The present book offers an essential but accessible introduction to the discoveries first made in the 1990s that the doubling condition is superfluous for most results for function spaces and the boundedness of operators. It shows the methods behind these discoveries, their consequences and some of their applications. It also provides detailed and comprehensive arguments, many typical and easy-to-follow examples, and interesting unsolved problems.

The theory of the Hardy space is a fundamental tool for Fourier analysis, with applications for and connections to complex analysis, partial differential equations, functional analysis and geometrical analysis. It also extends to settings where the doubling condition of the underlying measures may fail.

The arguments for the main results are detailed and self-contained At least one typical and easily explicable example is given for each important notion further clarifying the relationship between the known and the present notions Detailed references for the content of each chapter are given. Also, well-known related results and some unsolved problems, which will be of interest to the reader, are presented, which might be interesting to the reader Includes supplementary material: sn.pub/extras

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.

Inhalt

Preliminaries.- Approximations of the Identity.- The Hardy Space H 1 ( ** ).- The Local Atomic Hardy Space h 1 ( ** ).- Boundedness of Operators over (R D , ** ).- Littlewood-Paley Operators and Maximal Operators Related to Approximations of the Identity.- The Hardy Space H 1 (, )and Its Dual Space RBMO (, ).- Boundedness of Operators over((, ).- Bibliography.- Index.- Abstract.