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Functions of bounded variation in Carnot-Carathéodory spaces

2019

We study properties of functions with bounded variation in Carnot-Carathéodory spaces. In Chapter 2 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. In Chapter 3 we prove a rank-one theorem à la G. Alberti for the derivatives of vector-valued maps with bounded variation in a class of Carnot groups that includes all Heisenberg groups H n with n ≥ 2. Some important tools for the proof are properties linking the horizontal derivatives of a real-valued function with bounded variation to its subgraph. In Chapter 4 we prove a compactness result for bounded sequences (u j) of functions with bounded variation in metric spaces (X, d j) where the space X is fixed, but the metric may vary with j. We also provide an application to Carnot-Carathéodory spaces. The results of Chapter 4 are fundamental for the proofs of some facts of Chapter 2. dimension Q = s i=1 i(n i −n i−1) and the metric measure space (R n , d, L n) (where L n denotes the n-dimensional Lebesgue measure) is locally Ahlfors Q-regular (see Theorem 1.2.4), i.e., for every compact set K ⊆ R n there exist C ≥ 1 and R > 0 such that 1 C r Q ≤ L n (B(p, r) ≤ Cr Q , R will be discussed into details later on, together with the definition of rectifiability. Some of the main results about fine properties of BV functions presented in Chapter 2 need some fine blow-up analysis about intrinsic regular hypersurfaces (see Section 1.5). Chapter 2 and Section 1.5 are mostly new and contained in the work of the author and his supervisor Davide Vittone [30]. Part of the analysis of singular points for BV X functions requires some blow-up technique together with the nilpotent approximation of a CC space. Chapter 4 contains a technical but fundamental lemma (contained in [29]) that ensure compactness of equi-The aim of Chapter 2 is to establish "fine" properties of BV functions in CC spaces. A first non-trivial part of this Chapter consists in fixing the appropriate language in a consistent and robust manner. Section 2.1 is therefore devoted to the introduction of approximate notions of continuity, jump point and differentiability point for generic L 1 loc maps in CC spaces. The notion of approximate continuity has been already worked out in the literature (see e.g. [48, Section 2.7]) by the extension of the Lebesgue Theorem The proof of Theorem 1 is based on Lemma 2.2.6, i.e., on a suitable extension to CC spaces of the inequalitŷ B(p,r) |u(q) − u(p)| |q − p| dL n (q) ≤ Cˆ1 0 |Du|(B(p, tr)) t n dt valid for a classical BV function u on R n. Lemma 2.2.6 answers an open problem stated in [8] and it is new even in Carnot groups. Theorem 1 was proved in the setting of Carnot groups in [8] together with the following result, which we also extend to our more general setting. We denote by H Q−1 the Hausdorff measure of dimension Q − 1 and by S u the set of points where a function u does not possess an approximate limit in the sense of Definition 2.1.1. However, for classical BV functions much stronger results than Theorems 1 and 3 are indeed known: some of them are proved in Section 2.2 for BV X functions under the additional assumption that the space (R n , X) satisfies the following condition. Definition 1 (Property R). Let (R n , X) be an equiregular CC space with homogeneous dimension Q. We say that (R n , X) satisfies the property R if, for every open set Ω ⊆ R n and every E ⊆ R n with locally finite X-perimeter in Ω, the essential boundary ∂ * E ∩ Ω of E in Ω is countably X-rectifiable, i.e., there exists a countable family {S i : i ∈ N} of C 1 X hypersurfaces such that H Q−1 (∂ * E ∩ Ω \ ∞ i=0 S i) = 0. D X u to any countably X-rectifiable set R. D s G u |D s G u| (x) has rank one for |D s G u|-a.e. x ∈ Ω. It is worth pointing out that Theorem 7 applies to the n-th Heisenberg group H n , provided n ≥ 2. Heisenberg groups are defined in Example 1.3.24 and they represent some of the most simple non-trivial examples of Carnot groups. Notice also that to the rectifiable set ∂ * E u coincides H Q-almost everywhere with the measure-theoretic horizontal normal to E u. As already pointed out, by Theorem 8, property w-R is weaker than property R but we conjecture they are actually equivalent. Property w-R is a non-trivial technical obstruction one has to face when following the strategy of [67]: the rectifiability of sets with finite G-perimeter in Carnot groups is indeed a major open problem, which has been solved only in step 2 Carnot groups (see [38, 39]) and H Q−2 (Σ) might be either 0 or +∞ (even locally) as shown by A. Kozhevnikov [53]. See also the recent paper [63]. The fact that Theorem 9 does not apply to H 1 (actually, to H 1 × R × R, see the proof of Lemma 3.2.7) prevents us from proving the Rank-One Theorem for G = H 1. This does not follow from [26] either (see Remark 3.4.7) and, thus, it remains a very interesting open problem. BV X j functions with constant independent of j; these two results (Theorems 4.2.4 and 4.2.5, respectively) use in a crucial way some outcomes of the papers [18, 73]. As it is clear by the techniques used in Chapter 2, in the study of fine properties of BV X functions in CC spaces, and in particular of their local properties, one often needs to perform a blow-up procedure around a fixed point p: as explained in Theorem 1.4.5, this produces a sequence of CC metric spaces (R n , X j) that converges to (a quotient of) a Carnot group structure G. In this blow-up, the original BV X function u 0 gives rise to a sequence (u j) of functions in BV X j which, up to subsequences, will converge

Geometry of Carnot-Carathéodory Spaces, Differentiability, Coarea and Area Formulas

Analysis and Mathematical Physics, 2009

We compare geometries of two different local Lie groups in a Carnot-Carathéodory space, and obtain quantitative estimates of their difference. This result is extended to Carnot-Carathéodory spaces with C 1,α -smooth basis vector fields, α ∈ [0, 1], and the dependence of the estimates on α is established. From here we obtain the similar estimates for comparing geometries of a Carnot-Carathéodory space and a local Lie group. These results base on Gromov's Theorem on nilpotentization of vector fields for which we give new and simple proof. All the above imply basic results of the theory: Gromov type Local Approximation Theorems, and for α > 0 Rashevskiǐ-Chow Theorem and Ball-Box Theorem, etc. We apply the obtained results for proving hc-differentiability of mappings of Carnot-Carathéodory spaces with continuous horizontal derivatives. The latter is used in proving the coarea formula for smooth contact mappings of Carnot-Carathéodory spaces, and the area formula for Lipschitz (with respect to sub-Riemannian metrics) mappings of Carnot-Carathéodory spaces.

Metric Geometry of Nonregular Weighted Carnot–Carathéodory Spaces

Journal of Dynamical and Control Systems, 2014

We investigate local and metric geometry of weighted Carnot-Carathéodory spaces which are a wide generalization of sub-Riemannian manifolds and arise in nonlinear control theory, subelliptic equations etc. For such spaces the intrinsic Carnot-Carathéodory metric might not exist, and some other new effects take place. We describe the local algebraic structure of such a space, endowed with a certain quasimetric (first introduced by A. Nagel, E.M. Stein and S. Wainger), and compare local geometries of the initial CC space and its tangent cone at some fixed (possibly nonregular) point. The main results of the present paper are new even for the case of sub-Riemannian manifolds. Moreover, they yield new proofs of such classical results as the Local approximation theorem and the Tangent cone theorem, proved for Hörmander vector fields by M. Gromov, A.Bellaiche, J.Mitchell etc.

Geometry of Carnot-Carath'eodory Spaces, Differentiability and Coarea Formula

2009

We give a simple proof of Gromov's Theorem on nilpotentization of vector fields, and exhibit a new method for obtaining quantitative estimates of comparing geometries of two different local Carnot groups in Carnot-Carathéodory spaces with C 1,α-smooth basis vector fields, α ∈ [0, 1]. From here we obtain the similar estimates for comparing geometries of a Carnot-Carathéodory space and a local Carnot group. These two theorems imply basic results of the theory: Gromov type Local Approximation Theorems, and for α > 0 Rashevskiǐ-Chow Theorem and Ball-Box Theorem, etc. We apply the obtained results for proving hc-differentiability of mappings of Carnot-Carathéodory spaces with continuous horizontal derivatives. The latter is used in proving the coarea formula for some classes of contact mappings of Carnot-Carathéodory spaces.

Geometric structures on loop and path spaces

Proceedings - Mathematical Sciences, 2010

Is is known that the loop space associated to a Riemannian manifold admits a quasi-symplectic structure. This article shows that this structure is not likely to recover the underlying Riemannian metric by proving a result that is a strong indication of the "almost" independence of the quasi-symplectic structure with respect to the metric. Finally conditions to have contact structures on these spaces are studied.

Theorems of Barth-Lefschetz type and Morse theory on the space of paths in homogeneous spaces

Geometriae Dedicata, 2015

Homotopy connectedness theorems for complex submanifolds of homogeneous spaces (sometimes referred to as theorems of Barth-Lefshetz type) have been established by a number of authors. Morse Theory on the space of paths lead to an elegant proof of homotopy connectedness theorems for complex submanifolds of Hermitian symmetric spaces. In this work we extend this proof to a larger class of compact complex homogeneous spaces.

Geometric and Harmonic Analysis on Homogeneous Spaces

2013

Lobna Abdelmoula : The Selberg-Weil-Kobayashi local rigidity Theorem for exponential Lie groups. A local rigidity Theorem proved by A. Selberg and A. Weil for Riemannian symmetric spaces and generalized by T. Kobayashi for a non-Riemannian homogeneous space G/H, asserts that there are no continuous deformations of a cocompact discontinuous subgroup Γ for G/H in the setting of a linear non-compact semi-simple Lie group G except some few cases : G is not locally isomorphic to SL2(R) for H compact or G is not locally isomorphic to SO(n, 1) or SU(n, 1) for G × G and H = ∆G. When in large contrast G is assumed to be exponential solvable and H ⊂ G a maximal subgroup, we prove an analogue of such a Theorem stating that the local rigidity holds on the parameter space R(Γ, G,H) if and only if G is isomorphic to the two-dimensional group of a ne transformations of the line ax+ b. Remarkably, we do also drop the assumption on Γ to be uniform for G/H. This is a joint work with Ali Baklouti and ...