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

TL;DR

This paper develops a real-variable theory for a new class of twisted multiparameter singular integrals on complex quadratic surfaces, with applications to Fourier multipliers and signal processing.

Contribution

It introduces twisted tube systems and maximal functions, establishing foundational tools like Littlewood--Paley theory and atomic decompositions for these integrals.

Findings

Established a reproducing formula for twisted multiparameter integrals.

Developed a Littlewood--Paley theory and atomic decompositions.

Constructed twisted Fourier multipliers with potential signal processing applications.

Abstract

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.