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

TL;DR

This paper proves the boundedness and existence of multilinear Littlewood--Paley operators with convolution kernels on products of BMO spaces, extending results to non-convolution kernels and showing boundedness into BLO spaces.

Contribution

It establishes the boundedness and almost everywhere finiteness of multilinear Littlewood--Paley operators on BMO spaces, including new results for non-convolution kernels.

Findings

Operators are bounded from BMO to BLO spaces.

Finiteness at a single point implies finiteness almost everywhere.

Results include non-convolution kernel cases.

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 $g^{\ast}_{\lambda}$-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 $\mathrm{BMO}(\mathbb R^n)\times\cdots\times \mathrm{BMO}(\mathbb R^n)$ 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.