Weighted BMO-BLO estimates for Littlewood–Paley square operators

Let $T(f)$ denote the Littlewood–Paley square operators, including the Littlewood–Paley G-function G(f), Lusin’s area integral S(f) and Stein’s function G^λ(f) with λ >2. We establish the boundedness of Littlewood–Paley square operators on the weighted spaces BMO(ω ) with ω ∈ A₁. The weighted space 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 BMO(ω ). It is proved that if T(f)(x₀) is finite for a single point x₀∈ Rⁿ, then $T(f)(x)$ is finite almost everywhere in Rⁿ. Moreover, it is shown that $T(f)$ is bounded from BMO(ω ) into BLO(ω ), provided that ω ∈ A₁. The corresponding John–Nirenberg inequality suitable for the space BLO(ω ) with ω ∈ A₁ is discussed. Based on this, the equivalent characterization of the space BLO(ω ) is also given.