Authors: ÖZNUR KULAK, AHMET TURAN GÜRKANLI

Abstract: Let $G$ be a compact abelian metric group with Haar measure $\lambda $ and $% \hat{G}$ its dual with Haar measure $\mu $. Assume that$~1 < p_{i}<\infty $, $% p_{i}^{\prime }=\frac{p_{i}}{p_{i}-1}$, $\left( i=1,2,3\right) $ and $\theta \geq 0$. Let $L^{(p_{i}^{\prime },\theta }\left( G\right) ,$ $\left( i=1,2,3\right) $ be small Lebesgue spaces. A bounded sequence $m\left( \xi ,\eta \right) $ defined on $\hat{G}\times \hat{G}$ is said to be a bilinear multiplier on $G$ of type $\left[ (p_{1}^{\prime };(p_{2}^{\prime };(p_{3}^{\prime }\right] _{\theta }$ if the bilinear operator $B_{m}$ associated with the symbol $m$ \begin{equation*} B_{m}\left( f,g\right) \left( x\right) =\sum\limits_{s\in \hat{G}% }\sum\limits_{t\in \hat{G}}\hat{f}\left( s\right) \hat{g}\left( t\right) m\left( s,t\right) \left\langle s+t,x\right\rangle \end{equation*}% defines a bounded bilinear operator from $L^{(p_{1}^{\prime },\theta }\left( G\right) \times L^{(p_{2}^{\prime },\theta }\left( G\right) $ into $% L^{(p_{3}^{\prime },\theta }\left( G\right) $. We denote by \ $BM_{\theta }% \left[ (p_{1}^{\prime };(p_{2}^{\prime };(p_{3}^{\prime }\right] $ the space of all bilinear multipliers of type $\left[ (p_{1}^{\prime };(p_{2}^{\prime };(p_{3}^{\prime }\right] _{\theta }$. In this paper, we discuss some basic properties of the space $BM_{\theta }\left[ (p_{1}^{\prime };(p_{2}^{\prime };(p_{3}^{\prime }\right] $ and give examples of bilinear multipliers.

Keywords: Bilinear multipliers, grand Lebesgue spaces, small Lebesgue spaces

Full Text: PDF