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Extensions of Hardy spaces and their use in analysis. (English) Zbl 0358.30023
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 $$H^p (X)$$.
Reviewer: R. D. Carmichael

##### MSC:
 30D55 $$H^p$$-classes (MSC2000) 42A45 Multipliers in one variable harmonic analysis 42A50 Conjugate functions, conjugate series, singular integrals 43A85 Harmonic analysis on homogeneous spaces 32A07 Special domains in $${\mathbb C}^n$$ (Reinhardt, Hartogs, circular, tube) (MSC2010)
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