# Equivalent definitions of $BV$ space and of total variation on metric measure spaces

created by ambrosio on 20 Jun 2012
modified by dimarino on 16 Sep 2014

[BibTeX]

Published Paper

Inserted: 20 jun 2012
Last Updated: 16 sep 2014

Journal: J. Funct. Anal.
Volume: 266
Number: 7
Pages: 4150-4188
Year: 2014
In this paper we introduce a definition of $BV$ based on measure upper gradients and prove the equivalence of this definition, and the coincidence of the corresponding notions of total variation, with the definitions based on relaxation of L1 norm of the slope of Lipschitz functions or upper gradients. As in the previous work by the first author with Gigli and Savaré in the Sobolev case, the proof requires neither local compactness nor doubling and Poincaré