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.