Littlewood-Paley Theory for Triangle Buildings.

For the natural two-parameter filtration $\left(F_{\lambda },:,\lambda \in P\right)$ on the boundary of a triangle building, we define a maximal function and a square function and show their boundedness on $L^p\left({\Omega }_0\right)$ for $p\in (1,\infty )$ . At the end, we consider $L^p\left({\Omega }_0\right)$ boundedness of martingale transforms. If the building is of $\text{GL}(3,Q_p)$ , then ${\Omega }_0$ can be identified with p-adic Heisenberg group.