论文标题
小勒布斯格空间的双线性乘数
Bilinear Multipliers of Small Lebesgue spaces
论文作者
论文摘要
让$ g $是一个本地紧凑的Abelian度量集团，其中HAAR度量$λ$和$ \ hat {g} $与HAAR量度$μ，$和$λ（g）$是有限的。假设$ 〜1 <p_ {i} <\ infty $，$ p_ {i}^{\ prime} = \ frac {p_ {i}} {p_ {i} -1} $，$（i = 1,2,3）$和$θ\ geq 0 $。令$ l^{（p_ {i}^{\ prime}，θ}（g），$ $（i = 1,2,3）$是小的lebesgue空间。一个有界的可测量函数$ m（ξ，η）$在$ \ hat {g} （p_ {1}^{\ prime};（p_ {2}^{\ prime};（p_ {3}^{\ prime}] _ {θ} $，如果biinear operator $ b_ {m {m {m = \ sum_ {s \ in \ hat {g}} \ sum_ {t \ in \ hat {g}} \ hat {f}（s）\ hat {g}（t）m（s，s，t） $ l^{（p_ {1}^{\ prime}，θ}（g）\ times l^{（p_ {2}^{\ prime}，θ}}（g）$ in $ l^{（p_ {p_ {3}^{3}^{\ prime}，θ}，θ}，θ}（g）$。 （p_ {1}^{\ prime};（p_ {2}^{\ prime};（p_ {3}^{\ prime}] $类型$ [（p_}^{1}^{\ prime}^{\ prime};（p_};（p_ {2}} {\ prime} { _ {θ} $。
Let $G$ be a locally compact abelian metric group with Haar measure $λ$ and $\hat{G}$ its dual with Haar measure $μ,$ and $λ( G) $ is finite. Assume that$~1<p_{i}<\infty $, $p_{i}^{\prime }=\frac{ p_{i}}{p_{i}-1}$, $( i=1,2,3) $ and $θ\geq 0$. Let $ L^{(p_{i}^{\prime },θ}( G) ,$ $( i=1,2,3) $ be small Lebesgue spaces. A bounded measurable function $m( ξ,η) $ defined on $\hat{G}\times \hat{G}$ is said to be a bilinear multiplier on $G$ of type $[ (p_{1}^{\prime };(p_{2}^{\prime };(p_{3}^{\prime }] _{θ}$ 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 },θ}( G) \times L^{(p_{2}^{\prime },θ}( G) $ into $ L^{(p_{3}^{\prime },θ}(G) $. We denote by $BM_{θ} [ (p_{1}^{\prime };(p_{2}^{\prime };(p_{3}^{\prime }] $ the space of all bilinear multipliers of type $[ (p_{1}^{\prime };(p_{2}^{\prime };(p_{3}^{\prime }] _{θ}$. In this paper, we discuss some basic properties of the space $BM_{θ}[ (p_{1}^{\prime };(p_{2}^{\prime };(p_{3}^{\prime }] $ and give examples of bilinear multipliers.
