Real-Variable Theory of Hardy Spaces Associated with Generalized Herz Spaces of Rafeiro and Samko

The real-variable theory of function spaces has always been at the core of harmonic analysis. In particular, the real-variable theory of the Hardy space is a fundamental tool of harmonic analysis, with applications and connections to complex analysis, partial differential equations, and functional analysis. This book is devoted to exploring properties of generalized Herz spaces and establishing a complete real-variable theory of Hardy spaces associated with local and global generalized Herz spaces via a totally fresh perspective. This means that the authors view these generalized Herz spaces as special cases of ball quasi-Banach function spaces. In this book, the authors first give some basic properties of generalized Herz spaces and obtain the boundedness and the compactness characterizations of commutators on them. Then the authors introduce the associated Herz–Hardy spaces, localized Herz–Hardy spaces, and weak Herz–Hardy spaces, and develop a complete real-variable theory of these Herz–Hardy spaces, including their various maximal function, atomic, molecular as well as various Littlewood–Paley function characterizations. As applications, the authors establish the boundedness of some important operators arising from harmonic analysis on these Herz–Hardy spaces. Finally, the inhomogeneous Herz–Hardy spaces and their complete real-variable theory are also investigated. With the fresh perspective and the improved conclusions on the real-variable theory of Hardy spaces associated with ball quasi-Banach function spaces, all the obtained results of this book are new and their related exponents are sharp. This book will be appealing to researchers and graduate students who are interested in function spaces and their applications. Preface 6 Abstract 16 Keywords and Phrases 16 Contents 18 1 Generalized Herz Spaces of Rafeiro and Samko 21 1.1 Matuszewska–Orlicz Indices 22 1.2 Generalized Herz Spaces 32 1.3 Convexities 76 1.4 Absolutely Continuous Quasi-Norms 78 1.5 Boundedness of Sublinear Operators 93 1.6 Fefferman–Stein Vector-Valued Inequalities 104 1.7 Dual and Associate Spaces of Local Generalized Herz Spaces 107 1.8 Extrapolation Theorems 120 2 Block Spaces and Their Applications 128 2.1 Block Spaces 128 2.2 Duality 136 2.3 Boundedness of Sublinear Operators 154 3 Boundedness and Compactness Characterizations of Commutators on Generalized Herz Spaces 165 3.1 Boundedness Characterizations 167 3.2 Compactness Characterizations 176 4 Generalized Herz–Hardy Spaces 186 4.1 Maximal Function Characterizations 188 4.2 Relations with Generalized Herz Spaces 198 4.3 Atomic Characterizations 206 4.4 Generalized Finite Atomic Herz–Hardy Spaces 233 4.5 Molecular Characterizations 241 4.6 Littlewood–Paley Function Characterizations 261 4.7 Dual Space of Hω,0p,q(Rn) 269 4.8 Boundedness of Calderón–Zygmund Operators 277 4.9 Fourier Transform 307 5 Localized Generalized Herz–Hardy Spaces 318 5.1 Maximal Function Characterizations 320 5.2 Relations with Generalized Herz–Hardy Spaces 331 5.3 Atomic Characterizations 340 5.4 Molecular Characterizations 355 5.5 Littlewood–Paley Function Characterizations 371 5.6 Boundedness of Pseudo-Differential Operators 397 6 Weak Generalized Herz–Hardy Spaces 415 6.1 Maximal Function Characterizations 418 6.2 Relations with Weak Generalized Herz Spaces 428 6.3 Atomic Characterizations 432 6.4 Molecular Characterizations 454 6.5 Littlewood–Paley Function Characterizations 466 6.6 Boundedness of Calderón–Zygmund Operators 475 6.7 Real Interpolations 506 7 Inhomogeneous Generalized Herz Spaces and Inhomogeneous Block Spaces 517 7.1 Inhomogeneous Generalized Herz Spaces 517 7.1.1 Convexities 528 7.1.2 Absolutely Continuous Quasi-Norms 530 7.1.3 Boundedness of Sublinear Operators and Fefferman–Stein Vector-Valued Inequalities 534 7.1.4 Dual and Associate Spaces of Inhomogeneous Local Generalized Herz Spaces 542 7.1.5 Extrapolation Theorems 545 7.2 Inhomogeneous Block Spaces and Their Applications 548 7.2.1 Inhomogeneous Block Spaces 549 7.2.2 Duality Between Inhomogeneous Block Spaces and Global Generalized Herz Spaces 552 7.2.3 Boundedness of Sublinear Operators 557 7.3 Boundedness and Compactness Characterizations of Commutators 560 7.3.1 Boundedness Characterizations 560 7.3.2 Compactness Characterizations 564 8 Hardy Spaces Associated with Inhomogeneous Generalized Herz Spaces 568 8.1 Inhomogeneous Generalized Herz–Hardy Spaces 568 8.1.1 Maximal Function Characterizations 569 8.1.2 Relations with Inhomogeneous Generalized Herz Spaces 573 8.1.3 Atomic Characterizations 574 8.1.4 Inhomogeneous Generalized Finite Atomic Herz–Hardy Spaces 582 8.1.5 Molecular Characterizations 585 8.1.6 Littlewood–Paley Function Characterizations 588 8.1.7 Dual Space of HKω,0p,q(Rn) 590 8.1.8 Boundedness of Calderón–Zygmund Operators 593 8.1.9 Fourier Transform 595 8.2 Inhomogeneous Localized Generalized Herz–Hardy Spaces 601 8.2.1 Maximal Function Characterizations 602 8.2.2 Relations with Inhomogeneous Generalized Herz–Hardy Spaces 606 8.2.3 Atomic Characterizations 607 8.2.4 Molecular Characterizations 610 8.2.5 Littlewood–Paley Function Characterizations 614 8.2.6 Boundedness of Pseudo-Differential Operators 616 8.3 Inhomogeneous Weak Generalized Herz–Hardy Spaces 617 8.3.1 Maximal Function Characterizations 618 8.3.2 Relations with Inhomogeneous Weak Generalized Herz Spaces 623 8.3.3 Atomic Characterizations 625 8.3.4 Molecular Characterizations 627 8.3.5 Littlewood–Paley Function Characterizations 631 8.3.6 Boundedness of Calderón–Zygmund Operators 634 8.3.7 Real Interpolations 643 Bibliography 645 Index 657