Certain Multi(sub)linear square functions

Abstract

Let $d\ge 1,\ell \in {\mathbb{Z}}^d$, $m\in {\mathbb{Z}}^{+}$ 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 $$T(f_1,\dots ,f_m)(x)=(\sum _{\ell \in {\mathbb{Z}}^d}|{\int }_{{\mathbb{R}}^d}{f}_1(x-{\theta }_{1}y)\cdots f_m(x-{\theta }_{m}y)e^{2\pi i\ell \cdot y}K(y)dy{|}^2{)}^{1/2}$$ 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.

10 pages