INHOMOGENEOUS 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 $C_0 \gt 0$ and $\theta \in (0,1]$ such that for all $x,\ x',\ y \in X$, $$ |\rho(x,y)-\rho(x',y)| \le C_0 \rho(x,x')^\theta [\rho(x,y) + \rho(x',y)]^{1-\theta}, $$ and $\mu$ is a nonnegative Borel regular measure on $X$ satisfying that for some $d \gt 0$, all $x \in X$ and all $0 \lt r \lt \operatorname{diam} X$, \[ \mu(\{ y \in X: \rho(x,y) \lt r \}) \sim r^{d}. \] In this paper, we first obtain the boundedness of Calderón-Zygmund operators on spaces of test functions; and using this, we then establish the continuous Calderón reproducing formulae associated with a given para-accretive function, which is a key tool for developing the theory of Besov and Triebel-Lizorkin spaces associated with para-accretive functions. By the Calderón reproducing formulae, we finally obtain a Littlewood-Paley theorem on the inhomogeneous $g$-function which gives a new characterization of Lebesgue spaces $L^{p}(X)$ for $p \in (1,\infty)$ and generalizers a corresponding result of David, Journé and Semmes.