Tilings of the hyperbolic space and Lipschitz functions

Bargetz, Christian  ; Luggin, Franz ; Russo, Tommaso 

Canadian journal of mathematics, Tome 77 (2025) no. 6, pp. 1985-2005

Résumé

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 $\mathrm {Lip}(\mathbb {R}^d)$.

Détail

DOI : 10.4153/S0008414X24000804

Mots-clés : Lipschitz-free space, Banach space of Lipschitz functions, hyperbolic d-space, tiling, polytope, linear extension operator, Schauder basis