A compactness result for BV functions in metric spaces

Don, Sebastiano; Vittone, Davide

doi:10.5186/aasfm.2019.4415

Cited by 4 publications

(5 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This proves the existence of with the above properties. Since we know that for , the curves are contained in , arguing as in the proof of [ 17 , Lemma 3.3], for , we obtain for all , with and any and .…”

Section: Uniform Nilpotent Approximationmentioning

confidence: 93%

“…If , , , and , it is convenient to define as the absolutely continuous curves such that that almost everywhere on [0, T ] satisfy We divide the proof into several steps. The next step can be seen as “uniform version” of [ 17 , Lemma 3.2] with respect to , where another difference is that the vector fields are only defined on the open ball .…”

Section: Uniform Nilpotent Approximationmentioning

confidence: 99%

“…The previous theorem in some respects can be seen as a uniform version of [ 17 , Theorem 3.4] with respect to an “extra parameter” q varying in an abstract compact metric space. Another difference is that we consider vector fields which are only defined on an open bounded set.…”

Section: Uniform Nilpotent Approximationmentioning

confidence: 99%

See 2 more Smart Citations

Surface measure on, and the local geometry of, sub-Riemannian manifolds

Don,

Magnani

2023

Calc. Var.

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Uniform Nilpotent Approximationmentioning

confidence: 93%

Section: Uniform Nilpotent Approximationmentioning

confidence: 99%

See 1 more Smart Citation

Surface measure on, and the local geometry of, sub-Riemannian manifolds

Don,

Magnani

2023

Calc. Var.

Self Cite

View full text Add to dashboard Cite

show abstract

“…The proof of Theorem 2.43 given in [16] implicitly contains also the following result's proof, that we however provide for the sake of completeness. Proposition 2.44.…”

Section: 3mentioning

confidence: 99%

“…Theorem 1.6 is part of Theorem 3. 16. We also mention that, assuming property R, one can define a precise representative u p of u (see (45)) and prove that the convergence of the mean values ffl B(p,r) u dL n to u p (p) holds, as r → 0, for H Q−1 -almost every p. See Theorem 3.14.…”

mentioning

confidence: 99%

Fine properties of functions with bounded variation in Carnot-Carathéodory spaces

Don

Vittone

2019

Journal of Mathematical Analysis and Applications

Self Cite

View full text Add to dashboard Cite

We study properties of functions with bounded variation in Carnot-Carathéodory spaces. We prove their almost everywhere approximate differentiability and we examine their approximate discontinuity set and the decomposition of their distributional derivatives. Under an additional assumption on the space, called property R, we show that almost all approximate discontinuities are of jump type and we study a representation formula for the jump part of the derivative. IntroductionA lot of effort was devoted in the last decades to the development of Analysis and Geometry in general metric spaces and, in particular, to the study of functions with bounded variation (BV ). Carnot-Carathéodory (CC) spaces are among the most fruitful settings where BV functions have been introduced ([10, 20]), see also [8,12,19,22,23,24] and the more recent [3,5,6,9,11,15,31,35,44]. The aim of this paper is to give some contributions to this research lines by establishing "fine" properties of BV functions in CC spaces. A non-trivial part of our work consists in fixing the appropriate language in a consistent and robust manner.A CC space is the space R n endowed with the Carnot-Carathéodory distance d (see (1)) arising from a fixed family X = (X 1 , . . . , X m ) of smooth, linearly independent vector fields (called horizontal) in R n satisfying the Hörmander condition, see (2). As customary in the literature, we always assume that metric balls are bounded with respect to the Euclidean topology. Moreover, we work in equiregular CC spaces, where a homogeneous dimension Q, usually larger than the topological dimension n, can be defined; recall that any CC space can be lifted to an equiregular one, see e.g. [42].The space BV X of function with bounded X-variation consists of those functions u whose derivatives X 1 u, . . . , X m u in the sense of distributions are represented by a vector-valued measure D X u with finite total variation |D X u|. These functions have been extensively studied in the literature and important properties have been proved, like coarea formulae, approximation theorems, Poincaré inequalities.We now describe some of the results we prove in this paper. The first one, Theorem 1.1 below, concerns the almost everywhere approximate X-differentiability (see Section 2.3) of BV X functions; its classical counterpart is very well-known, see e.g. [2, Theorem 3.83]. As customary, we denote by D a X u and D s X u, respectively, the absolutely continuous and singular part of D X u with respect to the Lebesgue measure L n . (Italy) project "Campi vettoriali, superfici e perimetri in geometrie singolari". The second named author wishes to ackowledge the support and hospitality of FBK-CIRM (Trento), where part of this paper was written.

show abstract

Lipschitz Carnot-Carathéodory Structures and their Limits

2022

View full text Add to dashboard Cite

In this paper we discuss the convergence of distances associated to converging structures of Lipschitz vector fields and continuously varying norms on a smooth manifold. We prove that, under a mild controllability assumption on the limit vector-fields structure, the distances associated to equi-Lipschitz vector-fields structures that converge uniformly on compact subsets, and to norms that converge uniformly on compact subsets, converge locally uniformly to the limit Carnot-Carathéodory distance. In the case in which the limit distance is boundedly compact, we show that the convergence of the distances is uniform on compact sets. We show an example in which the limit distance is not boundedly compact and the convergence is not uniform on compact sets. We discuss several examples in which our convergence result can be applied. Among them, we prove a subFinsler Mitchell’s Theorem with continuously varying norms, and a general convergence result for Carnot-Carathéodory distances associated to subspaces and norms on the Lie algebra of a connected Lie group.

show abstract

A compactness result for BV functions in metric spaces

Cited by 4 publications

References 8 publications

Surface measure on, and the local geometry of, sub-Riemannian manifolds

Surface measure on, and the local geometry of, sub-Riemannian manifolds

Fine properties of functions with bounded variation in Carnot-Carathéodory spaces

Lipschitz Carnot-Carathéodory Structures and their Limits

Contact Info

Product

Resources

About