Boundedness of multilinear Littlewood--Paley operators with convolution type kernels on products of BMO spaces

In this paper, the authors establish the existence and boundedness of multilinear Littlewood--Paley operators on products of BMO spaces, including the multilinear g-function, multilinear Lusin's area integral and multilinear g^λ-function. The authors prove that if the above multilinear operators are finite for a single point, then they are finite almost everywhere. Moreover, it is shown that these multilinear operators are bounded from BMO( Rⁿ)×⋯× BMO( Rⁿ) into BLO( Rⁿ) (the space of functions with bounded lower oscillation), which is a proper subspace of BMO( Rⁿ) (the space of functions with bounded mean oscillation). The corresponding estimates for multilinear Littlewood--Paley operators with non-convolution type kernels are also discussed.