Littlewood-Paley theory for triangle buildings

Abstract

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 $L^p({\Omega }_0)$ for $p\in (1,\infty )$. At the end we consider $L^p({\Omega }_0)$ boundedness of martingale transforms. If the building is of $\text{GL}(3,{\mathbb{Q}}_p)$ then ${\Omega }_0$ can be identified with $p$-adic Heisenberg group.

Accepted for publication in The Journal of Geometric Analysis, 18 pages, 5 figures