell (u\circ \gamma \beta )-\ell (u\circ \beta )+ \varepsilon =\ell (u\circ \gamma )+\varepsilon \le \int _\gamma g+\varepsilon . \end{aligned}$$
It follows that \(g\in L^p_\mathrm {loc}(X)\) is a Q-weak upper gradient for f. By [2, Corollary 1.70] we have that \(f(y)<\infty \) for Q-quasievery \(y\in B\), and \(f\in N^{1,Q}_{loc}(X)\); cf. [22, Theorem 9.3.4].
Moreover, since \(g\in L^p_\mathrm {loc}(X)\), f has a continuous representative \({\bar{f}}\in N^{1,p}_{loc}(X)\) which satisfies (5.11), cf. [22, Theorem 9.2.14]. \(\square \)
Proof of Proposition 5.12
To prove the triangle inequality, let \(x,y,z\in X\) be distinct. Take \(\delta >0\) small, \(\rho \in L^Q(X)\) non-negative, and let \(E\subset X\) be a set of Q-capacity zero such that \(f_{x,\delta ,\Gamma _\rho }\) and \(f_{y,\delta ,\Gamma _\rho }\) agree with their continuous representatives outside E. Remember that \(\alpha \beta \notin \Gamma _\rho \) whenever \(\alpha ,\beta \notin \Gamma _\rho \). We have that
$$\begin{aligned} d'_{u,Q}(x,z)+d'_{u,Q}(z,y)&\ge \inf _{w\in {\bar{B}} (z,\delta )}f_{x,\delta ,\Gamma _\rho \cup \Gamma _E}(w)+ \inf _{v\in {\bar{B}}(z,\delta )}f_{y,\delta ,\Gamma _\rho \cup \Gamma _E}(v)\\&\ge \inf _{w,v\in {\bar{B}}(z,\delta )\setminus E} [f_{x,\delta ,\rho }(w)+f_{y,\delta ,\rho }(v)]. \end{aligned}$$
Together with the estimate (5.11) this yields
$$\begin{aligned} d'_{u,Q}(x,z)+d'_{u,Q}(z,y)&\ge \inf _{w\in {\bar{B}}(z,\delta )\setminus E}[f_{x,\delta ,\rho }(w)+f_{y,\delta ,\rho }(w)]-C\delta ^{1-Q/p}\\&\ge \inf \{ \ell (u\circ \gamma ): \gamma \in \Gamma ({\bar{B}}(x,\delta ),{\bar{B}}(y,\delta ))\setminus \Gamma _\rho \}- C\delta ^{1-Q/p}, \end{aligned}$$
where C depends on u, x and y as well as the data. Since \(\delta >0\) and \(\rho \) are arbitrary it follows that
$$\begin{aligned} d'_{u,Q}(x,z)+d'_{u,Q}(z,y)\ge d'_{u,Q}(x,y). \end{aligned}$$
Moreover, for any \(y'\in {\bar{B}}(y,\delta )\setminus E\),
Taking supremum over \(\rho \), and letting \(\delta \) tend to zero, we obtain (5.10). \(\square \)
Remark 5.14
A slight variation of the proof of Proposition 5.12 shows that also in the setting of Theorem 1.4 the essential pull-back distance defines a metric. In particular it satisfies the triangle inequality and one does not have to pass to the maximal semimetric below. In this case in the argument instead of the Morrey embedding one applies Proposition 3.2.
Fix a continuous map \(u\in N^{1,p}_{loc}(X;Y)\), where \(p>Q\), and denote by \(Y_u\) the set of equivalence classes [x] of points \(x\in X\), where x and y are set to be equivalent if \(d_{u,Q}'(x,y)=0\). The pseudometric \(d_{u,Q}'\) defines a metric \(d_u\) on \(Y_u\) by
$$\begin{aligned} d_u([x],[y]):=d_{u,Q}'(x,y),\quad x,y\in X. \end{aligned}$$
The natural projection map
$$\begin{aligned} \widehat{P}_u:X\rightarrow Y_u,\quad x\mapsto [x] \end{aligned}$$
is continuous by (5.10). The map u factors as \(u={\widehat{u}}\circ {\widehat{P}}_u\), where
$$\begin{aligned} {\widehat{u}}:Y_u\rightarrow Y,\quad [x]\mapsto u(x) \end{aligned}$$
is well-defined and 1-Lipschitz; cf. Lemma 5.6.
Lemma 5.15
Under the given assumptions we obtain the following properties.
-
(1)
\({\widehat{P}}_u\in N^{1,p}_{loc}(X;Y_u)\) and \(g_{{\widehat{P}}_u}=g_u\) \(\mu \)-almost everywhere.
-
(2)
If u is proper then \(Y_u\) is proper and the projection \({\widehat{P}}_u:X\rightarrow Y_u\) is proper and monotone.
Recall here that a map is called proper if the preimage of every compact set is compact or equivalently if the preimage of every singleton is compact.
Proof
(1) Suppose \(\gamma \) is a (u, Q)-regular curve such that \(g_u\) is an upper gradient of u along \(\gamma \). Then by Proposition 5.7
$$\begin{aligned} d_u(\widehat{P}_u\circ \gamma (1),\widehat{P}_u\circ \gamma (0))\le \ell (u\circ \gamma )\le \int _\gamma g_u. \end{aligned}$$
By Proposition 5.5 this implies that \({\widehat{P}}_u\in N^{1,Q}_{loc}(X;Y_u)\) and \(g_{{\widehat{P}}_u}\le g_u\). The opposite inequality follows because \(\widehat{u}\) is 1-Lipschitz.
(2) The factorization implies that \(K\subset {\widehat{P}}_u( u^{-1}(\overline{{\widehat{u}}(K)}))\), for \(K\subset Y_u\) and hence \(Y_u\) is proper. Similarly \({\widehat{P}}_u^{-1}(K)\subset u^{-1}(\widehat{u}(K))\) and hence \(\widehat{P}_u\) is proper. The proof of monotonicity is an adaptation of the proof of [33, Lemma 6.3].
Assume \({\widehat{P}}_u^{-1}(y)\) is not connected for some \(y\in Y_u\). Then there are compact sets \(K_1,K_2\) for which \({\text {dist}}(K_1,K_2)>0\) and \({\widehat{P}}_u^{-1}(y)=K_1\cup K_2\). Let S denote the closed and non-empty set of points X whose distance to \(K_1\) and \(K_2\) agree. Let a be the minimum of \(x\mapsto {\text {dist}}_u(y,\widehat{P}_u(x))\) on S. Note here that \(\widehat{P}_u(S)\) is closed as \(Y_u\) and \(\widehat{P}_u\) are proper and hence the infimum is attained and positive. Let \(k_i\in K_i\) for \(i=1,2\). Since \(d_{u,Q}'(k_1,k_2)=0\), for every \(\varepsilon >0\) and small enough \(\delta >0\), Proposition 5.5 implies the existence of a (u, Q)-regular curve \(\gamma \in \Gamma (B(k_1,\delta ),B(k_2,\delta ))\) with \(\ell (u\circ \gamma )<\varepsilon \). If \(\delta \) is chosen small enough the curve must intersect S at some point \(s:=\gamma (t)\). Since \(\gamma \) is (u, Q)-regular, \(\gamma |_{[0,t]}\) is (u, Q)-regular and it follows that
$$\begin{aligned} a\le d_{u,Q}'(s,k_1)\le d_{u,Q}'(s,\gamma (0))+C\delta ^{1-\frac{Q}{p}}\le \ell (u\circ \gamma )+ C\delta ^{1-\frac{Q}{p}}\le \varepsilon +C\delta ^{1-\frac{Q}{p}}; \end{aligned}$$
cf. Proposition 5.7 and (5.10). Choosing \(\varepsilon >0\) and \(\delta >0\) small enough this yields a contradiction. Thus \({\widehat{P}}_u^{-1}(y)\) is connected for every \(y\in Y_u\). \(\square \)
Assume additionally that Y is endowed with a measure \(\nu \) such that \(Y=(Y,d,\nu )\) is a metric measure space. We equip \((Y_u,d_u)\) with the measure \(\nu _u:={\widehat{u}}^*\nu \), which is characterized by the property
$$\begin{aligned} \nu _u(E)=\int _Y\#({\widehat{u}}^{-1}(y)\cap E)\mathrm {d}\nu (y),\quad E\subset Y_u \text { Borel}; \end{aligned}$$
(5.12)
cf. [13, Theorem 2.10.10]. Note that in general \(\nu _u\) is not a \(\sigma \)-finite measure. For the next theorem, we say that a Borel map \(u:X\rightarrow Y\) has Jacobian Ju, if there exists a Borel function \(Ju:X\rightarrow [0,\infty ]\) for which
$$\begin{aligned} \int _EJu\mathrm {d}\mu =\int _Y\#(u^{-1}(y)\cap E)\mathrm {d}\nu (y) \end{aligned}$$
(5.13)
holds for every Borel set \(E\subset X\). The Jacobian Ju, if it exists, is unique up to sets of \(\mu \)-measure zero.
Theorem 5.16
Assume u is proper, nonconstant and has a locally integrable Jacobian Ju which satisfies
$$\begin{aligned} g_u^Q\le KJu \end{aligned}$$
(5.14)
\(\mu \)-almost everywhere for some \(K\ge 1\). Then the following properties hold.
-
(1)
\(Y_u\) is a metric measure space and has the Q-Sobolev-to-Lipschitz property.
-
(2)
Ju is the locally integrable Jacobian of \(\widehat{P}_u\) and \(\#{\widehat{P}}_u^{-1}(y)=1\) for \(\nu _u\)-almost every \(y\in Y_u\).
Proof
Note that
$$\begin{aligned} \int _{{\widehat{P}}_u^{-1}E}Ju\mathrm {d}\mu =\int _Y\#(u^{-1}(y)\cap \widehat{P}_u^{-1}E)\mathrm {d}\nu (y)\ge \int _Y\#({\widehat{u}}^{-1}(y)\cap E)\mathrm {d}\nu (y)=\nu _u(E) \end{aligned}$$
for every Borel set \(E\subset Y_u\). Thus \(\nu _u\) is a locally finite measure, and the estimate above implies that \({\widehat{P}}_u\) has a locally integrable Jacobian \(J{\widehat{P}}_u\le Ju\). For any Borel set \(E\subset X\) we have
$$\begin{aligned} \int _EJ {\widehat{P}}_u\mathrm {d}\mu&=\int _{Y_u} \#({\widehat{P}}_u^{-1}(y)\cap E)\mathrm {d}\nu _u(y)=\int _Y\left( \sum _{y\in {\widehat{u}}^{-1}(z)}\#({\widehat{P}}_u^{-1}(y)\cap E)\right) \mathrm {d}\nu (z)\\&=\int _Y\#(u^{-1}(z)\cap E)\mathrm {d}\nu (z)=\int _EJu\mathrm {d}\mu , \end{aligned}$$
which yields \(J\widehat{P}_u=J u\) almost everywhere. Since \(\widehat{P}_u\) is monotone and satisfies (5.13) we have that \(\#P_u^{-1}(y)=1\) for \(\nu _u\)-almost every \(y\in Y_u\).
By Lemma 5.15 to see that \(Y_u\) is a metric measure space it remains to show that balls in \(Y_u\) have positive measure. The idea of of proof is borrowed from [33, Lemma 6.11]. Assume \(B=B(z,r)\) is a ball in \(Y_u\) such that \(\nu _u(B)=0\). Then we would have \(J {\widehat{P}}_u=J_u=0\) almost everywhere on \(U:=\widehat{P}_u^{-1}(B)\). By (5.14) and Lemma 5.15\(g_{\widehat{P}_u}=g_u=0\) almost everywhere on U. Thus \(\widehat{P}_u\) is locally constant on U. But U is connected and hence \(\widehat{P}_u\) is constant.
To see that U is connected let \(x\in U\) satisfy \(\widehat{P}_u(x)=z\). Then, for \(y\in U\) and \(\delta >0\) arbitrary, there is a (u, Q)-regular curve \(\gamma \in \Gamma (B(x,\delta ),B(y,\delta ))\) such that \(\ell (u\circ \gamma )<r\). For \(\delta \) small enough by Proposition 5.7 the image of \(\gamma \) is contained in U. As this holds for all small enough \(\delta \), the points x and y must lie in the same connected component of the open set U. Since y was arbitrary, U must be connected.
We have deduced that \(B=\widehat{P}_u(U)\) consists only of the single point z. So either \(Y_u\) is disconnected or consists of a single point. The former is impossible because X is connected and the later by the assumption that u is nonconstant.
Before showing the Sobolev-to-Lipschitz property we first claim
$$\begin{aligned} {\text {Mod}}_Q\Gamma \le K{\text {Mod}}_Q{\widehat{P}}_u(\Gamma ) \end{aligned}$$
(5.15)
for every curve family \(\Gamma \) in X. Indeed, suppose \(\rho \in L^Q(Y_u)\) is admissible for \({\widehat{P}}_u(\Gamma )\), and set \(\rho _1=\rho \circ {\widehat{P}}_u\). Let \(\Gamma _0\) a curve family with \({\text {Mod}}_Q\Gamma _0=0\) for which \(\gamma \) is \(({\widehat{P}}_u,Q)\)- and (u, Q)-regular, and \(g_u\) is an upper gradient of \({\widehat{P}}_u\) and u along \(\gamma \), whenever \(\gamma \notin \Gamma _0\). For any \(\gamma \in \Gamma \setminus \Gamma _0\) we have
$$\begin{aligned} 1\le \int _{\widehat{P}_u\circ \gamma }\rho =\int _0^1\rho _1(\gamma (t))|(\widehat{P}_u\circ \gamma )_t'|\mathrm {d}t\le \int _0^1\rho _1(\gamma (t))g_u(\gamma (t))|\gamma _t'|\mathrm {d}t, \end{aligned}$$
i.e. \(\rho _1g_u\) is admissible for \(\Gamma \setminus \Gamma _0\). We obtain
$$\begin{aligned} {\text {Mod}}_Q\Gamma ={\text {Mod}}_Q(\Gamma \setminus \Gamma _0)&\le \int _X\rho _1^Qg_u^Q\mathrm {d}\mu \le K\int _X\rho ^Q\circ {\widehat{P}}_u Ju\mathrm {d}\mu \\&=K\int _{Y_u} \rho ^Q\mathrm {d}\nu _u. \end{aligned}$$
The last equality follows, since Lemma 5.15 implies \(\widehat{P}_{u*}(Ju\ \mu )=\nu _u\). Taking infimum over admissible \(\rho \) yields (5.15).
We prove that \(Y_u\) has the Q-Sobolev-to-Lipschitz property. Suppose \(f\in N^{1,Q}(Y_u)\) satisfies \(g_f\le 1\). There is a curve family \(\Gamma _0\) in \(Y_u\) with \({\text {Mod}}_{Q}(\Gamma _0)=0\) such that
$$\begin{aligned} |f(\gamma (1))-f(\gamma (0))|\le \ell _u(\gamma ) \end{aligned}$$
whenever \(\gamma \notin \Gamma _0\). By (5.15) we have
$$\begin{aligned} {\text {Mod}}_{Q}{\widehat{P}}_u^{-1}\Gamma _0\le K{\text {Mod}}_{Q}\Gamma _0=0. \end{aligned}$$
Here \({\widehat{P}}_u^{-1}\Gamma \) denotes the family of curves \(\gamma \) in X such that \({\widehat{P}}_u\circ \gamma \in \Gamma _0\). Together with Propositions 5.5 and 5.7 this implies that, for \({\text {Mod}}_{Q}\)-almost every curve \(\gamma \) in X, we have
$$\begin{aligned} |f\circ {\widehat{P}}_u(\gamma (1))-f\circ {\widehat{P}}_u(\gamma (0))|\le \ell _u(\gamma )=\ell (u\circ \gamma )\le \int _\gamma g_u. \end{aligned}$$
Since \(g_u\in L^p_{\mathrm {loc}}(X)\) for \(p>Q\), it follows that \(f\circ {\widehat{P}}_u\) has a continuous representative \({\bar{f}}\). Let \(x,y\in X\) be distinct, \(\varepsilon >0\) arbitrary, and \(\delta >0\) such that
$$\begin{aligned} |{\bar{f}}(x)-{\bar{f}}(z)|+|{\bar{f}}(y)-{\bar{f}}(w)|<\varepsilon \end{aligned}$$
whenever \(z\in {\bar{B}}(x,\delta )\) and \(w\in {\bar{B}}(y,\delta )\). Denote by \(\Gamma _1\) the curve family with \({\text {Mod}}_{X,Q}\Gamma _1=0\) so that
$$\begin{aligned} |{\bar{f}}(\gamma (1))-{\bar{f}}(\gamma (0))|\le \ell (u\circ \gamma )\quad \text {whenever }\gamma \notin \Gamma _1. \end{aligned}$$
Then
$$\begin{aligned} |{\bar{f}}(x)-{\bar{f}}(y)|\le \varepsilon + \underset{\gamma \in \Gamma ({\bar{B}}(x,\delta ),{\bar{B}}(y,\delta ))\setminus \Gamma _1}{\inf \ell (u\circ \gamma )}\le \varepsilon +d_{u,Q}'(x,y). \end{aligned}$$
Since \(x,y\in X\) and \(\varepsilon >0\) are arbitrary it follows that \({\bar{f}}(x)={\bar{f}}(y)\) whenever \(d_{u,Q}'(x,y)=0\) and that the map \([x]\mapsto {\bar{f}}(x)\) is 1-Lipschitz with respect to the metric \(d_u\), and is a \(\nu _u\)-representative of f. \(\square \)