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

TL;DR

This book develops a comprehensive real-variable theory of Hardy spaces linked to generalized Herz spaces, introducing new characterizations and operator boundedness results, with sharp exponents and broad applications in harmonic analysis.

Contribution

The authors present the first complete real-variable theory for Hardy spaces associated with generalized Herz spaces, including novel characterizations and operator boundedness results.

Findings

Established boundedness of key harmonic analysis operators on Herz-Hardy spaces.

Developed atomic, molecular, and Littlewood-Paley characterizations for these spaces.

Proved sharpness of exponents in all main results.

Abstract

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 which means that the authors view these generalized Herz spaces as special cases of ball quasi-Banach function spaces. To be precise, in this book, the authors first study some basic properties of generalized Herz spaces and obtain boundedness and compactness characterizations of commutators on them. Then the authors introduce the associated Herz-Hardy spaces, local 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, finite 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. Due to the deficiency of the associate space of the global Herz space, the known real-variable characterizations about Hardy-type spaces associated with ball quasi-Banach function spaces are not applicable to Hardy spaces associated with global generalized Herz spaces which need an improved generalization of the existing one, done by the authors also in this book and having more additional anticipating applications. The authors should also point out that, with the fresh perspective and the improved conclusions on the real-variable theory of Hardy spaces associated with ball quasi-Banach function spaces, the exponents in all the obtained results of this book are sharp. Moreover, all of these results in this book are new and have never been published before.