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

Inhalt

Preliminaries.- Approximations of the Identity.- The Hardy Space H 1().- The Local Atomic Hardy Space h 1().- Boundedness of Operators over (RD, **).- Littlewood-Paley Operators and Maximal Operators Related to Approximations of the Identity.- The Hardy Space H1 (, )and Its Dual Space RBMO (, ).- Boundedness of Operators over((, ).- Bibliography.- Index.- Abstract.