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, 2[g(f)] 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 -function.