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

By Luigi Ambrosio, Simone Di Marino. 2014

161 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/ef141e4d72333132da7135a667b2beab278e22cb

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows (2018, background): “We further introduce, following [5], the space of functions of bounded variation over .”
Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates (2018, methodology): “Initially this was done in [77] under the assumption that the measure on X is doubling and X is sufficiently well-connected by paths to support a 1-Poincaré inequality controlling f by |∇f |, but it was subsequently recognized that weaker assumptions still ensure the existence of a rich theory [9].”
The globalization theorem for the Curvature-Dimension condition (2016, background): “10 (1)), possibly using a cut-off Lipschitz function over Br(o), we may assume that φs has bounded total variation measure ‖Dφs‖ (we refer to [53] and [9] for all missing notions and background regarding BV-functions on metric-measure spaces).”
Nonsmooth differential geometry - An approach tailored for spaces with Ricci curvature bounded from below (2014, background): “We also remark that the space H1,1(M) coincides with what in the literature on metric measure spaces has often been calledW 1,1(M): see in particular [5] and [33].”
Rectifiability of the reduced boundary for sets of finite perimeter over $\RCD(K,N)$ spaces. (2019, background): “In [3] (see also [41] for the case of locally compact spaces) it is proven that this set function is the restriction to open sets of a finite Borel measure that we call total variation of f and still denote by |Df |.”
Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope (2012, background): “It is on one hand a survey paper on the most recent developments of the theory occurred in [3], [4] (see also [5] for analogous results in the space BV of functions of bounded variation), but it contains also new results on the reflexivity of W , 1 < q <∞, improving those of [7].”
A characterization of the convergence in variation for the generalized sampling series (2017, unknown): “Notations and preliminaries We will work in the frame of the space of functions of bounded variation on R (see [27, 4, 7, 28]), namely BV (R) := {f : R −→ R : V [f ] < +∞}.”
Independence on p of weak upper gradients on RCD spaces (2014, background): “Recalling the notation and results of [1] about BV functions and denoting by |Df | the total variation measure of f ∈ BV(X), assume for a moment that (X, d,m) is a proper (=bounded closed sets are compact) RCD(K,∞) space.”