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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 Hardy Space H1 with Non-doubling Measures and Their Applications

Preliminaries.- Approximations of the Identity.- The Hardy Space H¹(mi).- The Local Atomic Hardy Space h¹(mi).- Boundedness of Operators over (R^D, mi).- 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.