New Multilinear Littlewood--Paley $g_{\lambda }^{\ast }$ Function and Commutator on weighted Lebesgue Spaces

Via the new weight function $A_{\vec{p}}^{\theta }(\phi )$, the authors introduce a new class of multilinear Littlewood–Paley $g_{\lambda }^{\ast }$ 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 $g_{\lambda }^{\ast }$ function and the new $BMO$ function on weighted Lebesgue spaces. The results in this article include the known results in \cite{XY2015,SXY2014}. When $m=1$, that is, in the case of one-linear, our conclusions are also new, further extending the results in \cite{S1961}.