Haar bases for multi-parameter twisted structures
Motivated by the Cauchy--Szeg\H{o} projections on a broad class of Siegel domains and the geometric quotient structures of nilpotent Lie groups observed by Nagel, Ricci, and Stein, we develop a martingale and Haar wavelet framework for twisted multi-parameter geometries. We introduce twisted dyadic filtrations and construct adapted Haar bases on Euclidean spaces $\mathbb{R}^{2m}$. Each of the resulting dyadic systems forms a complete orthonormal basis of $L^2(\mathbb R^{2m})$, and their union yields a tight frame with frame bound $3$. We establish $L^p$-equivalences for the associated discrete twisted Littlewood--Paley square functions. Furthermore, we extend this discrete real-variable theory to the non-abelian setting of a nilpotent Lie group of step two, $\mathscr{N}$, which serves as the Shilov boundary of certain fundamental Siegel domains. By projecting product fractal tiles from a lifting group of Heisenberg products, we define twisted dyadic shards and construct twisted nilpotent Haar frames. More precisely, we first introduce raw projected shards that reflect the quotient geometry, and then pass to analytic dyadic shards which are exactly rectifiable and remain uniformly comparable to the raw quotient structure in the relevant scale regimes. This yields a discrete framework adapted to twisted quotient geometries in both the Euclidean and nilpotent settings, providing a basic dyadic infrastructure for further developments in twisted real-variable theory.