New multilinear Littlewood–Paley gλ∗ function and commutator on weighted Lebesgue spaces

Abstract

Via the new weight function $A_{\overrightarrow{p}}^{\theta }\left(\phi \right)$, 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 previous studies. When $m=1$, that is, in the case of one linear, our conclusions are also new, further extending the results in previous work.