On boundedness of discrete multilinear singular integral operators

TL;DR

This paper establishes a transference principle connecting the boundedness of certain bilinear Fourier multiplier operators on real line spaces with their discrete analogs on integer sequences, under specific Lebesgue space conditions.

Contribution

It provides a novel equivalence between the boundedness of continuous bilinear operators and their discrete counterparts, extending classical transference results to multilinear singular integrals.

Findings

Boundedness of the operator on real spaces is equivalent to boundedness on discrete sequences.

The result applies to a broad class of multipliers with measurable, locally bounded functions.

The paper characterizes the conditions under which these operators extend to bounded bilinear maps.

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 $t>0$