Suppose that (X, d, µ) is a space of homogeneous type, with upper dimension

Let X be an RD-space, i.e., a space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property. Assume that X has a "dimension" n. Hardy space of Coifman and Weiss coincide. Furthermore, it is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from H p (X ) to some quasi-Banach space B if and only if T maps all (p, q)-atoms when q ∈ (p, ∞)∩[1, ∞) or continuous (p, ∞)-atoms into uniformly bounded elements of B. Keywords: space of homogeneous type, Calderón reproducing formula, space of test function, maximal function, Hardy space, atom, Littlewood-Paley function, sublinear operator, quasiBanach space MSC(2000): 42B25, 42B30, 47B38, 47A30

Assume that (X, d, µ) is a space of homogeneous type in the sense of Coifman and Weiss. In this article, motivated by the breakthrough work of P. Auscher and T. Hytönen on orthonormal bases of regular wavelets on spaces of homogeneous type, the authors introduce a new kind of approximations of the identity with exponential decay (for short, exp-ATI). Via such an exp-ATI, motivated by another creative idea of Y. Han et al. to merge the aforementioned orthonormal bases of regular wavelets into the frame of the existed distributional theory on spaces of homogeneous type, the authors establish the homogeneous continuous/discrete Calderón reproducing formulae on (X, d, µ), as well as their inhomogeneous counterparts. The novelty of this article exists in that d is only assumed to be a quasi-metric and the underlying measure µ a doubling measure, not necessary to satisfy the reverse doubling condition. It is well known that Calderón reproducing formulae are the cornerstone to develop analysis and, especially, harmonic analysis on spaces of homogeneous type.2010 Mathematics Subject Classification. Primary 42C40; Secondary 42B20, 42B25, 30L99. Key words and phrases. space of homogeneous type, Calderón reproducing formula, approximation of the identity, wavelet, space of test functions, distribution.

The Fefferman-Stein vector-valued maximal function inequality is proved for spaces of homogeneous type. The approach taken here is based on the theory of vector-valued Calderón-Zygmund singular integral theory in this context, which is appropriately developed.

A multivariable version of the strong maximal function is introduced and a sharp distributional estimate for this operator in the spirit of the Jessen, Marcinkiewicz, and Zygmund theorem is obtained. Conditions that characterize the boundedness of this multivariable operator on products of weighted Lebesgue spaces equipped with multiple weights are obtained. Results for other multi(sub)linear maximal functions associated with bases of open sets are studied too. Bilinear interpolation results between distributional estimates, such as those satisfied by the multivariable strong maximal function, are also proved.