Abstract
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-Poincaré inequality. In doing so, we also prove that if the measure is doubling and supports a 1-Poincaré inequality, then the metric space supports a Semmes family of curves structure.
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-1704215
Award Identifier / Grant number: DMS-1500440
Funding source: Suomen Akatemia
Award Identifier / Grant number: 308063
Funding statement: Sylvester Eriksson-Bique was partially supported by grant DMS-1704215 of the National Science Foundation, Riikka Korte was supported by Academy of Finland grant number 308063, Nageswari Shanmugalingam was partially supported by grant DMS-1500440 of the National Science Foundation, and Estibalitz Durand-Cartagena’s research is partially supported by the grants MTM2015-65825-P (MINECO of Spain) and 2018-MAT14 (ETSI Industriales, UNED).
Acknowledgements
Part of the research was done during the visit of the second and fourth authors to Aalto University and Linköping University, and during the visit of the third and fourth authors to Universidad Complutense de Madrid and UNED; the authors wish to thank these institutions for their kind hospitality. We also thank Olli Martio for many valuable discussions related to this subject and for sharing his early manuscript [11] and the anonymous referee for valuable comments that helped improve Example 3.5.
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