一类多（次）线性平方函数
Certain Multi(sub)linear square functions
摘要：
设$d\ge 1, \ell\in\Z^d$，$m\in \mathbb Z^+$和$θ_i$，$i=1,\dots,m $是固定的不同的非零实数.我们证明了当$2\le p_i<\infty$满足$1/p=1/p_1+\cdots+1/p_m$和$1\le p<\infty$时，Ratnakumar和Shrivastava \cite{RS1} Littlewood-Paley平方函数$$T(f_1,\dots , f_m)(x)=\Big(\sum\limits_{\ell\in\Z^d}|\int_{\mathbb{R}^d}f_1(x-θ_1 y)\cdots f_m(x-θ_m y)e^{2πi \ell \cdot y}K (y)dy|^2\Big)^{1/2} $$的$m$ -（子）线性版本从$L^{p_1}(\mathbb{R}^d) \times\cdots\times L^{p_m}(\mathbb{R}^d) $到$L^p(\mathbb{R}^d) $是有界的。我们的证明是基于修改的不等式古利耶夫和Nazirova \cite{GN}关于多线性卷积。
英文摘要：
Let $d\ge 1, \ell\in\Z^d$, $m\in \mathbb Z^+$ and $θ_i$, $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 $$T(f_1,\dots , f_m)(x)=\Big(\sum\limits_{\ell\in\Z^d}|\int_{\mathbb{R}^d}f_1(x-θ_1 y)\cdots f_m(x-θ_m y)e^{2πi \ell \cdot y}K (y)dy|^2\Big)^{1/2} $$ 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.
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