On boundedness of discrete multilinear singular integral operators

Abstract

Let $m(\xi,\eta)$ be a measurable locally bounded function defined in $\mathbb R^2$. Let $1\leq p_1,q_1,p_2,q_2<\infty $ such that $p_i=1$ implies $q_i=\infty $. Let also $0<p_3,q_3<\infty $ and $1/p=1/p_1+1/p_2-1/p_3$. We prove the following transference result: the operator $$ {\mathcal C}_m(f,g)(x)=\int_{\bbbr} \int_{\bbbr} \hat f(\xi) \hat g(\eta) m(\xi,\eta) e^{2\pi i x(\xi +\eta)}d\xi d\eta $$ initially defined for integrable functions with compact Fourier support, extends to a bounded bilinear operator from $L^{p_1,q_1}(\bbbr)\times L^{p_2,q_2}(\bbbr)$ into $L^{p_3,q_3}(\bbbr)$ if and only if the family of operators $$ {\mathcal D}_{\widetilde{m}_{t,p}} (a,b)(n) =t^{\frac{1}{p}}\int_{-\12}^{\12}\int_{-\12}^{\12}P(\xi) Q(\eta) m(t\xi,t\eta) e^{2\pi in(\xi +\eta)}d\xi d\eta $$ initially defined for finite sequences $a=(a_{k_{1}})_{k_{1}\in \bbbz}$, $b=(b_{k_{2}})_{k_{2}\in \bbbz}$, where $P(\xi)=\sum_{k_{1}\in \bbbz}a_{k_{1}}e^{-2\pi i k_{1}\xi}$ and $Q(\eta)=\sum_{k_{2}\in \bbbz}b_{k_{2}}e^{-2\pi i k_{2}\eta}$, extend to bounded bilinear operators from $l^{p_1,q_1}(\bbbz)\times l^{p_2,q_2}(\bbbz)$ into $l^{p_3,q_3}(\bbbz)$ with norm bounded by uniform constant for all $t>0$