The authors state the principal purpose of this paper as follows: “We shall examine some of the properties of \(H^p\) for \(0<p\leq 1\) 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.” The authors give a brief history of the major points in \(H^p\) and \(Re\; H^p\), \(0<p\leq\infty\), theory prior to the paper of D. L. Burkholder, R. F. Gundy and M. L. Silverstein [Trans. Amer. math. Soc. 157, 137-153 (1971; Zbl 0223.30048)]. The real beginning of the present exposition is the introduction of the spaces \(Re \; H^1\) with respect to the real variable characterization of \(Re\; H^1\) by Burkholder, Gundy and Silverstein in the above mentioned paper and the characterization of the dual \(\{ Re\; H^1 \}\)* of \(Re\; H^1\) as the class of functions of Bounded Mean Oscillation (BMO) given by Ch. Fefferman [Bull. Amer. math. Sec. 77, 587-588 (1971; Zbl 0229.46051)]. Certain Characterizations of \(Re\; H^1\) and then of \(Re \; H^p\) , \(0 < p \leq 1\), are obtained and consequences and extensions are derived from them. These characterizations of \(Re\; H^p\), \(0 < p \leq 1\), are in terms of \(p\)-atoms which are “building blocks” for elements in \(Re\; H^p\). The authors briefly describe the Hardy spaces associated with the real line and their atomic characterization and extend the discussion to \(n\)-dimensions. Now let \(X\) be a topological space endowed with a Borel measure \(\mu\), and a quasi-metric \(d\). Here \(d\) is a mapping
\[ d:\;X\times X\rightarrow \{ tz\in\mathbb{R}:\; t\geq 0 \} \]
satisfying (a) \(d(x,y) =d(y,x)\), (b) \(d(x,y) >0\) if and only if \(x\neq y\) and (c) there exists a constant \(K\) such that \(d(x,y) \leq K(d(x,z)+d(z,y))\) for all \(x,y,z\) in \(X\). lt is postulated that the spheres \(B_r (x) = \{y \in X:\; d(x,y) <r\}\) centered at \(x\) and of radius \(r>0\) form a basis of open neighborhoods of the point \(x\) and \(\mu (B_r (x)) >0\) whenever \(r>0\). It is assumed that there exists a constant \(A\) such that
\[ (B_r (x)) \leq A\mu (B_{r/2}(x)) . \]
The topological space \(X\) together with \(\mu\), and \(d\) satisfying the above assumptions is called a space of homogeneous type. Using the notion of atoms in this more general setting, the authors construct Hardy spaces \(H^p (X)\), \(0<p \leq 1\), associated with spaces of homogeneous type. Duality results for \(H^p (X)\), \(0< p \leq 1\), are obtained and interpolation theorems for operators acting on \(H^p (X)\) and \(L^q (X)\) are discussed. Proofs are given for the principal results concerning \(H^p (X)\).