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Maximal functions with twisted structures, distribution inequality and applications

Ji Li, Chong-Wei Liang, Chaojie Wen, Qingyan Wu

Abstract

Motivated by the geometric reduction of Cauchy--Szegő 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.

Maximal functions with twisted structures, distribution inequality and applications

Abstract

Motivated by the geometric reduction of Cauchy--Szegő 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 . 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 boundedness of the twisted maximal function on the twisted atoms and complete the maximal function characterization of the twisted Hardy space.

Paper Structure

This paper contains 15 sections, 14 theorems, 246 equations.

Table of Contents

  1. Introduction and statement of main results
  2. Preliminaries and auxiliary lemmas
  3. Tube structures and tube maximal function
  4. Miscellaneous types of dyadic rectangles/ tubes
  5. Covering Lemma adapted to the miscellaneous dyadic rectangles
  6. Twisted differential operators
  7. The recursive argument and the twisted good-$\lambda$ inequality
  8. Setup and the separation by the Poisson extension
  9. First iteration: passing $\nabla_1$ to $g$.
  10. Second iteration: passing $\nabla_2$ to $g$.
  11. Third iteration: passing $\nabla_3$ to $g$.
  12. $L^2$ Littlewood--Paley theory and the Chebyshev splitting
  13. Application of good-$\lambda$ inequality: maximal function and proof of Theorem \ref{['secondmainthm']}
  14. Miscellaneous Hardy spaces
  15. Twisted non-tangential characterization of Hardy space

Key Result

Theorem 1.4

There exist constants $C>0$ and $\beta>1$ such that for all $f \in C_0^\infty(\mathbb{R}^{2m})$ and all $\lambda>0$, $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (27)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Proposition 2.1
  • Theorem 2.2: FLLWW
  • Remark 2.3
  • Lemma 2.4: FLLWW
  • ...and 17 more