Hardy Classes, Integral Operators, and Duality on Spaces of Homogeneous Type

Function spaces play a significant role in harmonic analysis and partial differential equations. The integral operators that form a bridge between function spaces and partial differential equations are the Calderón-Zygmund operators. It is well known that Calderón-Zygmund operators are bounded on the Lebesgue space L(R) for 1 < p < ∞. It is also known that the Calderón-Zygmund operators are not bounded on L(R) for any 0 < p ≤ 1. It is natural to ask what are the substitutes for L(R) when p is in this range; this circle of ideas has received considerable attention in harmonic analysis during the past thirty years (see, for example, [COI], [CHR2], [COW1, 2], [FS], [KRA2] and [STE], etc.). We now understand that the best substitutes for the L spaces are the atomic Hardy spaces. Of course the holomorphic Hardy spaces on a domain in C and the real variable harmonic Hardy spaces on R are, at least on a formal level, quite different. It is natural to wish to find a way to connect them. With this end in view, the abstract Hardy spaces on a space of homogeneous type have been introduced and studied by Coifman and Weiss [COW1, 2] and others. For the case 0 < p ≤ 1, the atomic Hardy spaces H(X) on a space of homogeneous type were introduced in [COI] and [COW1, 2]. With their definition of H(X), it cannot be guaranteed that the Calderón-Zygmund operators are bounded on H(X) when p < 1/2—even when X is the real line. One of the main purposes of this paper is to find a natural way to define Hardy spaces H(X) on a space of homogeneous type with 0 < p < 1 in such a way that singular integrals will be bounded on all of these Hardy spaces. This will extend the work of Coifman andWeiss in [COW1], [COW2]. In order to achieve this goal we shall have to address several important ancillary issues: how to define higher order moment