Estimates for Littlewood–Paley operators in [formula omitted

Abstract

Let g ( f ) be the Littlewood–Paley g-function of f on R n . In this paper, the authors prove that if f ∈ BMO ( R n ) (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 ( R n ) into BLO ( R n ) (the space of functions with bounded lower oscillation), which is a proper subspace of BMO ( R n ) . Moreover, the authors also establish similar results for the Lusin-area function and the Littlewood–Paley g λ * -function.