Abstract:

In this paper, the authors investigate some multilinear square operator with generalized kernel. They prove that the multilinear square operator $T$ is bounded from $(L^{p_1}(\omega_1)\times\cdots\times L^{p_m}(\omega_m))$ into $L^{p}(\nu_{\omega})$, where $\frac{1}{p_1}+ \cdots+\frac{1}{p_m}=\frac{1}{p}, \nu_{\omega}=\prod\limits_{i=1}^m\omega_i^{\frac{p_i}{p}} $, the authors proved the commutator $T_{\sum b}$, generalized by multilinear square operator $T$ and BMO function, is also bounded from$(L^{p_1}(\omega_1)\times\cdots\times L^{p_m}(\omega_m))$ into $L^{p}(\nu_{\omega})$上. Finally, the authors also prove the multilinear square operator $T$ is bounded from $L^\infty\times\cdots\times L^{\infty}$ into $BMO$. Some known results are improved.

Key words: Generalized integral kernel, Multilinear square operator, Sharp maximal function, Commutator, Weight