Certain Multi(sub)linear square functions

Let $d\ge 1,\mathrm{\ell }\in {\mathbb{Z}}^d$, $m\in {\mathbb{Z}}^{+}$ and ${\theta }_i$, $i=1,\dots ,m$ are fixed, distinct and nonzero real numbers. We show that the $m$-(sub)linear version below of the Ratnakumar and Shrivastava \cite{RS1} Littlewood-Paley square function $$T(f_1,\dots ,f_m)(x)=(\sum _{\mathrm{\ell }\in {\mathbb{Z}}^d}\mathrm{\mid }{\int }_{{\mathbb{R}}^d}{f}_1(x-{\theta }_{1}y)\cdots f_m(x-{\theta }_{m}y)e^{2\pi i\mathrm{\ell }\cdot y}K(y)dy{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$$ is bounded from $L^{p_1}({\mathbb{R}}^d)\times \cdots \times L^{p_m}({\mathbb{R}}^d)$ to $L^p({\mathbb{R}}^d)$ when $2\le p_i<\mathrm{\infty }$ satisfy $1\mathrm{/}p=1\mathrm{/}{p}_1+\cdots +1\mathrm{/}{p}_m$ and $1\le p<\mathrm{\infty }$. Our proof is based on a modification of an inequality of Guliyev and Nazirova \cite{GN} concerning multilinear convolutions.