Calderón reproducing formulae on product spaces of homogeneous type and their applications

Let $(X_1,d_1,{\mu }_1)$ and $(X_2,d_2,{\mu }_2)$ be two spaces of homogeneous type in the sense of R. R. Coifman and G. Weiss. In this article, the authors first introduce spaces of product test functions and product approximations of the identity with exponential decay on the product space $X_1\times X_2$. Using these, the authors establish product continuous/discrete Calderón reproducing formulae. As applications, the Littlewood–Paley characterizations, respectively, in terms of the Lusin area function, the Littlewood–Paley $g$-function, and the Littlewood–Paley $g_{\lambda }^{\ast }$-function, of the Lebesgue space $L^p(X_1\times X_2)$ with any given $p\in (1,\infty )$ are also given. Besides, the authors also obtain the boundedness of Calderón–Zygmund operators on product Lebesgue spaces. The novelty of this article is that all the results circumvent the reverse doubling condition of ${\mu }_1$ and ${\mu }_2$, $d_1$ and $d_2$ are only assumed to be quasi-metrics, and these results lay a foundation for the further development of the real-variable theory of function spaces on product spaces of homogeneous type.