Littlewood-Paley theory for triangle buildings

For the natural two parameter filtration $($\mathcal{F}$_\lambda : \lambda \in P)$ on the boundary of a triangle building we define a maximal function and a square function and show their boundedness on $L^p({\mathrm{\Omega }}_0)$ for $p \in (1, \infty)$\mathrm{.}Attheendweconsider$L^p(\Omega_0)$ boundedness of martingale transforms. If the building is of $\text{GL}(3,{\mathbb{Q}}_p)$ then ${\mathrm{\Omega }}_0$ can be identified with $p$-adic Heisenberg group.