Sergey Vodopyanov - Academia.edu (original) (raw)

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Papers by Sergey Vodopyanov

Comptes Rendus Mathematique

Bulletin des Sciences Mathématiques, 2006

In the present paper we define quasimeromorphic mappings on homogeneous groups and study their pr... more In the present paper we define quasimeromorphic mappings on homogeneous groups and study their properties. We prove an analogue of results of L. Ahlfors, R. Nevanlinna and S. Rickman, concerning the value distribution for quasimeromorphic mappings on H-type Carnot groups for parabolic and hyperbolic type domains.

Russian Mathematical Surveys

We study the properties of the mappings on a Carnot group which induce, via the changeof-variable... more We study the properties of the mappings on a Carnot group which induce, via the changeof-variables rule, the isomorphisms of Sobolev spaces with the summability exponent different from the Hausdorff dimension of the group.

$Research paper thumbnail of Approximate Differentiability of Mappings of Carnot-Carath\'eodory Spaces$

We study the approximate differentiability of measurable mappings of Carnot--Carath\'eodory s... more We study the approximate differentiability of measurable mappings of Carnot--Carath\'eodory spaces. We show that the approximate differentiability almost everywhere is equivalent to the approximate differentiability along the basic horizontal vector fields almost everywhere. As a geometric tool we prove the generalization of Rashevsky--Chow theorem for C1C^1C1-smooth vector fields. The main result of the paper extends theorems on approximate differentiability proved by Stepanoff (1923, 1925) and Whitney (1951) in Euclidean spaces and by Vodopyanov (2000) on Carnot groups.

Doklady Mathematics

ABSTRACT The authors consider the space BMO of functions with a bounded mean oscillation on a Car... more ABSTRACT The authors consider the space BMO of functions with a bounded mean oscillation on a Carnot group G and obtain necessary and sufficient conditions on a domain D⊂G under which a continuous extension operator ext:BMO(D)→BMO exists. A sufficient condition for the existence of a linear bounded extension operator ext:W p 1 (D)→W p 1 (G) (ext:L p 1 (D)→L p 1 (G)), p∈[1,∞], W p 1 (D) denotes the Sobolev space, is also established. Another topic of the article is a relation between BMO spaces and quasiconformal mappings in the case of a Carnot group. The following theorem contains sufficiency in the superposition problem: If f:D→D ' is a q-quasiconformal homeomorphism of domains in Carnot groups, then the operator φ:v→v∘f -1 is a bijective isomorphism of BMO spaces and ∥v∘f -1 ∥ * ≤∥v∥ * , where ∥·∥ * denotes the BMO-norm.

Russian Mathematical Surveys, 2014

Contemporary Mathematics, 2011

We provide an axiomatic approach to the theory of local tangent cones of regular sub-Riemannian m... more We provide an axiomatic approach to the theory of local tangent cones of regular sub-Riemannian manifolds and the differentiability of mappings between such spaces. This axiomatic approach relies on a notion of a dilation structure which is introduced in the general framework of quasimetric spaces. Considering quasimetrics allows us to cover a general case including, in particular, minimal smoothness assumptions on the vector fields defining the sub-Riemannian structure. It is important to note that the theory existing for metric spaces can not be directly extended to quasimetric spaces.

Springer INdAM Series, 2014

ABSTRACT We prove the Local Approximation Theorem on equiregular Carnot-Carathéodory spaces with ... more ABSTRACT We prove the Local Approximation Theorem on equiregular Carnot-Carathéodory spaces with C 1-smooth basis vector fields.

Analysis and Mathematical Physics, 2009

We compare geometries of two different local Lie groups in a Carnot-Carathéodory space, and obtai... more 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.

Doklady Mathematics, 2007

Siberian Mathematical Journal, 2007

We prove differentiability of the mappings of the Sobolev classes and BV-mappings of Carnot-Carat... more We prove differentiability of the mappings of the Sobolev classes and BV-mappings of Carnot-Carathodory spaces in the topology of these classes. We infer from these results a generalization of the Caldern-Zygmund theorems for mappings of the Carnot-Carathodory spaces and other facts.

Siberian Mathematical Journal, 1979

Siberian Mathematical Journal, 1995

In a series of recent articles, the properties of nilpotent Lie groups and related objects have u... more In a series of recent articles, the properties of nilpotent Lie groups and related objects have undergone intensive study in connection with various problems of sub-Riemannian geometry, analysis, and subelliptic differential equations. The analytic questions in such problems are primarily connected with the presence of nontrivial commutation relations which, as a rule, prohibit straightforward translation of the technique developed for similar problems in Euclidean space. Such difficulties appear, for instance, in the problem concerning the differential properties of quasiconformal mappings on Carnot groups which is studied in the present article. A metric definition of a quasiconformal mapping can be given in an arbitrary metric space (see, for instance, [1]). However, for developing the theory of quasiconformal mappings, in particular for establishing their analytic properties, the domain of definition must possess some extra structure.