Maximal functions with twisted structures, distribution inequality and applications

TL;DR

This paper develops a distribution inequality for twisted maximal functions, advancing the theory of twisted Hardy spaces and their boundedness properties using recursive integration techniques.

Contribution

It establishes a Fefferman--Stein type distribution inequality for twisted structures, linking area functions and maximal functions in a new geometric setting.

Findings

Proved the distribution inequality for twisted area and maximal functions.

Established the uniform L^1 boundedness of the twisted maximal function.

Completed the maximal function characterization of twisted Hardy spaces.

Abstract

Motivated by the geometric reduction of Cauchy--Szeg\H{o} projections on quadratic surfaces of higher codimension (Nagel--Ricci--Stein, 2001) and recent developments on the real-variable theory adapted to twisted multiparameter structures (arXiv:2603.26119), we establish the Fefferman--Stein type distribution inequality relating the twisted area function and the twisted non-tangential maximal function over $\mathbb{R}^{2m}$. By deploying a recursive integration-by-parts argument involving the twisted gradient and Laplacian, and constructing smooth, compactly supported weight functions to absorb cross-derivative errors, we obtain the required estimate. As an application, we prove the uniform $L^1$ boundedness of the twisted maximal function on the twisted atoms and complete the maximal function characterization of the twisted Hardy space.