On boundedness of discrete multilinear singular integral operators

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

A Homomorphism Theorem for Bilinear Multipliers

In this paper we prove an abstract homomorphism theorem for bilinear multipliers in the setting of locally compact Abelian (LCA) groups. We also provide some applications. In particular, we obtain a bilinear abstract version of K. de Leeuw's theorem for bilinear multipliers of strong and weak type. We also obtain necessary conditions on bilinear multipliers on non-compact LCA groups, yielding boundedness for the corresponding operators on products of rearrangement invariant spaces. Our investigations extend some existing results in Euclidean spaces to the framework of general LCA groups, and yield new boundedness results for bilinear multipliers in quasi Banach spaces

Multilinear Fourier multipliers on variable Lebesgue spaces

In this paper, we study properties of the bilinear multiplier space. We give a necessary condition for a continuous integrable function to be a bilinear multiplier on variable exponent Lebesgue spaces. And we prove the localization theorem of multipliers on variable exponent Lebesgue spaces. Moreover, we present a Mihlin-H\"ormander type theorem for multilinear Fourier multipliers on weighted variable Lebesgue spaces and give some applications.Comment: 16 page

Fourier multipliers in Banach function spaces with UMD concavifications

We prove various extensions of the Coifman-Rubio de Francia-Semmes multiplier theorem to operator-valued multipliers on Banach function spaces. Our results involve a new boundedness condition on sets of operators which we call $\ell^{r}(\ell^{s})$-boundedness, which implies $\mathcal{R}$-boundedness in many cases. The proofs are based on new Littlewood-Paley-Rubio de Francia-type estimates in Banach function spaces which were recently obtained by the authors

Bilinear littlewood-paley for circle and transference

In this paper we have obtained the boundedness of bilinear Littlewood-Paley operators on the circle group T by using appropriate transference techniques. In particular, bilinear analogue of Carleson's Littlewood-Paley result for all possible indices has been obtained. Also, we prove some bilinear analogues of de Leeuw's results concerning multipliers of Rn

Spectral multipliers on 22-step groups: topological versus homogeneous dimension

Let $G$ be a $2$-step stratified group of topological dimension $d$ and homogeneous dimension $Q$. Let $L$ be a homogeneous sub-Laplacian on $G$. By a theorem due to Christ and to Mauceri and Meda, an operator of the form $F(L)$ is of weak type $(1,1)$ and bounded on $L^p(G)$ for all $p \in (1,\infty)$ whenever the multiplier $F$ satisfies a scale-invariant smoothness condition of order $s > Q/2$. It is known that, for several $2$-step groups and sub-Laplacians, the threshold $Q/2$ in the smoothness condition is not sharp and in many cases it is possible to push it down to $d/2$. Here we show that, for all $2$-step groups and sub-Laplacians, the sharp threshold is strictly less than $Q/2$, but not less than $d/2$.Comment: 17 page