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Conformal Geometry and Dynamics

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The space of functions of bounded variation on curves in metric measure spaces

Author: O. Martio
Journal: Conform. Geom. Dyn. 20 (2016), 81-96
MSC (2010): Primary 26A45; Secondary 26B30, 30L99
Published electronically: April 19, 2016
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Abstract: The approximation modulus, $ AM$-modulus for short, was defined in an earlier paper by the author. In this paper it is shown that an $ L^1(X)$-function $ u$ in a metric measure space $ (X,d,\nu )$ can be defined to be of bounded variation on $ AM$-a.e. curve in $ X$ without an approximation of $ u$ by Lipschitz or Newtonian functions. The essential variation of $ u$ on $ AM$ almost every curve is bounded by a sequence of non-negative Borel functions in $ L^1(X)$. The space of such functions is a Banach space and there is a Borel measure associated with $ u$. For $ X = \mathbf {R}^n$ this gives the classical space of BV functions.

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Additional Information

O. Martio
Affiliation: Department of Mathematics and Statistics, FI–00014 University of Helsinki, Finland

Keywords: Metric measure space, $AM$--modulus, functions of bounded variation
Received by editor(s): September 24, 2015
Received by editor(s) in revised form: February 10, 2016
Published electronically: April 19, 2016
Dedicated: Dedicated to Jan Malý on his $60^{th}$ birthday
Article copyright: © Copyright 2016 American Mathematical Society