Some new function spaces and their applications to harmonic analysis. (Q1061337)

This paper is devoted to the definition of a new family of function spaces and to the investigation of their fundamental properties. These spaces, called ``tent spaces'' are of functions on \(X\times {\mathbb{R}}_+\) where X is a Euclidean space and the spaces are so defined that the functions have ``good'' boundary values on the boundary \(X\) of this space. Such boundary values play a central role in harmonic analysis and the theory developed in this paper systemises a great deal of the earlier work. It is so rich in material that it is hardly possible in a short review to summarize the results in detail. To show the range of these methods the authors give a number of applications at the close of this paper, to maximal functions, to the Hilbert transform and to the theory of Hardy spaces.