A Note on the Kolmogorov Condition of a Weak Type Operator

Abstract

本文主要的目的在予完整地探討弱型算子的Kolmogorov 條件，Kolmogorov條件在研究弱型算子上是相當重要的性質；而算子的強、弱型之性質在實分析學上具有廣泛應用的工具，因此對算子 Kolmogorov 條件之研究具有其實質意義。本文主要的結果是推廣弱型算子與滿足Kolmogorov條件的等價性，並指出其推廣時所限制因素的一些實例。

In this article we are mainly concerned with the Kolmogorov condition which is a very nice tool to study the type of an operator. It is proved that an operator T from m(Ω1) to (Ω2), is of o-finite, is of weak type (p,s), l≦ p ≦∞, 1≦s≦∞, if and only if T satisfies the Kolmogorov condition. It is also pointed out that the hypothesis "Ω2 is of a-finite", is not redundant.