AI 中文总结
本文以体积函数的Cesàro均值给出度量测度空间上Maz'ya-Shaposhnikova极限存在的几何充要条件，并应用于无界Sierpiński垫片，同时构造反例说明Ahlfors正则性不蕴含该极限。
AI 中文摘要
我们给出了一个纯几何的充分必要条件，该条件以体积函数的某种Cesàro均值表述，用于保证在度量测度空间上（在温和的测度条件下）关于Gagliardo半范数的Maz'ya-Shaposhnikova泛函极限的存在性。作为应用，我们在无界Sierpiński垫片上建立了Maz'ya-Shaposhnikova收敛性。我们还通过构造一个具体例子表明，Ahlfors正则性并不蕴含该泛函极限的存在性。
英文摘要
We provide a purely geometric necessary and sufficient condition, in terms of a certain Cesàro mean of the volume function, for the existence of the Maz'ya-Shaposhnikova functional limit with respect to Gagliardo seminorms on metric measure spaces (under a mild measure condition). As an application, we establish the Maz'ya-Shaposhnikova convergence on the unbounded Sierpiński gasket. We also show that Ahlfors regularity does not imply the existence of this functional limit by constructing a concrete example.
Comments10 pages, 1 figures. Comments are welcome
