Let $\mu$ be a nonnegative Radon measure on ${ℝ}^d$ which only satisfies $\mu \left(B(x,r)\right)\le C_0{r}^n$ for all $x\in {ℝ}^d$, $r>0$, with some fixed constants $C_0>0$ and $n\in (0,d].$ In this paper, a new characterization for the space $\mathrm{RBMO}\left(\mu \right)$ of Tolsa in terms of the John-Strömberg sharp maximal function is established.

Given a doubling measure μ on Rd, it is a classical result of harmonic analysis that Calderón-Zygmund operators which are bounded in L2(μ) are also of weak type (1,1). Recently it has been shown that the same result holds if one substitutes the doubling condition on μ by a mild growth condition on μ. In this paper another proof of this result is given. The proof is very close in spirit to the classical argument for doubling measures and it is based on a new Calderón-Zygmund decomposition adapted...