Some new function spaces and their applications to Harmonic analysis

Abstract

In this paper a family of spaces is introduced which seems well adapted for the study of a variety of questions related to harmonic analysis and its applications. These spaces are the “tent spaces.” They provide the natural setting for the study of such things as maximal functions (the relevant space here is T ∞ p ), and also square functions (where the space T 2 p is relevant). As such these spaces lead to unifications and simplifications of some basic techniques in harmonic analysis. Thus they are closely related to L p and Hardy spaces, important parts of whose theory become corollaries of the description of tent spaces. Also, as (“Proc. Conf. Harmonic Analysis, Cortona,” Lect. Notes in Math. Vol. 992, Springer-Verlag, Berlin/New York,1983), already indicated where these spaces first appeared explicitly, the tent spaces can be used to simplify some of the results related to the Cauchy integral on Lipschitz curves, and multilinear analysis. In retrospect one can recognize that various ideas important for tent spaces had been used, if only implicitly, for quite some time. Here one should mention Carleson's inequality, its simplifications and extensions, the theory of Hardy spaces, and atomic decompositions.