LEADER 05247nam 22006613 450 001 9910960743903321 005 20231110234130.0 010 $a9781470472276 010 $a1470472279 035 $a(MiAaPQ)EBC29731900 035 $a(Au-PeEL)EBL29731900 035 $a(CKB)24767186500041 035 $a(OCoLC)1343250653 035 $a(PPN)270359117 035 $a(EXLCZ)9924767186500041 100 $a20220904d2022 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aMaximal Functions, Littlewood-Paley Theory, Riesz Transforms and Atomic Decomposition in the Multi-Parameter Flag Setting 205 $a1st ed. 210 1$aProvidence :$cAmerican Mathematical Society,$d2022. 210 4$d©2022. 215 $a1 online resource (118 pages) 225 1 $aMemoirs of the American Mathematical Society ;$vv.279 311 08$aPrint version: Han, Yongsheng Maximal Functions, Littlewood-Paley Theory, Riesz Transforms and Atomic Decomposition in the Multi-Parameter Flag Setting Providence : American Mathematical Society,c2022 9781470453459 327 $aCover -- 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: | _{ }( )|?, | _{ }( )|? and | _{ }( )|? -- 2.1. Discrete Calderón Reproducing Formula -- 2.2. Flag Plancherel-Pólya Type Inequalities -- 2.3. The Equivalence of | _{ }( )|? and | _{ }( )|? -- 2.4. The Estimate | _{ }( )|??| _{ }( )|? -- Chapter 3. Estimates of Area Functions, Maximal Functions and Riesz Transforms via Flag Poisson Integral Technique -- 3.1. The Estimate | _{ }( )|??| *|? -- 3.2. The Estimate | *|??| ?|? -- 3.3. The Estimate | ?|??????^{ + }?_{ =1}^{ }| _{ , }( )|?+| |? -- Chapter 4. Flag Maximal Functions: from Poisson Kernel to General Schwartz Kernels -- 4.1. The Equivalence | *|??| *?( )|? -- 4.2. The Equivalence | ?|??| ??( )|? -- Chapter 5. Atomic Decompositions of Flag Hardy Spaces -- 5.1. Heat Kernel and Finite Speed Propagation -- 5.2. Atomic Decomposition for ¹_{ }(??×?^{ }). -- 5.3. Proof of the Atomic Decomposition -- Chapter 6. Estimates of Riesz Transforms and Area Function via Atomic Decomposition -- Bibliography -- Back Cover. 330 $a"In this monograph, we develop via real variable methods various characterisations of the Hardy spaces in the multi-parameter flag setting. These characterisations include those via, the non-tangential and radial maximal function, the Littlewood- Paley square function and area integral, Riesz transforms and the atomic decomposition in the multi-parameter flag setting. The novel ingredients in this paper include (1) establishing appropriate discrete Calderon reproducing formulae in the flag setting and a version of the Plancherel-Polya inequalities for flag quadratic forms; (2) introducing the maximal function and area function via flag Poisson kernels and flag version of harmonic functions; (3) developing an atomic decomposition via the finite speed propagation and area function in terms of flag heat semigroups. As a consequence of these real variable methods, we obtain the full characterisations of the multi-parameter Hardy space with the flag structure"--$cProvided by publisher. 410 0$aMemoirs of the American Mathematical Society 606 $aHardy spaces 606 $aMaximal functions 606 $aLittlewood-Paley theory 606 $aSingular integrals 606 $aHarmonic analysis on Euclidean spaces -- Harmonic analysis in several variables -- $H^p$-spaces$2msc 606 $aHarmonic analysis on Euclidean spaces -- Harmonic analysis in several variables -- Maximal functions, Littlewood-Paley theory$2msc 606 $aHarmonic analysis on Euclidean spaces -- Harmonic analysis in several variables -- Singular and oscillatory integrals (Caldero?n-Zygmund, etc.)$2msc 615 0$aHardy spaces. 615 0$aMaximal functions. 615 0$aLittlewood-Paley theory. 615 0$aSingular integrals. 615 7$aHarmonic analysis on Euclidean spaces -- Harmonic analysis in several variables -- $H^p$-spaces. 615 7$aHarmonic analysis on Euclidean spaces -- Harmonic analysis in several variables -- Maximal functions, Littlewood-Paley theory. 615 7$aHarmonic analysis on Euclidean spaces -- Harmonic analysis in several variables -- Singular and oscillatory integrals (Caldero?n-Zygmund, etc.). 676 $a515/.2433 676 $a515.2433 686 $a42B30$a42B25$a42B20$2msc 700 $aHan$b Yongsheng$0505470 701 $aLee$b Ming-Yi$01801839 701 $aLi$b Ji$0654528 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910960743903321 996 $aMaximal Functions, Littlewood-Paley Theory, Riesz Transforms and Atomic Decomposition in the Multi-Parameter Flag Setting$94347235 997 $aUNINA