$h^1$, bmo, blo and Littlewood-Paley $g$-functions with non-doubling measures

Abstract

Let $\mu$ be a nonnegative Radon measure on ${\mathbb{R}}^d$ which satisfies the growth condition that there exist constants $C_0>0$ and $n\in (0,d]$ such that for all $x\in {\mathbb{R}}^d$ and $r>0$, $\mu (B(x,r))\le C_0{r}^n$, where $B(x,r)$ is the open ball centered at $x$ and having radius $r$. In this paper, we introduce a local atomic Hardy space $h_{\mathrm{atb}}^{1,\infty }(\mu )$, a local BMO-type space $\mathrm{rbmo}(\mu )$ and a local BLO-type space $\mathrm{rblo}(\mu )$ in the spirit of Goldberg and establish some useful characterizations for these spaces. Especially, we prove that the space $\mathrm{rbmo}(\mu )$ satisfies a John-Nirenberg inequality and its predual is $h_{\mathrm{atb}}^{1,\infty }(\mu )$. We also establish some useful properties of $\mathrm{RBLO}(\mu )$ and improve the known characterization theorems of $\mathrm{RBLO}(\mu )$ in terms of the natural maximal function by removing the assumption on the regularity condition. Moreover, the relations of these local spaces with known corresponding function spaces are also presented. As applications, we prove that the inhomogeneous Littlewood-Paley $g$-function $g(f)$ of Tolsa is bounded from $h_{\mathrm{atb}}^{1,\infty }(\mu )$ to $L^1(\mu )$, and that $[g(f){]}^2$ is bounded from $\mathrm{rbmo}(\mu )$ to $\mathrm{rblo}(\mu )$.