3.1 Tilings of ${\mathbb {H}}^d$

We call a subset of $\mathbb {R}^d$ or ${\mathbb {H}}^d$ a polyhedron if it is a finite intersection of closed half-spaces. A bounded (or equivalently compact) polyhedron is called a polytope. Note that polyhedra are convex sets, as is, for example, obvious in the Beltrami–Klein model. Hence, the nearest point projection – namely, the mapping which assigns to a point x the point in the polyhedron which minimizes the distance to x – is well defined and $1$ -Lipschitz; see Proposition 2.4 in [Reference Bridson and Haefliger14, p. 176]. In particular, polyhedra are $1$ -Lipschitz retracts of both $\mathbb {R}^d$ and ${\mathbb {H}}^d$ .

In order to investigate the local and global structure of the Lipschitz functions on ${\mathbb {H}}^d$ separately, we will need a suitable tiling of the hyperbolic space ${\mathbb {H}}^d$ . A sequence $(P_n)_{n\in \mathbb {N}}$ of polytopes is a tiling, or tessellation, of ${\mathbb {H}}^d$ if $\bigcup _{n\in \mathbb {N}}P_n= {\mathbb {H}}^d$ , and the intersection of any two distinct polytopes is either empty or a face of both polytopes (in particular, the interiors of the polytopes $P_n$ are mutually disjoint).

In our arguments, we will use the existence in ${\mathbb {H}}^d$ for $d=2,3,4$ of a regular orthogonal tiling – namely, a tiling $(P_n)_{n\in \mathbb {N}}$ consisting of mutually isometric polytopes and where each $P_n$ satisfies the following definitions (see, for example, [Reference Davis22, Chapter 6] for more details):

1. (T1) A polytope P in ${\mathbb {H}}^d$ is regular if the group of isometries of P is flag-transitive. More precisely, a flag in P is a chain $F_0\subset F_1\subset \dots \subset F_{d-1}$ , where $F_0$ is a vertex of P and each $F_k$ is a k-dimensional face of P. Then P is regular if for every two flags $F_0\subset F_1\subset \dots \subset F_{d-1}$ and $F^{\prime }_0\subset F^{\prime }_1\subset \dots \subset F^{\prime }_{d-1}$ , there is an isometry of P that maps one flag onto the other.

2. (T2) A polytope is right-angled, or orthogonal, if all dihedral angles are exactly $\pi /2$ .

Before passing to the explanation of the existence of the tiling, let us mention two more properties that follow from (T1) and (T2) (two further properties will be proved in Lemma 3.2 and Lemma 3.3 below).

1. (T3) Every polytope $P_n$ has an inscribed circle, whose center we denote by $p_n$ and whose radius is by definition the in-radius of $P_n$ .

   Indeed, every regular polytope P admits a center, whose existence can be shown as follows. If F is a maximal face of P, then $x\mapsto \operatorname {dist}(x,F)$ is a convex function (as follows easily from Proposition 2.2 in [Reference Bridson and Haefliger14, p. 176]). Thus, letting $\{F_1,\dots ,F_k\}$ be the maximal faces of P, the map $x\mapsto \operatorname {dist}(x,F_1)^2 +\dots + \operatorname {dist}(x,F_k)^2$ is strictly convex and hence has a unique minimum p. Since the above map is defined only by metric properties, every isometry of P must fix p. Finally, by regularity, p has the same distance to all maximal faces. For further details, we refer to [Reference Bridson and Haefliger14, pp. 178–179].

2. (T4) There is a number $N(d)$ such that every $P_n$ intersects at most $N(d)$ polytopes from the tiling.

   Figure 1:

   The first seventeen isometric octagons tiling the hyperbolic plane.

   

   Indeed, at each vertex of $P_n$ , there are exactly $2^d$ polytopes intersecting $P_n$ , by the right-angle property. So a (rough) upper bound for $N(d)$ is $2^d$ times the number of vertices of $P_n$ (such a number of vertices is independent of n, as the polytopes are mutually isometric).

We now pass to explaining the existence of regular orthogonal tilings $(P_n)_{n\in \mathbb {N}}$ for $d=2,3,4$ . By Proposition 6.3.2 and 6.3.9 in [Reference Davis22], every polytope P whose dihedral angles are of the form $\pi /m$ for some integer $m\geqslant 2$ (these polytopes are sometimes called Coxeter polytopes) is simple. Hence, by Theorem 6.4.3 in [Reference Davis22], P is a strict fundamental domain of the reflection group generated by reflections across the faces of P. In other words, P intersects every orbit in exactly one point. This implies in particular that the images of P by the elements of the reflection group tessellate ${\mathbb {H}}^d$ . Thus, every element of the reflection group maps the faces of a polytope $P_n$ to the faces of some other polytope of the tiling.

If we restrict our attention to right-angled polytopes, by the above observations, the existence of a right-angled polytope P directly implies that there is a tessellation of ${\mathbb {H}}^d$ by isometric copies of P. Several explicit constructions of such polytopes for dimensions $d=2,3,4$ are available in the literature, and we mention a few of them in the next paragraph. On the other hand, E. B. Vinberg proved in [Reference Vinberg44] that there is no right-angled polytope in ${\mathbb {H}}^d$ when $d\geqslant 5$ ; a short proof of this fact can also be found in [Reference Potyagailo and Vinberg42, Section 2]. Consequently, an orthogonal tiling of ${\mathbb {H}}^d$ exists precisely for dimensions $d=2,3,4$ .

As it turns out, in dimension $d=2,3,4$ , an orthogonal tiling can be constructed that is additionally regular. In dimension $d=2$ , there are infinitely many regular orthogonal tilings, one for each integer $p\geqslant 5$ . They are obtained by taking a regular right-angled p-gon in ${\mathbb {H}}^2$ and gluing together $4$ of them at each vertex; by means of the so-called Schläfli symbol, such a configuration is denoted by $\{p,4\}$ . For instance, $\{8, 4\}$ is a tiling via four regular octagons meeting at each vertex, illustrated in Figure 1. In dimension $d=3$ , a tiling exists described by $\{5,3,4\}$ , which is to be read as follows: start with a regular right-angled pentagon in ${\mathbb {H}}^2$ and glue three of them at each vertex. This gives a (hyperbolic) dodecahedron $\{5,3\}$ and gluing $4$ on each edge gives the tiling. In dimension $d=4$ , instead, one glues together $3$ dodecahedra on each edge to obtain a $4$ -dimensional polytope $\{5,3,3\}$ whose maximal faces are the dodecahedra $\{5,3\}$ . Then, gluing $4$ such polytopes at each $2$ -dimensional face, one gets the tiling $\{5,3,3,4\}$ . A list of all regular tilings of ${\mathbb {H}}^d$ can be found in [Reference Vinberg and Shvartsman45, Section 5.3.3] or [Reference Coxeter19, Sections 2–4] (the orthogonal ones among them are exactly those with $4$ as the last number in their Schläfli symbol).

One notable feature of orthogonal tilings, which we will use in the next lemma as well as Section 5, is the following:

As one can already glean from Figure 1, the edges of the octagons combine into geodesic lines, and this is true in general for any regular orthogonal tiling.

Lemma 3.3 Let $\mathcal {P}=(P_n)_{n\in \mathbb {N}}$ be a regular orthogonal tiling of ${\mathbb {H}}^d$ , $d\leqslant 4$ . Then, there exists a sequence $(H_k)_{k\in \mathbb {N}}$ of hyperplanes in ${\mathbb {H}}^d$ such that

$$ \begin{align*} \bigcup_{n\in\mathbb{N}}\partial P_n = \bigcup_{k\in\mathbb{N}}H_k. \end{align*} $$

These $H_k$ are exactly all the supporting hyperplanes for any maximal face of any $P_n$ . As a consequence, if R is a reflection across some maximal face,

$$\begin{align*}R\left[ \bigcup_{k\in\mathbb{N}}H_k \right]= \bigcup_{k\in\mathbb{N}}H_k. \end{align*}$$

5.1 Extension from the net $\mathcal N$

In this part, we construct a bounded, weak $^*$ -to-weak $^*$ continuous linear extension operator from the net $\mathcal {N}$ to ${\mathbb {H}}^d$ . In order to achieve this, we will exploit a Lipschitz partition of unity in the spirit of [Reference Albiac, Ansorena, Cúth and Doucha1, Reference Bruè, Di Marino and Stra16, Reference Johnson, Lindenstrauss and Schechtman35]. Notice that said results cannot be applied directly because ${\mathbb {H}}^d$ is not a doubling metric space. However, the specific shape of the net, inherited from the tiling, allows a very simple construction of the partition of unity.

Since the polytopes $P_n$ are mutually isometric, the in-radius $\mathrm {dist}(p_n, \partial P_n)$ of $P_n$ is the same for all n; thus, we can set . Therefore, when $k\neq n$ , $\mathrm {dist}(p_k,P_n)\geqslant \delta $ . Define functions $\varrho _n\colon {\mathbb {H}}^d \to \mathbb {R}$ by . Then $\varrho _n(p_k)= \delta _{k,n}$ and $\varrho _n$ is $1/\delta $ -Lipschitz. Moreover, the restriction of $\varrho _n$ to $P_k$ is nonzero only when $P_k$ intersects $P_n$ . Hence, by (T4), the series $\sum _{n=1}^\infty \varrho _n$ is locally a sum of at most $N(d)$ terms, and therefore, it defines a bounded Lipschitz function. Additionally, we have $\sum _{n=1}^\infty \varrho _n(x)\geqslant 1$ for all $x\in {\mathbb {H}}^d$ , as $\varrho _n$ equals $1$ on $P_n$ . Consequently, the functions

are uniformly Lipschitz, with constant, say, $L_{\mathcal {N}}$ . Clearly, $(\varphi _n)_{n=1}^\infty $ is the desired partition of unity and $\varphi _n(p_k)=\delta _{k,n}$ . We can now proceed to the construction of the extension operator from the net $\mathcal {N}$ .

Lemma 5.1 The operator $E_{\mathcal N}$ defined as

$$ \begin{align*} E_{\mathcal N}\colon \mathrm{Lip}_0(\mathcal N)&\to \mathrm{Lip}_0({\mathbb{H}}^d)\colon f\mapsto \sum_{n=1}^\infty f(p_n)\varphi_n \end{align*} $$

is a bounded linear extension operator from the net $\mathcal N$ to ${\mathbb {H}}^d$ which is weak $^*$ -to-weak $^*$ continuous.

Proof The sum defining $E_{\mathcal N}f$ is locally finite; thus, $E_{\mathcal N}$ is clearly a well-defined linear extension operator, and $E_{\mathcal {N}}f$ is continuous. In order to bound the Lipschitz constant of $E_{\mathcal N}f$ , we use Lemma 2.1. Thus, fix $f\in \mathrm {Lip}_0(\mathcal N)$ and $x,y\in P_n$ , for some $n\in \mathbb {N}$ . Let $\{n_1,\dots , n_{N(d)}\}$ be the set of indices corresponding to the polytopes $P_k$ that intersect $P_n$ . Hence, we have

$$ \begin{align*} |E_{\mathcal{N}}f(x) - E_{\mathcal{N}}f(y)| =& \left|\sum_{j=1}^{N(d)} \big(\varphi_{n_j}(x) - \varphi_{n_j}(y)\big) f(p_{n_j})\right|\\ =& \left|\sum_{j=1}^{N(d)} \big(\varphi_{n_j}(x) - \varphi_{n_j}(y)\big) \big(f(p_{n_j})-f(p_n)\big)\right|\\ \leqslant& \sum_{j=1}^{N(d)} L_{\mathcal{N}}\ \rho(x,y)\ \mathrm{Lip}(f)\ \rho(p_{n_j},p_n)\leqslant C\ \mathrm{Lip}(f)\ \rho(x,y). \end{align*} $$

Here, $C= 2\operatorname {diam}(P_n)\ L_{\mathcal {N}}\ N(d)$ , and we used the fact that $\rho (p_{n_j},p_n)\leqslant 2\operatorname {diam}(P_n)$ , since $P_{n_j}\cap P_n\neq \emptyset $ . According to Lemma 2.1, this inequality proves that $\mathrm {Lip}(E_{\mathcal {N}}f)\leqslant C\mathrm {Lip} (f)$ , whence the boundedness of $E_{\mathcal {N}}$ .

Lastly, to prove weak $^*$ -to-weak $^*$ continuity, we need only show pointwise-to-pointwise continuity. So assume that $(f_k)_{k\in \mathbb {N}}$ is a sequence of Lipschitz functions converging pointwise to some function $f\in \mathrm {Lip}_0(\mathcal N)$ . Then, using again local finiteness of the sum,

$$ \begin{align*} \kern-12pt\lim_{k\to\infty}E_{\mathcal N}(f_k)(x) &= \sum_{n=1}^\infty \lim_{k\to\infty}f_k(p_n) \varphi_n(x) = \sum_{n=1}^\infty f(p_n) \varphi_n(x) = E_{\mathcal N}(f)(x).\\[-41pt] \end{align*} $$

5.2 Decomposition into functions on tiles

The aim of this section is to decompose a Lipschitz function on ${\mathbb {H}}^d$ into a sequence of functions defined on the tiles of our tiling. As a first step, we use the results of the previous section to remove the part of the function defined on the net $\mathcal {N}$ . More precisely, Lemma 5.1 implies that the mapping

$$\begin{align*}\mathrm{Lip}_0({\mathbb{H}}^d)\to \mathrm{Lip}_0(\mathcal{N}) \oplus \mathrm{Lip}_{0,\mathcal{N}} ({\mathbb{H}}^d)\colon \qquad f \mapsto \big(f|_{\mathcal{N}}, f-E_{\mathcal{N}}(f|_{\mathcal{N}}) \big) \end{align*}$$

is a weak $^*$ -to-weak $^*$ continuous isomorphism (with inverse $(g,h)\mapsto E_{\mathcal {N}}(g) + h$ ).

Hence, our goal now is to decompose a function in $\mathrm {Lip}_{0,\mathcal {N}}({\mathbb {H}}^d)$ into a sequence of functions defined on the tiles. We start with some geometric preliminaries.

Let us now fix some notation and parameters. Let $\mathcal P$ be our tiling of ${\mathbb {H}}^d$ and $(H_m)_{m\in \mathbb {N}}$ be the sequence of hyperplanes supporting the faces of the tiles $P_n$ as in Lemma 3.3. Let $\varepsilon>0$ be a fixed number which is smaller than the in-radius of $P_m$ and smaller than half of the minimal distance of nonadjacent faces of $P_m$ . For the hyperplane $H_m$ , we denote by $R_m\colon {\mathbb {H}}^d\to {\mathbb {H}}^d$ the reflection across $H_m$ .

In order to start the construction of the decomposition, for each hyperplane $H_n$ , we define a cutoff function

$$\begin{align*}\psi_n\colon {\mathbb{H}}^d\to [0,1]\colon \qquad x\mapsto \max\left\{1-\frac{ \mathrm{dist}(x,H_n)}{\varepsilon},0\right\}, \end{align*}$$

which clearly satisfies $\mathrm {Lip}(\psi _n)\leqslant 1/\varepsilon $ , $\psi _n(x)=1$ if and only if $x\in H_n$ , and $\psi _n(x)=0$ if and only if $\operatorname {dist}(x,H_n)\geqslant \varepsilon $ .

We now construct operators on the space $(\bigoplus _{m\in \mathbb {N}}\mathrm {Lip}_0(P_m))_{\ell _\infty }$ . In order to simplify the notation, we often interpret this sum as the space of all functions g defined on $\bigcup _{m\in \mathbb {N}} \mathrm {int}(P_m)\subset {\mathbb {H}}^d$ which are Lipschitz on every $\mathrm {int}(P_m)$ with $\sup _{m\in \mathbb {N}} \mathrm {Lip}(g|_{P_m})< \infty $ and that vanish on the net $\mathcal {N}$ . More precisely, to a sequence $(g_m)_{m\in \mathbb {N}}$ , we associate the function g defined to be equal to $g_m$ on $\mathrm {int}(P_m)$ , for every $m\in \mathbb {N}$ ; vice versa, to a function g, we associate the sequence $(g_m)_{m\in \mathbb {N}}$ , where $g_m$ is the unique continuous extension of $g|_{\mathrm {int} (P_m)}$ to $P_m$ . The important advantage of this interpretation is that we do not consider values on the boundaries of the tiles, which simplifies several formulas. As a first instance of this, this interpretation allows us to identify $\mathrm {Lip}_{0,\mathcal {N}}({\mathbb {H}}^d)$ with the subspace of all functions admitting a continuous extension to ${\mathbb {H}}^d$ .

With these preparations, we are now able to define the operator

$$\begin{align*}\chi_n\colon \Big(\bigoplus_{m\in\mathbb{N}}\mathrm{Lip}_0(P_m)\Big)_{\ell_\infty} \to \Big(\bigoplus_{m\in\mathbb{N}}\mathrm{Lip}_0(P_m)\Big)_{\ell_\infty} \end{align*}$$

by setting

$$\begin{align*}(\chi_n g)(x) = \begin{cases} g(x) & \text{for}\;x\in H_n^{-}, \\ g(x) - \psi_n(x) g(R_n(x)) &\text{for}\; x\in H_n^+. \end{cases} \end{align*}$$

Notice that in the definition of $\chi _n$ , we used the interpretation explained above. Also, according to Lemma 3.3, $R_n$ maps $\bigcup _{m\in \mathbb {N}} \mathrm {int}(P_m)$ to itself; thus, we can legitimately evaluate g at the point $R_n(x)$ .

Since reflections are isometries and g is bounded, the operator $\chi _n$ is bounded and it is clearly also weak $^*$ -to-weak $^*$ continuous. Also note that since $\chi _n$ does not change, the values of g on $H_n^-$ and $R_n(x)\in H_n^-$ whenever $x\in H_n^+$ , its inverse is the operator defined by

$$\begin{align*}(\chi_n^{-1} g)(x) = \begin{cases} g(x) & \text{for}\;x\in H_n^{-}, \\ g(x) + \psi_n(x) g(R_n(x)) &\text{for}\; x\in H_n^+. \end{cases} \end{align*}$$

We denote by $\chi _{1,\ldots ,n}:=\chi _n\circ \cdots \circ \chi _1$ the composition of $\chi _1,\ldots , \chi _n$ and note that it satisfies $\chi _{1,\ldots ,n}^{-1} = \chi _1^{-1}\circ \cdots \circ \chi _n^{-1}$ .

Given a function $g\in \mathrm {Lip}_{0,\mathcal N}({\mathbb {H}}^d)$ , we define a function $g_m\colon P_m\to \mathbb {R}$ by

$$\begin{align*}g_m(x) := \lim_{n\to\infty} \chi_{1,\ldots,n}(g)(x) \qquad (x\in P_m). \end{align*}$$

Note that for each m, there is an $N_m$ such that $\chi _{1,\ldots ,n}(g)(x)$ does not change for $n\geqslant N_m$ . Hence, the above limit exists since the sequence is eventually constant, and the index where it becomes constant only depends on $P_m$ and not on the individual point $x\in P_m$ .

Let us now consider the linear mapping

$$\begin{align*}\mathrm{Lip}_{0,\mathcal N}({\mathbb{H}}^d)\to \Big(\bigoplus_{m\in\mathbb{N}}\mathrm{Lip}_0(P_m)\Big)_{\ell_\infty}\colon \quad g\mapsto (g_m)_{m\in\mathbb{N}}, \end{align*}$$

which will eventually be the main ingredient for our isomorphism. Since it is not surjective between the above spaces, we have to determine a suitable codomain. With this in mind, for each $m\in \mathbb {N}$ , we define

(5.1) $$ \begin{align} S_m:=\bigcup_{\substack{n\in\mathbb{N}\\P_m\subset H_n^+}}H_n\cap P_m. \end{align} $$

Now we are able to define the desired operator

$$\begin{align*}\Phi\colon \mathrm{Lip}_{0,\mathcal N}({\mathbb{H}}^d)\to \Big(\bigoplus_{m\in\mathbb{N}}\mathrm{Lip}_{0,S_m}(P_m)\Big)_{\ell_\infty}\colon g \mapsto (g_m)_{m\in\mathbb{N}} \end{align*}$$

and start checking that it has the desired properties.

Proof We first check that $\Phi (g)$ is a sequence of Lipschitz functions whose Lipschitz constants are bounded by a multiple of the Lipschitz constant of g. Since $g\circ R_n$ and $\psi _n$ are Lipschitz and bounded, the standard product formula gives us that $\chi _n$ is bounded with $\|\chi _n\|= \|\chi _1\|$ for all n. Moreover, $\chi _n$ only changes its argument on an $\varepsilon $ -neighborhood of $H_n$ , and $P_m$ intersects said neighborhood if and only if $H_n$ supports a face of $P_m$ . Hence, the number of faces $N(d)$ of the polytopes $P_m$ gives us the number of times the function g can at most be changed on that tile during the limit process that produces $\Phi (g)$ . Thus, the Lipschitz constant of $\Phi (g)_m$ is at most

$$\begin{align*}\mathrm{Lip}(\Phi(g)_m)\leqslant \|\chi_1\|^{N(d)} \mathrm{Lip}(g). \end{align*}$$

Also, $\Phi (g)(p_m)=g(p_m)=0$ because $p_m\in \mathcal N$ and the center-point $p_m$ of any $P_m$ is not in an $\varepsilon $ -neighborhood of any $H_n$ . It follows that $\Phi (g)_m\in \mathrm {Lip}_0(P_m)$ . Note that the above inequality also shows the boundedness of $\Phi $ . By the same reason, the weak $^*$ -to-weak $^*$ continuity of $\chi _n$ implies that $\Phi $ is also weak $^*$ -to-weak $^*$ continuous.

To conclude, we need to check that $(\Phi (g))_m$ vanishes on $S_m$ . For this aim, assume $x\in P_m\cap H_n$ and $P_m\subset H_n^+$ and observe that

$$\begin{align*}\chi_{1,\ldots,n}(g)(x)= \chi_{1,\ldots,n-1}(g)(x) - \underset{=1}{\underbrace{\psi_n(x)}}(\chi_{1,\ldots,n-1}(g)(x))(\underset{=x}{\underbrace{R_nx}}) = 0. \end{align*}$$

So at the n-th step of our limit process, the values of our function $\chi _{1,\dots ,n}g$ are set to zero everywhere along $H_n$ on the tiles that lie within $H_n^+$ . To express this briefly, we say that $\chi _{1,\dots ,n}g$ vanishes on $\partial H_n^+$ . We now have to show that this value of zero is retained by all the subsequent functions $\chi _{1,\dots ,k}g$ , $k>n$ . We will do this by induction.

So fix $k>n$ and assume that $\chi _{1,\dots ,k-1}g(x)=0$ for $x\in \partial H_n^+$ . To show that $\chi _{1,\dots ,k}g(x)=0$ as well, we distinguish two cases. If $H_k$ and $H_n$ are not orthogonal, it follows from our choice of $\varepsilon $ that $\psi _k$ is zero on $H_n$ . Hence, $\chi _{1,\dots ,k}g(x)=\chi _{1,\dots ,k-1}g(x)=0$ for $x\in \partial H_n^+$ . However, if $H_k$ is orthogonal to $H_n$ , Lemma 3.2 implies that $R_k[H_n]=H_n$ and $R_k[H_n^+]= H_n^+$ . Therefore, $\chi _{1,\dots ,k-1}g(R_k x)=0$ when $x\in \partial H_n^+$ . As before, it follows that $\chi _{1,\dots ,k}g$ vanishes on $\partial H_n^+$ , which finishes the proof.

In the following subsection, we will construct an inverse operator for $\Phi $ , thus showing that $\Phi $ is an isomorphism.

5.3 Extension operators from the tiles and inverse of $\Phi $

The goal of this section is to construct a linear extension operator from $\mathrm {Lip}_{0,S_m}(P_m)$ to $\mathrm {Lip}_{0,\mathcal {N}}({\mathbb {H}}^d)$ which works well with the decomposition considered in the previous section and allows us to show that the operator $\Phi $ is an isomorphism. The interpretation of $(\bigoplus _{k\in \mathbb {N}}\mathrm {Lip}_{0}(P_k))_{\ell _\infty }$ and its subspace $(\bigoplus _{k\in \mathbb {N}}\mathrm {Lip}_{0,S_k}(P_k))_{\ell _\infty }$ considered in the previous section allows us to view the space $\mathrm {Lip}_{0,S_m}(P_m)$ as a subset of $(\bigoplus _{k\in \mathbb {N}}\mathrm {Lip}_{0,S_k}(P_k))_{\ell _\infty }$ , by setting the function equal to zero outside of $P_m$ .

For the construction of the extension operator for $\mathrm {Lip}_{0,S_m}(P_m)$ , let $n_1>\dotsb >n_s$ be the natural numbers such that the hyperplane $H_{n_j}$ supports a face of $P_m$ which is not contained in $S_m$ . As a first step, we define the operator

$$\begin{align*}\mathrm{Lip}_{0,S_m}(P_m)\to\left(\bigoplus_{m\in\mathbb{N}}\mathrm{Lip}_{0}(P_m)\right)_{\ell_\infty}\colon \qquad h \mapsto(\chi_{n_s}^{-1}\circ \dots\circ \chi_{n_1}^{-1})(h). \end{align*}$$

The next lemma shows that $(\chi _{n_s}^{-1}\circ \dots \circ \chi _{n_1}^{-1})(h)$ admits a continuous extension to ${\mathbb {H}}^d$ , and hence, we may define the extension operator

$$\begin{align*}E_m\colon \mathrm{Lip}_{0,S_m}(P_m)\to \mathrm{Lip}_{0,\mathcal{N}}({\mathbb{H}}^d)\colon \qquad h\mapsto E_mh, \end{align*}$$

where $E_mh$ is the unique continuous extension of $(\chi _{n_s}^{-1}\circ \dots \circ \chi _{n_1}^{-1})(h)$ to ${\mathbb {H}}^d$ .

Proof In order to show that $E_m$ is well defined, we have to show that $g:=(\chi _{n_s}^{-1}\dots \chi _{n_1}^{-1})(h)$ admits a continuous extension to ${\mathbb {H}}^d$ . We show this in an inductive manner. Note that no hyperplane $H_n$ intersects the interior of $P_m$ and, by definition of $S_m$ (see (5.1)), $H_{n_j}^{+}$ is the half-space separated from $P_m$ by $H_{n_j}$ (otherwise, $H_{n_j}$ would support a face contained in $S_m$ , which it does not by definition of $n_1,\dots ,n_s$ ).

Since we think of $C_k:=\bigcup _{j=k+1}^{s} \mathrm {int}(H_{n_j}^{+})$ as the set to where we have not yet extended our function in the k-th step, we now show that $g_k:=(\chi _{n_{k}}^{-1}\circ \dots \circ \chi _{n_{1}}^{-1})(h)$ has an extension to ${\mathbb {H}}^d$ which is zero on $C_k$ and continuous on ${\mathbb {H}}^d\setminus C_k$ . Since $g=g_s$ and $C_s=\emptyset $ this will prove our claim.

The function $g_0=h$ is zero outside of $P_m$ and hence on $C_0$ and $({\mathbb {H}}^d\setminus C_0)\setminus P_m$ . Since the (relative) boundary of $P_m$ in the set ${\mathbb {H}}^d\setminus C_0$ is precisely $S_m$ and $h_m$ vanishes there, $g_0$ has a continuous extension to ${\mathbb {H}}^d\setminus C_0$ .

Let us assume now that we have already shown that $g_{k-1}$ has an extension to ${\mathbb {H}}^d$ which is zero on $C_{k-1}$ and continuous on ${\mathbb {H}}^d\setminus C_{k-1}$ . By abuse of notation, we call this extension $g_{k-1}$ as well. We now have to check that $g_{k}$ also has the desired properties.

We first check that $g_k$ vanishes on $C_k$ and recall that $g_k(x)=g_{k-1}(x)$ for $x\in H_{n_k}^{-}$ and $g_{k}(x) = g_{k-1}(x) + \psi _n(x) g_{k-1}(R_{n_k}x)$ for $x\in H_{n_k}^+$ . We distinguish between two cases: If for $j>k$ we have $H_{n_j}\cap H_{n_{k}}\neq \emptyset $ , these hyperplanes are orthogonal, and hence, $x\in H_{n_j}^+$ if and only if $R_{n_{k}}x \in H_{n_j}^+$ , and hence, $g_k(x)=0$ for $x\in \mathrm {int}\,(H_{n_j}^+)$ . For the other case, note that $H_{n_{k}}$ cannot be in $H_{n_j}^+$ since otherwise the face of $P_m$ contained in $H_{n_{k}}$ would have to be in $H_{n_j}^+$ , which is only possible if it was in $H_{n_j}$ , which contradicts $H_{n_{k}}\cap H_{n_j}=\emptyset $ . Hence, we have $\mathrm {dist}\,(H_{n_j}^+, H_{n_{k}})>2\varepsilon $ , and therefore, $g_k(x) = g_{k-1}(x)=0$ for $x\in H_{n_{s-k+1}}$ .

In order to show that $g_k$ has a continuous extension to ${\mathbb {H}}^d\setminus C_k$ , let $z\in H_{n_k}\setminus C_k$ and note that $g_{k-1}(x)=0$ for $x\in \mathrm {int}(H_{n_k}^{-})$ . Hence,

$$\begin{align*}\lim_{x\to z} g_{k}(x) = \lim_{x\to z} \psi_{n_k}(x) g_{k-1}(R_{n_k}(x)) = g_{k-1}(z), \end{align*}$$

where x converges to z in the interior of $H_{n_k}^{-}$ since $\psi _{n_k}(z)=1$ and $R_{n_k}(z)=z$ . This finishes the proof of the well-definedness of $E_m$ .

Linearity and pointwise-to-pointwise continuity are clear. Since $\chi _{n_s}^{-1}\dots \chi _{n_1}^{-1}$ is the composition of bounded operators on $(\bigoplus _{m\in \mathbb {N}}\mathrm {Lip}_{0}(P_m))_{\ell _\infty }$ and $E_mh_m$ is continuous, boundedness of $E_m$ follows from Lemma 2.1 and $\|E_m\|\leqslant \|\chi _{n_s}^{-1}\circ \dots \circ \chi _{n_1}^{-1}\|$ . Finally, note that this norm only depends on $\varepsilon $ , s, and the diameter of $P_m$ and that s is at most the number of faces of $P_m$ .

We are now in position to define the operator

$$\begin{align*}\Psi\colon \Big(\bigoplus_{m\in\mathbb{N}}\mathrm{Lip}_{0,S_m}(P_m)\Big)_{\ell_\infty} \longrightarrow \mathrm{Lip}_{0,\mathcal N}({\mathbb{H}}^d)\colon \qquad (h_m)_{m\in\mathbb{N}}\mapsto \sum_{m\in\mathbb{N}}E_mh_m \end{align*}$$

(where the sum is meant to be taken pointwise) and show that it is the inverse of $\Phi $ .

Proof The sum in the definition of $\Psi $ is locally finite with a uniform bound $N(d)$ on the number of summands; thus, $\Psi $ is a bounded linear operator by Lemma 5.4. Since weak $^*$ -to-weak $^*$ continuity of $\Psi $ is easy to see, we are left to show that $\Psi $ is the inverse of $\Phi $ .

We start by showing that $\Phi \circ \Psi = \mathrm {Id}$ . For this, let $(h_m)_{m\in \mathbb {N}} \in (\bigoplus _{m\in \mathbb {N}} \mathrm {Lip}_{0,S_m}(P_m))_{\ell _\infty }$ and fix an arbitrary $k\in \mathbb {N}$ . It is enough to check that $(\Phi (\Psi ((h_m)_{m\in \mathbb {N}})))_k(x) = h_k(x)$ for x in the interior of $P_k$ . Recall that in Section 5.2, we observed that there is an index $N_k\in \mathbb {N}$ such that $(\chi _{1,\ldots ,n}g)(x)=(\chi _{1,\dots ,N_k}g)(x)$ for $n\geqslant N_k$ , every $g\in \mathrm {Lip}_{0,\mathcal {N}}({\mathbb {H}}^d)$ , and all $x\in P_k$ . Using this observation together with the definition of these operators and Remark 5.5, we have

$$ \begin{align*} (\Phi(\Psi((h_m)_{m\in\mathbb{N}})))_k(x) &= \Big(\Phi\Big(\sum_{m\in\mathbb{N}} E_mh_m\Big)\Big)_k(x) = \sum_{m\in\mathbb{N}}\chi_{1,\ldots,N_k}(E_mh_m)(x) \\ &= \sum_{m\in\mathbb{N}} (\chi_{1,\ldots,N_k} (\chi_{1,\ldots,N_k}^{-1}(h_m))) (x) = \sum_{m\in\mathbb{N}} h_m(x) = h_k(x) \end{align*} $$

for all $x\in \mathrm {int}\,(P_k)$ (i.e. $(\Phi (\Psi ((h_m)_{m\in \mathbb {N}})))_k=h_k$ ).

In order to show that $\Psi \circ \Phi = \mathrm {Id}$ , let $g\in \mathrm {Lip}_{0,\mathcal {N}}({\mathbb {H}}^d)$ be given. It is enough to check that $\Psi \circ \Phi (g)(x)= g(x)$ for all x in the interior of $P_k$ and for all k. Since $\chi _n^{-1}((\Phi (g))_k)(x)$ for $k\neq n$ is only nonzero if $x\in H_{n}^+$ and within $\varepsilon $ -distance of $H_n$ and there are at most d hyperplanes $H_{n_1},\ldots , H_{n_d}$ with this property, there are numbers $M,N\in \mathbb {N}$ such that

since we may interpret the sum locally around x as a representation of $\chi _{1,\ldots ,N}(g)$ as a function similar to the previous sections, and the operator $\chi _{1,\ldots , N}$ and its inverse are defined pointwise and only depend on “nearby” values of the function.

5.4 Conclusion of the proof and consequences

In this section, we use the arguments of the previous sections and combine them with the results of Section 4 to prove our main result. Moreover, we state a number of direct consequences of the main result and compare it to the case of the space of bounded Lipschitz functions $\mathrm {Lip}({\mathbb {H}}^d)$ .

Theorem 5.7 For $d=2,3,4$ , we have

$$\begin{align*}\mathrm{Lip}_0({\mathbb{H}}^d)\simeq\Big(\mathrm{Lip}_0(\mathcal N) \oplus \bigoplus_{m\in \mathbb{N}} \mathrm{Lip}_0(P_m)\Big)_{\ell_\infty} \end{align*}$$

and

$$\begin{align*}\mathcal{F}({\mathbb{H}}^d)\simeq \Big(\mathcal{F}(\mathcal N) \oplus \bigoplus_{m\in \mathbb{N}} \mathcal{F}(P_m)\Big)_{\ell_1}. \end{align*}$$

Proof We consider the mapping

$$\begin{align*}\mathrm{Lip}_0({\mathbb{H}}^d) \to \mathrm{Lip}_0(\mathcal{N}) \oplus \Big(\bigoplus_{m\in\mathbb{N}}\mathrm{Lip}_{0,S_m}(P_m)\Big)_{\ell_\infty}\colon \qquad f\mapsto \big(f|_{\mathcal{N}}, \Phi(f-f|_{\mathcal{N}})\big), \end{align*}$$

which by Lemma 5.1, Lemma 5.3, and Proposition 5.6 is a weak $^*$ -to-weak $^*$ continuous isomorphism. By Lemma 3.1, $P_m$ is bi-Lipschitz equivalent to the same polytope $P_m'$ considered in $\mathbb {R}^d$ . Combining this with Proposition 4.3 implies that $\mathrm {Lip}_{0,S_m}(P_m)$ and $\mathrm {Lip}_0(P_m)$ are weak $^*$ -to-weak $^*$ isomorphic, with a uniform bound on the distortion (because the polytopes $P_m$ are mutually isometric). This gives the first of the claims above and shows that the isomorphisms are also weak $^*$ -to-weak $^*$ continuous. Hence, we may pass to the preduals and arrive at the second claim.

Remark 5.11 Note that a similar construction using a tessellation with cubes works in $\mathbb {R}^d$ . Additionally, if we look at the spaces of bounded Lipschitz functions $\mathrm {Lip}(\mathbb {R}^d)$ and $\mathrm {Lip}({\mathbb {H}}^d)$ rather than the ones which are zero at a given base point, then we can use an analogous construction to the one we employ in Section 5.2 and Section 5.3 to decompose any function $f\in \mathrm {Lip}({\mathbb {H}}^d)$ and any $g\in \mathrm {Lip}(\mathbb {R}^d)$ into a sequence of bounded Lipschitz functions $(f_n)_{n\in \mathbb {N}}$ and $(g_n)_{n\in \mathbb {N}}$ on the tiles $(P_n)_{n\in \mathbb {N}}$ tessellating ${\mathbb {H}}^d$ and the cubes $(Q_n)_{n\in \mathbb {N}}$ tesselating $\mathbb {R}^d$ , respectively. Since the functions in $\mathrm {Lip}(\mathbb {R}^{d})$ and $\mathrm {Lip}(\mathbb {H}^d)$ are already bounded and functions in $\mathrm {Lip}(M)$ do not require a base point within the domain on which they are zero, removing the values on a net is not needed in this case. Thus, for $d=2,3,4$ ,

$$ \begin{gather*} \mathrm{Lip}(\mathbb{H}^d) \simeq \Big( \bigoplus_{n\in \mathbb{N}} \mathrm{Lip}_{S_n}(P_n)\Big)_{\ell_\infty} \qquad\mbox{and}\qquad \mathrm{Lip}(\mathbb{R}^d) \simeq \Big(\bigoplus_{n\in \mathbb{N}} \mathrm{Lip}_{T_n}(Q_n) \Big)_{\ell_\infty}, \end{gather*} $$

where $S_n$ is defined as in Definition 5.1 and $T_n$ , analogously, is the union of all boundary (hyper-)surfaces of the (hyper-)cubes $Q_n$ which are supported by a hyperplane which does not separate $Q_n$ from the origin $0_{\mathbb {R}^d}$ . Equivalently, the union of all faces of $Q_n$ which contain the point(s) of $Q_n$ that is (are) furthest from the origin.

From this, we are able to obtain that $\mathrm {Lip}(\mathbb {H}^d)\simeq \mathrm {Lip}(\mathbb {R}^d)$ – this time, in a completely explicit manner.

Indeed, since $S_n$ is the part of the boundary of $P_n$ which is invisible from the origin (i.e., the geodesics connecting a point from $S_n$ with the origin intersect the polytope in other points) an elementary geometric argument shows that it is simply connected. Similarly, $T_n$ is a simply connected subset of $Q_n$ . It follows that one can find explicit bi-Lipschitz mappings between any pair $(P_n,S_n)$ and $(Q_m,T_m)$ . The only two tiles for which this is not true are the two tiles $P_1\subset {\mathbb {H}}^d$ and $Q_1\subset \mathbb {R}^d$ which contain their spaces’ respective base points $0$ , since those are the only two tiles whose “boundary condition” $S_n$ is the entire boundary $\partial P_1$ and $\partial Q_1$ , respectively. However, this simply means that we map $P_1$ to $Q_1$ in a bi-Lipschitz way, and then carry on with the rest of the sequence arbitrarily since any other $(P_n,S_n)$ can be bi-Lipschitz mapped to any, say, $(Q_m,T_m)$ .

Finally, there are only finitely many different pairs $(P_n,S_n)$ , which are not pairwise congruent (and similarly, finitely many pairs $(Q_m,T_m)$ ). Therefore, these bi-Lipschitz maps have uniformly bounded distortion and hence induce and isomorphism $\mathrm {Lip}(\mathbb {H}^d)\simeq \mathrm {Lip}(\mathbb {R}^d)$ .

Remark 5.12 The isomorphism $\mathrm {Lip}(\mathbb {S}^d)\simeq \mathrm {Lip}(\mathbb {R}^d)$ can be deduced from the corresponding isomorphism for the $\mathrm {Lip}_0$ -spaces. First note that since $\mathbb {S}^d$ has a finite diameter, the space $\mathrm {Lip}_0(\mathbb {S}^{d})$ is a hyperplane in $\mathrm {Lip}(\mathbb {S}^d)$ . Moreover, we have

$$\begin{align*}\mathrm{Lip}_0(\mathbb{S}^d) \simeq \mathrm{Lip}_0(\mathbb{R}^d) \simeq \mathrm{Lip}_0([0,1]^d) \end{align*}$$

by [Reference Albiac, Ansorena, Cúth and Doucha1] or [Reference Freeman and Gartland25]. As for the sphere, the space $\mathrm {Lip}_0([0,1]^d)$ is a hyperplane in $\mathrm {Lip}([0,1]^d)$ . By [Reference Cúth, Doucha and Wojtaszczyk20, Theorem 5] all the $\mathrm {Lip}_0$ -spaces above and hence also all spaces of bounded Lipschitz functions contain a (complemented) copy of $\ell _\infty $ ; hence, they are isomorphic to their hyperplanes. Therefore, we may conclude that $\mathrm {Lip}(\mathbb {S}^{d}) \simeq \mathrm {Lip}([0,1]^d)$ . Using Proposition 4.3 together with an argument similar to the one in Remark 5.11, we obtain that $\mathrm {Lip}(\mathbb {S}^{d})\simeq \mathrm {Lip}(\mathbb {R}^d)$ . In contrast to the case of the hyperbolic space, this argument works for arbitrary d.