Maximal Functions, Littlewood–Paley Theory, Riesz Transforms and Atomic Decomposition in the Multi-parameter Flag Setting

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 a...

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Hlavní autoři: Han, Yongsheng, Lee, Ming-Yi, Li, Ji, Wick, Brett D.
Médium: E-kniha Kniha
Jazyk:angličtina
Vydáno: Providence, Rhode Island American Mathematical Society 2022
Vydání:1
Edice:Memoirs of the American Mathematical Society
Témata:
Hardy spaces
Littlewood-Paley theory
Maximal functions
flag Hardy space
flag Riesz transforms
maximal function
Littlewood–Paley square function
atomic decomposition
Lusin area integral
ISBN:1470453452, 9781470453459
ISSN:0065-9266, 1947-6221
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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.
AbstractList 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.
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Author Han, Yongsheng
Wick, Brett D.
Lee, Ming-Yi
Li, Ji
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  givenname: Brett D.
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BackLink https://cir.nii.ac.jp/crid/1130293545593070498$$DView record in CiNii
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Keywords flag Hardy space
flag Riesz transforms
maximal function
Littlewood–Paley square function
atomic decomposition
Lusin area integral
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Notes September 2022, volume 279, number 1373 (second of 6 numbers)
Includes bibliographical references (p. 101-102)
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Snippet In this paper, we develop via real variable methods various characterisations of the Hardy spaces in the multi-parameter flag setting. These characterisations...
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SubjectTerms Hardy spaces
Littlewood-Paley theory
Maximal functions
TableOfContents Acknowledgement -- Notation -- Introduction and Statement of Main Results, Applications -- Flag Littlewood–Paley Estimate: <inline-formula content-type="math/mathml"> ‖ g F ( f ) ‖ 1 \|g_F(f)\|_1 </inline-formula>, <inline-formula content-type="math/mathml"> ‖ S F ( f ) ‖ 1 \|S_F(f)\|_1 </inline-formula> and <inline-formula content-type="math/mathml"> ‖<!-- ‖ --> S F ( U ) ‖ 1 \|S_F(U)\|_1 </inline-formula> -- Estimates of Area Functions, Maximal Functions and Riesz Transforms via Flag Poisson Integral Technique -- Flag Maximal Functions: from Poisson Kernel to General Schwartz Kernels -- Atomic Decompositions of Flag Hardy Spaces -- Estimates of Riesz Transforms and Area Function via Atomic Decomposition
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 -- Back Cover
Title Maximal Functions, Littlewood–Paley Theory, Riesz Transforms and Atomic Decomposition in the Multi-parameter Flag Setting
URI https://www.ams.org/memo/1373/
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Volume 279
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