Weighted BMO-BLO estimates for Littlewood--Paley square operators
Hua Wang
TL;DR
This work develops a weighted theory for Littlewood--Paley square operators by introducing the weighted BLO space and proving that Littlewood--Paley operators map weighted BMO into weighted BLO for $\omega\in A_1$, while preserving finiteness almost everywhere. It establishes weighted $L^p$-boundedness for the classical and generalized square operators and derives a BLO control for $\mathcal{G},\mathcal{S},\mathcal{G}^{*}_{\lambda}$ when acting on $\mathrm{BMO}(\omega)$, under precise conditions on $\lambda$ and the kernel. A weighted John--Nirenberg inequality for BLO(\omega) is proved, leading to the equivalence of the family $\mathrm{BLO}^{p}(\omega)$ with $\mathrm{BLO}(\omega)$ for $1<p<\infty$ and to an equivalent BLO characterization. The results generalize unweighted Littlewood--Paley theory to the weighted setting, providing new tools for harmonic analysis and PDEs in weighted spaces.
Abstract
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 $\mathcal{G}^{\ast}_λ(f)$ with $λ>2$. We establish the boundedness of Littlewood--Paley square operators on the weighted spaces $\mathrm{BMO}(ω)$ with $ω\in A_1$. The weighted space $\mathrm{BLO}(ω)$ (the space of functions with bounded lower oscillation) is introduced and studied in this paper. This new space is a proper subspace of $\mathrm{BMO}(ω)$. 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 $\mathrm{BMO}(ω)$ into $\mathrm{BLO}(ω)$, provided that $ω\in A_1$. The corresponding John--Nirenberg inequality suitable for the space $\mathrm{BLO}(ω)$ with $ω\in A_1$ is discussed. Based on this, the equivalent characterization of the space $\mathrm{BLO}(ω)$ is also given.