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Maximal Functions, Littlewood–Paley Theory, Riesz Transforms and Atomic Decomposition in the Multi-parameter Flag Setting
About this Title
Yongsheng Han, Ming-Yi Lee, Ji Li and Brett D. Wick
Publication: Memoirs of the American Mathematical Society
Publication Year:
2022; Volume 279, Number 1373
ISBNs: 978-1-4704-5345-9 (print); 978-1-4704-7227-6 (online)
DOI: https://doi.org/10.1090/memo/1373
Published electronically: July 27, 2022
Keywords: maximal function,
Littlewood–Paley square function,
Lusin area integral,
flag Riesz transforms,
atomic decomposition,
flag Hardy space
Table of Contents
Chapters
- Acknowledgement
- Notation
- 1. Introduction and Statement of Main Results, Applications
- 2. Flag Littlewood–Paley Estimate: $\|g_F(f)\|_1$, $\|S_F(f)\|_1$ and $\|S_F(U)\|_1$
- 3. Estimates of Area Functions, Maximal Functions and Riesz Transforms via Flag Poisson Integral Technique
- 4. Flag Maximal Functions: from Poisson Kernel to General Schwartz Kernels
- 5. Atomic Decompositions of Flag Hardy Spaces
- 6. Estimates of Riesz Transforms and Area Function via Atomic Decomposition
Abstract
In this paper, we develop via real variable methods various characterisations of the Hardy spaces in the multi-parameter flag setting. These characterisations include those via, the non-tangential and radial maximal function, the Littlewood–Paley square function and area integral, Riesz transforms and the atomic decomposition in the multi-parameter flag setting. The novel ingredients in this paper include (1) establishing appropriate discrete Calderón reproducing formulae in the flag setting and a version of the Plancherel–Pólya inequalities for flag quadratic forms; (2) introducing the maximal function and area function via flag Poisson kernels and flag version of harmonic functions; (3) developing an atomic decomposition via the finite speed propagation and area function in terms of flag heat semigroups. As a consequence of these real variable methods, we obtain the full characterisations of the multi-parameter Hardy space with the flag structure.- Sun-Yung A. Chang and Robert Fefferman, Some recent developments in Fourier analysis and $H^p$-theory on product domains, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 1, 1–43. MR 766959, DOI 10.1090/S0273-0979-1985-15291-7
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