论文标题
整体操作员的紧凑性和测量空间上的统一集成性
Compactness of integral operators and uniform integrability on measure spaces
论文作者
论文摘要
令$（E，\ Mathcal E，μ）$为度量空间，$ g \ colon e \ times e \ to [0，\ infty] $是可测量的。此外，让$ \ MATHCAL f \！_ {ui} $表示所有$ q \ in \ Mathcal e^+$（$ e $上的可衡量的数值函数$ q \ ge 0 $）的集合，以便$ \ {g（x，x，\ cdot）q \ colon x \ in E \ in e $ \ c. f \！_ {co} $表示所有$ q \ in \ Mathcal e^+$的集合，使得映射$ f \ mapsto g（fq）：= \ int g（\ cdot，y）f（y）q（y）q（y）q（y）\，d -μ（y）\，d -（y）$ compactor compactor compactor compations $ c $ \ nituct $ e \ n y mathcal e y $ e y $ e y $ e y $ e _ $ e__ b $ e__ b $ e__ b $ e__ b $（ sup-norm）。 显示出$ \ Mathcal f \！_ {ui} = \ Mathcal f \！_ {Co} $提供了$ \ Mathcal f \！_ {ui} $和$ \ Mathcal f \！_ {CO} $都具有严格的正函数。
Let $(E,\mathcal E,μ)$ be a measure space and $G\colon E\times E\to [0,\infty]$ be measurable. Moreover, let $\mathcal F\!_{ui}$ denote the set of all $q\in\mathcal E^+$ (measurable numerical functions $q\ge 0$ on $E$) such that $\{G(x,\cdot)q\colon x\in E\}$ is uniformly integrable, and let $\mathcal F\!_{co}$ denote the set of all $q\in\mathcal E^+$ such that the mapping $f\mapsto G(fq) :=\int G(\cdot,y) f(y) q(y)\,dμ(y)$ is a compact operator on the space $\mathcal E_b$ of bounded measurable functions on $E$ (equipped with the sup-norm). It is shown that $\mathcal F\!_{ui}=\mathcal F\!_{co} $ provided both $\mathcal F\!_{ui}$ and $\mathcal F\!_{co} $contain strictly positive functions.
