# Littlewood-Paley theory for triangle buildings

## TL;DR

This paper develops Littlewood-Paley theory for triangle buildings, establishing boundedness of maximal and square functions on $L^p$ spaces and exploring martingale transforms, with applications to $p$-adic groups.

## Contribution

It introduces a Littlewood-Paley framework for triangle buildings and proves boundedness results for associated maximal, square, and martingale transform operators.

## Key findings

- Boundedness of maximal and square functions on $L^p$ for $p o (1, \, ext{infinity})$
- Extension of theory to $p$-adic Heisenberg group case
- Analysis of martingale transforms in this geometric setting

## 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.