  Via the new weight $A_{\vec p}^{\theta }(\varphi )$, the authors introduce a
new class of multilinear square operators. The boundedness on the weighted
Lebesgue space and the weighted Morrey space is obtained, respectively. Our
results include the known results of the standard multilinear square operator
and the weight $A_{\vec p}$. Moreover, the results in this article seem to be
new even for one-linear case.
