New Multilinear Littlewood--Paley gλ* Function and Commutator on weighted Lebesgue Spaces with New Weight

Via the new weight function Aₚθ(φ ), the authors introduce a new class of multilinear Littlewood--Paley gλ* functions and establish the boundedness on weighted Lebesgue spaces. In addition, the authors obtain the boundedness of the multilinear commutator and multilinear iterated commutator generated by the multilinear Littlewood--Paley gλ* function and the new $BMO$ function on weighted Lebesgue spaces. The results in this article include the known results in {XY2015,SXY2014}. When $m=1$, that is, in the case of one-linear, our conclusions are also new, further extending the results in {S1961}.