Estimates for Littlewood–Paley operators in BMO(Rn) ✩

Let g( f ) be the Littlewood–Paley g-function of f on Rn. In this paper, the authors prove that if f ∈ BMO(Rn) (the space of functions with bounded mean oscillation), then g( f ) is either infinite everywhere or finite almost everywhere, and in the latter case, [g( f )]2 is bounded from BMO(Rn) into BLO(Rn) (the space of functions with bounded lower oscillation), which is a proper subspace of BMO(Rn). Moreover, the authors also establish similar results for the Lusin-area function and the Littlewood–Paley g∗λ-function.