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 products of BMO spaces, including the multilinear $g$-function, multilinear Lusin's area integral and multilinear $g_{\hat{I}}^{\ast }»$-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 $\mathrm{BMO}({\mathbb{R}}^n)\times \cdots \times \mathrm{BMO}({\mathbb{R}}^n)$ into $\mathrm{BLO}({\mathbb{R}}^n)$ (the space of functions with bounded lower oscillation), which is a proper subspace of $\mathrm{BMO}({\mathbb{R}}^n)$ (the space of functions with bounded mean oscillation). The corresponding estimates for multilinear Littlewood--Paley operators with non-convolution type kernels are also discussed.

36 pages