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In this paper, the authors establish the existence and boundedness ofmultilinear Littlewood--Paley operators on products of BMO spaces, includingthe multilinear $g$-function, multilinear Lusin's area integral and multilinear$g^{\ast}_{\lambda}$-function. The authors prove that if the above multilinearoperators are finite for a single point, then they are finite almosteverywhere. Moreover, it is shown that these multilinear operators are boundedfrom $\mathrm{BMO}(\mathbb R^n)\times\cdots\times \mathrm{BMO}(\mathbb R^n)$into $\mathrm{BLO}(\mathbb R^n)$ (the space of functions with bounded loweroscillation), which is a proper subspace of $\mathrm{BMO}(\mathbb R^n)$ (thespace of functions with bounded mean oscillation). The corresponding estimatesfor multilinear Littlewood--Paley operators with non-convolution type kernelsare also discussed.

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