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.

Autorentext

Yinqin Li is a Ph.D. student of mathematics at Beijing Normal University, China and his advisor is Professor Dachun Yang. He received his B.S. from Beijing Normal University in 2022. His research interests now include the real-variable theory of function spaces and its applications in the boundedness of operators. Dachun Yang is a professor of mathematics at Beijing Normal University, China. He received his Ph.D. from Beijing Normal University in 1992 under the supervision of Shanzhen Lu. Since his Ph.D., real-variable theory about Herz-Hardy spaces has been one of Dachun Yang's research interests. His research interests now include real-variable theory of function spaces (associated with operators) on various underlying spaces including Euclidean spaces, metric measure spaces, and nonhomogeneous metric spaces, as well as their applications to the boundedness of (Riesz or singular integral) operators and multipliers. Dachun Yang and his co-authors have published 4 monographs and more than 400 journal articles. Long Huang is a postdoctoral researcher of mathematics at Guangzhou University, China. He received his Ph. D. from Beijing Normal University in 2021 under the supervision of Dachun Yang. His research interests now include the real-variable theory of function spaces and its applications in the boundedness of operators.