Inhomogeneous discrete Calderón reproducing formulae associated to para-accretive functions on metric measure spaces

Abstract

Let $(X,\rho,\mu)_{d,\theta}$ be a space of homogeneous type which includes metric measure spaces and some fractals, namely, $X$ is a set, $\rho$ is a quasi-metric on $X$ satisfying that there exist constants $C_0> 0$ and $\theta \in (0, 1]$ such that for all $x, x^{\prime}, y \in X$, $$|\rho(x, y) - \rho(x^{\prime}, y)| \leqslant C_{0}\rho(x, x^{\prime})^\theta[\rho(x, y) + \rho(x^{\prime}, y)]^{1-\theta},$$ and $\mu$ is a nonnegative Borel regular measure on $X$ satisfying that for some $d > 0$, all $x \in X$ and all $0 \lt r \lt \mathrm{diam} \ X$, $$\mu(\{y \in X \ : \ \rho(x,y) \lt r\}) \sim r^{d}.$$ In this paper, the authors establish the inhomogeneous discrete Calderón reproducing formulae on spaces of homogeneous type associated to a given special para-accretive function introduced by G. David, which will pave the way for developing the theory of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type associated to a given special para-accretive function.