1 results
Tilings of the hyperbolic space and Lipschitz functions
- Part of
-
- Journal:
- Canadian Journal of Mathematics , First View
- Published online by Cambridge University Press:
- 18 November 2024, pp. 1-21
-
- Article
-
- You have access
- Open access
- HTML
- Export citation
-
We use a special tiling for the hyperbolic d-space
$\mathbb {H}^d$ for 
$d=2,3,4$ to construct an (almost) explicit isomorphism between the Lipschitz-free space 
$\mathcal {F}(\mathbb {H}^d)$ and 
$\mathcal {F}(P)\oplus \mathcal {F}(\mathcal {N})$, where P is a polytope in 
$\mathbb {R}^d$ and 
$\mathcal {N}$ a net in 
$\mathbb {H}^d$ coming from the tiling. This implies that the spaces 
$\mathcal {F}(\mathbb {H}^d)$ and 
$\mathcal {F}(\mathbb {R}^d)\oplus \mathcal {F}(\mathcal {M})$ are isomorphic for every net 
$\mathcal {M}$ in 
$\mathbb {H}^d$. In particular, we obtain that, for 
$d=2,3,4$, 
$\mathcal {F}(\mathbb {H}^d)$ has a Schauder basis. Moreover, using a similar method, we also give an explicit isomorphism between 
$\mathrm {Lip}(\mathbb {H}^d)$ and 
$\mathrm {Lip}(\mathbb {R}^d)$.
