This article is the first part of two works by the same authors on the same topic. Let $(\mathcal{X},d,\mu )$ 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 $\mathcal{X}$ expressed by its equipped quasi-metric d, dyadic reference points and dyadic cubes.

中文翻译：

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同质 I 型空间上副产品的有界性

本文是同一作者关于同一主题的两部作品的第一部分。让$(\mathcal{X},d,\mu )$是 Coifman 和 Weiss 意义上的同质类型空间。在本文中，作者建立了一些基本估计和修正的非齐次离散 Calderón 再现公式，它们在本文第二部分的副产品有界证明中发挥了重要作用，也具有独立的兴趣。为了避免装备测度μ的反向倍增性质，作者充分利用了 μ 的几何性质。$\mathcal{X}$由其配备的准度量d、二元参考点和二元立方体表示。