  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.
