Using real variable methods, Han and colleagues develop characterizations of the Hardy spaces in the multi-parameter flag setting. Novel aspects of their study are establishing appropriate discrete Calderón reproducing formulae in the flag setting and a version of the Plancherel-Pólya inequalities for flag quadratic forms, introducing the maximal function and area functions via flag Poisson kernels and a flag version of harmonic functions, and 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, they say, they obtain the full characterizations of the multi-parameter Hardy space with the flag structure. Annotation ©2022 Ringgold, Inc., Portland, OR (protoview.com)
Author(s): Yongsheng Han, Ming-Yi Lee, Ji Li, Brett Wick
Series: Memoirs of the American Mathematical Society, 1373
Publisher: American Mathematical Society
Year: 2022
Language: English
Pages: 117
City: Providence
Cover
Title page
Acknowledgement
Notation
Chapter 1. Introduction and Statement of Main Results, Applications
1.1. Background and Main Results
1.2. Statement of Main Results
1.3. Strategy of Proofs of the Main Results
1.4. Applications and Related Open Questions
Chapter 2. Flag Littlewood–Paley Estimate: |?_{?}(?)|₁, |?_{?}(?)|₁ and |?_{?}(?)|₁
2.1. Discrete Calderón Reproducing Formula
2.2. Flag Plancherel–Pólya Type Inequalities
2.3. The Equivalence of |?_{?}(?)|₁ and |?_{?}(?)|₁
2.4. The Estimate |?_{?}(?)|₁≲|?_{?}(?)|₁
Chapter 3. Estimates of Area Functions, Maximal Functions and Riesz Transforms via Flag Poisson Integral Technique
3.1. The Estimate |?_{?}(?)|₁≲|?*|₁
3.2. The Estimate |?*|₁≲|?⁺|₁
3.3. The Estimate |?⁺|₁≲∑ⱼ₌₁^{?+?}∑_{?=1}^{?}|?_{?,?}(?)|₁+|?|₁
Chapter 4. Flag Maximal Functions: from Poisson Kernel to General Schwartz Kernels
4.1. The Equivalence |?*|₁≈|?*ᵩ(?)|₁
4.2. The Equivalence |?⁺|₁≈|?⁺ᵩ(?)|₁
Chapter 5. Atomic Decompositions of Flag Hardy Spaces
5.1. Heat Kernel and Finite Speed Propagation
5.2. Atomic Decomposition for ?¹_{?}(ℝⁿ×ℝ^{?}).
5.3. Proof of the Atomic Decomposition
Chapter 6. Estimates of Riesz Transforms and Area Function via Atomic Decomposition
Bibliography
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