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 ${\mathcal{G}}_{\lambda }^{\ast }(f)$ with $\lambda >2$. We establish the boundedness of Littlewood–Paley square operators on the weighted spaces $\mathrm{B}\mathrm{M}\mathrm{O}(\omega )$ with $\omega \in A_1$. The weighted space $\mathrm{B}\mathrm{L}\mathrm{O}(\omega )$ (the space of functions with bounded lower oscillation) is introduced and studied in this paper. This new space is a proper subspace of $\mathrm{B}\mathrm{M}\mathrm{O}(\omega )$. 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 $\mathrm{B}\mathrm{M}\mathrm{O}(\omega )$ into $\mathrm{B}\mathrm{L}\mathrm{O}(\omega )$, provided that $\omega \in A_1$. The corresponding John–Nirenberg inequality suitable for the space $\mathrm{B}\mathrm{L}\mathrm{O}(\omega )$ with $\omega \in A_1$ is discussed. Based on this, the equivalent characterization of the space $\mathrm{B}\mathrm{L}\mathrm{O}(\omega )$ is also given.