Weighted BMO-BLO estimates for Littlewood--Paley square operators

Let $T(f)$ denote the Littlewood--Paley square operators, including the Littlewood--Paley $\mathcal{G}$-function $\mathcal{G}(f)$, Lusin's area integral $\mathcal{S}(f)$ and Stein's function $\mathcal{G}^{\ast}_{\lambda}(f)$ with $\lambda>2$. We establish the boundedness of Littlewood--Paley square operators on the weighted spaces $\mathrm{BMO}(\omega)$ with $\omega\in A_1$. The weighted space $\mathrm{BLO}(\omega)$ (the space of functions with bounded lower oscillation) is introduced and studied in this paper. This new space is a proper subspace of $\mathrm{BMO}(\omega)$. It is proved that if $T(f)(x_0)$ is finite for a single point $x_0\in\mathbb R^n$, then $T(f)(x)$ is finite almost everywhere in $\mathbb R^n$. Moreover, it is shown that $T(f)$ is bounded from $\mathrm{BMO}(\omega)$ into $\mathrm{BLO}(\omega)$, provided that $\omega\in A_1$. The corresponding John--Nirenberg inequality suitable for the space $\mathrm{BLO}(\omega)$ with $\omega\in A_1$ is discussed. Based on this, the equivalent characterization of the space $\mathrm{BLO}(\omega)$ is also given.