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 $\theta \geq 0$. 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 $BM_{\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 $BM_{\theta }[ (p_{1}^{\prime };(p_{2}^{\prime };(p_{3}^{\prime }] $ and give examples of bilinear multipliers.