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Equivalence of two BV classes of functions in metric spaces, and existence of a Semmes family of curves under a 1-Poincaré inequality
Advances in Calculus of Variations ( IF 1.3 ) Pub Date : 2019-01-30 , DOI: 10.1515/acv-2018-0056
Estibalitz Durand-Cartagena 1 , Sylvester Eriksson-Bique 2 , Riikka Korte 3 , Nageswari Shanmugalingam 4
Affiliation  

We consider two notions of functions of bounded variation in complete metric measure spaces, one due to Martio and the other due to Miranda~Jr. We show that these two notions coincide, if the measure is doubling and supports a $1$-Poincar\'e inequality. In doing so, we also prove that if the measure is doubling and supports a $1$-Poincar\'e inequality, then the metric space supports a \emph{Semmes family of curves} structure.

中文翻译:

度量空间中两个 BV 类函数的等价性,以及 1-庞加莱不等式下 Semmes 曲线族的存在

我们考虑完全度量空间中的有界变差函数的两个概念,一个是由于 Martio,另一个是由于 Miranda~Jr。我们证明这两个概念是一致的,如果该度量加倍并且支持 $1$-Poincar\'e 不等式。这样做,我们还证明了如果度量加倍并支持 $1$-Poincar\'e 不等式,那么度量空间支持 \emph{Semmes 曲线族}结构。
更新日期:2019-01-30
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