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 $1\le p_1,q_1,p_2,q_2<\infty$ 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 $${\mathcal{C}}_m(f,g)(x)={\int }_{\mathbb{R}}{\int }_{\mathbb{R}}\hat{f}(\xi )\hat{g}(\eta )m(\xi ,\eta )e^{2\pi ix(\xi +\eta )}d\xi d\eta$$ initially defined for integrable functions with compact Fourier support, extends to a bounded bilinear operator from $L^{p_1,q_1}(\mathbb{R})\times L^{p_2,q_2}(\mathbb{R})$ into $L^{p_3,q_3}(\mathbb{R})$ if and only if the family of operators $$ {\mathcal D}_{\widetilde{m}_{t,p}} (a,b)(n) =t^{\frac{1}{p}}\int_{-\12}^{\12}\int_{-\12}^{\12}P(ξ) Q(η) m(tξ,tη) e^{2πin(ξ+η)}dξdη$$ initially defined for finite sequences $a=(a_{k_1}{)}_{k_1\in \mathbb{Z}}$, $b=(b_{k_2}{)}_{k_2\in \mathbb{Z}}$, where $P(\xi )={\sum }_{k_1\in \mathbb{Z}}{a}_{k_1}{e}^{-2\pi i{k}_1\xi }$ and $Q(\eta )={\sum }_{k_2\in \mathbb{Z}}{b}_{k_2}{e}^{-2\pi i{k}_2\eta }$, extend to bounded bilinear operators from $l^{p_1,q_1}(\mathbb{Z})\times l^{p_2,q_2}(\mathbb{Z})$ into $l^{p_3,q_3}(\mathbb{Z})$ with norm bounded by uniform constant for all $t>0$

References in corpus (1)

• Modulation invariant bilinear T(1) theorem