Bilinear Multipliers of Small Lebesgue spaces

TL;DR

This paper investigates the properties of bilinear multipliers on small Lebesgue spaces over locally compact abelian groups, providing foundational results and examples for these operators.

Contribution

It introduces and analyzes the space of bilinear multipliers on small Lebesgue spaces, establishing basic properties and providing explicit examples.

Findings

Characterization of bilinear multiplier spaces

Basic properties of these spaces

Examples of bilinear multipliers

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.