Extensions of Hardy spaces and their use in analysis

Introduction. It is well known that the theory of functions plays an important role in the classical theory of Fourier series. Because of this certain function spaces, the H p spaces, have been studied extensively in harmonic analysis. When p > 1, If and H p are essentially the same; however, when p < 1 the space H p {% much better adapted to problems arising in the theory of Fourier series. We shall examine some of the properties of H p for p < 1 and describe ways in which these spaces have been characterized recently. These characterizations enable us to extend their definition to a very general setting that will allow us to unify the study of many extensions of classical harmonic analysis.