Bilinear Multipliers of Small Lebesgue spaces

Abstract

Let $G$ be a locally compact abelian metric group with Haar measure $\lambda$ and $\hat{G}$ its dual with Haar measure $\mu ,$ and $\lambda (G)$ is finite. Assume that$1<p_i<\infty$, $p_i^{\prime }=\frac{p_i}{p_i-1}$, $(i=1,2,3)$ and $\theta \ge 0$. Let $L^{(p_i^{\prime },\theta }(G),$ $(i=1,2,3)$ be small Lebesgue spaces. A bounded measurable function $m(\xi ,\eta )$ defined on $\hat{G}\times \hat{G}$ is said to be a bilinear multiplier on $G$ of type $[(p_1^{\prime };(p_2^{\prime };(p_3^{\prime }{]}_{\theta }$ if the bilinear operator $B_m$ associated with the symbol $m$, \begin{equation} B_{m}(f,g) ( x) =\sum_{s\in \hat{G} }\sum_{t\in \hat{G}}\hat{f}(s) \hat{g}(t) m(s,t) \langle s+t,x\rangle \end{equation} defines a bounded bilinear operator from $L^{(p_1^{\prime },\theta }(G)\times L^{(p_2^{\prime },\theta }(G)$ into $L^{(p_3^{\prime },\theta }(G)$. We denote by $B{M}_{\theta }[(p_1^{\prime };(p_2^{\prime };(p_3^{\prime }]$ the space of all bilinear multipliers of type $[(p_1^{\prime };(p_2^{\prime };(p_3^{\prime }{]}_{\theta }$. In this paper, we discuss some basic properties of the space $B{M}_{\theta }[(p_1^{\prime };(p_2^{\prime };(p_3^{\prime }]$ and give examples of bilinear multipliers.

29 pages