Boundedness of paraproducts on spaces of homogeneous type I

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This article is the first part of two works by the same authors on the same topic. Let be a space of homogeneous type in the sense of Coifman and Weiss. In this article, the authors establish some basic estimates and a modified inhomogeneous discrete Calderón reproducing formula, which play essential roles in the proof of the boundedness of paraproducts in the second part of this work, and are also of independent interest. To escape the reverse doubling property of the equipped measure μ, the authors make full use of the geometrical properties of expressed by its equipped quasi-metric d, dyadic reference points and dyadic cubes.