Certain Multi(sub)linear square functions

Let $d\ge 1, \ell\in\Z^d$, $m\in \mathbb Z^+$ 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 $$T(f_1,\dots , f_m)(x)=\Big(\sum\limits_{\ell\in\Z^d}|\int_{\mathbb{R}^d}f_1(x-\theta_1 y)\cdots f_m(x-\theta_m y)e^{2\pi i \ell \cdot y}K (y)dy|^2\Big)^{1/2}$$ 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 $2\le p_i<\infty$ satisfy $1/p=1/p_1+\cdots+1/p_m$ and $1\le p<\infty$. Our proof is based on a modification of an inequality of Guliyev and Nazirova \cite{GN} concerning multilinear convolutions.