发表机构
Ankara University; Institute of Mathematics of the Academy of Sciences of the Czech Republic; Chuo University（安卡拉大学; 捷克科学院数学研究所; 中央大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了齐次型空间上Banach格的Kolmogorov-Riesz紧性定理，用有界性、紧性和平均算子逼近刻画紧性，无需Vitali覆盖等假设，并推广至重排不变空间和Morrey空间。
AI 中文摘要
我们证明了齐次型空间上的Banach格的一个Kolmogorov-Riesz紧性定理。特别地，我们考虑了$L^1$的情形。我们的刻画用有界性、紧性（tightness）以及通过平均算子（averaging operators）的逼近来表述，这些平均算子在没有群结构时替代了平移。该结果既不要求Vitali覆盖性质，也不要求对球的测度的连续性假设、下体积界或底测度空间的有限性，从而推广了文献中若干最近的紧性准则。我们的方法始于对$L^1$中Kolmogorov-Riesz定理的一个新证明，该证明避免了自反性，并对$1\le p<\infty$的$L^p$-空间给出了统一的处理。然后我们建立了一个基于Christ二进立方体的逼近定理，并用它来刻画一般球Banach函数空间中紧支撑连续函数的闭包。应用包括重排不变Banach函数空间和Morrey空间的紧性准则。在欧几里得情形下，我们还将所得的逼近空间与经典热半群（等价地，磨光算子）闭包等同起来。
英文摘要
We prove a Kolmogorov--Riesz compactness theorem for Banach lattices over spaces of homogeneous type. Especially, we consider the one for $L^1$. Our characterization is formulated in terms of boundedness, tightness, and approximation by averaging operators, which replace translations in the absence of a group structure. The result requires neither the Vitali covering property nor continuity assumptions on the measure of balls, lower volume bounds, or finiteness of the underlying measure space, thereby extending several recent compactness criteria in the literature. Our approach begins with a new proof of the Kolmogorov--Riesz theorem in $L^1$, which avoids reflexivity and yields a unified treatment of $L^p$-spaces for $1\le p<\infty$. We then establish an approximation theorem based on Christ's dyadic cubes and use it to characterize the closure of compactly supported continuous functions in general ball Banach function spaces. Applications include compactness criteria for rearrangement-invariant Banach function spaces and Morrey spaces. In the Euclidean setting, we also identify the resulting approximation space with the classical heat-semigroup (equivalently, mollifier) closure.
Comments21 pages
