Twisted Multiparameter singular integrals -- real variable methods and applications, I

In this paper, we introduce a class of twisted multiparameter singular integrals on $\mathbb{R}^{2m}$, motivated by the Cauchy--Szeg\H{o} projections and the solving operators for $\bar{\partial}_b$ on a broad family of quadratic surfaces of higher codimension in $\mathbb{C}^n$. These surfaces are represented as suitable quotients of products of Heisenberg groups, a framework illustrated by Stein (Notices Amer. Math. Soc., 1998). While classical multiparameter product and flag theories are well-developed, Nagel, Ricci, and Stein observed a critical limitation: the class of product operators is not closed under passage to a quotient subgroup. To handle the geometric reduction that models these quotient structures, we take the first step in developing an adapted real-variable theory. We achieve this by introducing twisted tube systems and tube maximal functions, establishing a reproducing formula, Littlewood--Paley theory, a Journ\'e-type covering lemma, and atomic decompositions. As particular examples, we obtain twisted Fourier multipliers -- which emerge as novel, direction-sensitive, and anisotropic phase-shift converters with potential applications in signal and image processing.