Boundedness of Multilinear Littlewood–Paley Operators With Convolution Type Kernels on Products of BMO Spaces

Abstract

In this paper, the authors establish the existence and boundedness of multilinear Littlewood–Paley operators on the product of BMO spaces, including the multilinear $g$-function, multilinear Lusin's area integral and multilinear $g_{\lambda }^{*}$-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 $\mathrm{BMO}\left(R^n\right)\times \cdots \times \mathrm{BMO}\left(R^n\right)$ into $\mathrm{BLO}\left(R^n\right)$ (the space of functions with bounded lower oscillation), which is a proper subspace of $\mathrm{BMO}\left(R^n\right)$ (the space of functions with bounded mean oscillation). The corresponding estimates for multilinear Littlewood–Paley operators with non-convolution type kernels are also discussed.