Witt-Burnside functor attached to ${\mathbf{Z}}_p^2$ and $p$-adic Lipschitz continuous functions

Dress and Siebeneicher gave a significant generalization of the construction of Witt vectors, by producing for any profinite group $G$, a ring-valued functor ${\mathbf{W}}_G$. This paper gives a concrete interpretation of the rings ${\mathbf{W}}_{{\mathbf{Z}}_p^2}(k)$ where $k$ is a field of characteristic $p>0$ in terms of rings of Lipschitz continuous functions on the $p$-adic upper half plane ${\mathbf{P}}^1({\mathbf{Q}}_p)$. As a consequence we show that the Krull dimensions of the rings ${\mathbf{W}}_{{\mathbf{Z}}_p^d}(k)$ are infinite for $d \geq 2$ and we show the Teichm"uller representatives form an analogue of the van der Put basis for continuous functions on ${\mathbf{Z}}_p$.