Approximation of Hölder mappings on Carnot–Carathéodory spaces

The general nature of approximation of certain Hölder mappings of Carnot–Carathéodory spaces is described. The local existence of a basis in the image which preserves sub-Riemannian Hausdorff dimension is also proved.     

Graph surfaces of codimension two over three-dimensional Carnot–Carathéodory spaces

An area formula for graph surfaces of codimension two over three-dimensional Carnot–Carathéodory spaces is derived and applied to obtain basic properties of minimal surfaces.     

Polynomial Sub-Riemannian Differentiability on Carnot–Carathéodory Spaces

We show that the classes of Hölder mappings of Carnot–Carathéodory spaces are polynomially differentiable in the sub-Riemannian sense. Moreover, we prove the existence of intrinsic (or adapted) bases, which enable us to match the nonholonomic structures of the images of mappings and target spaces.     

Three-Dimensional Graph Surfaces on Five-Dimensional Carnot–Carathéodory Spaces

We study the class of codimension 2 graph surfaces over some three-dimensional Lie groups and establish some analogs of the differential properties of the corresponding graph mappings. Moreover, we derive the area formula and describe the classes of minimal surfaces of codimension 1 and 2.     

Local Metric Properties of Level Surfaces on Carnot–Carathéodory Spaces

Doklady Mathematics - An adequate local metric characteristic is introduced for level sets of $$C_{H}^{1}$$ -mappings of Carnot manifolds to Carnot–Carathéodory spaces. Moreover, for...     

Existence of tangent lines to Carnot–Carathéodory geodesics

We show that length minimizing curves in Carnot–Carathéodory spaces possess at any point at least one tangent curve (i.e., a blow-up in the nilpotent approximation) equal to a straight horizontal line.     

Area of Some Hölder Surfaces on Carnot–Carathéodory Spaces

An area formula for classes of Hölder mappings on Carnot–Carathéodory spaces is proved. As important particular cases, general and classical versions of surface measure calculation for graph surfaces are considered.     

Metric Properties of Graphs on Carnot–Carathéodory Spaces with Sub-Lorentzian Structure

Doklady Mathematics - The notion of a sub-Lorentzian structure with multidimensional time on Carnot–Carathéodory spaces is introduced. It is proved that classes of graph surfaces are...     

On the Class of Hölder Surfaces in Carnot—Carathéodory Spaces

Considering two classes of Hölder mappings of Carnot-Carathéodory spaces, graph mappings and smooth mappings in the Riemannian sense, we obtain an area formula for image surfaces. In particular, we describe the structure of the polynomial sub-Riemannian differential for graph mappings.

     

A Coarea Formula for Smooth Contact Mappings of Carnot–Carathéodory Spaces

We prove the coarea formula for sufficiently smooth contact mappings of Carnot manifolds to Carnot–Carathéodory spaces. In particular, we investigate level surfaces of these mappings, and compare Riemannian and sub-Riemannian measures on them. Our main tool is the sharp asymptotic behavior of the Riemannian measure of the intersection of a tangent plane to a level surface and a sub-Riemannian ball. This calculation in particular implies that the sub-Riemannian measure of the set of characteristic points (i.e., the points at which the sub-Riemannian differential is degenerate) equals zero on almost every level set.     

On the Carathéodory superposition of multifunctions and an existence theorem

Let I ? ? be an interval and Y a reflexive Banach space. We introduce the (H) property of a multifunction F from I × Y to Y and prove that the Carathéodory superposition of F with each continuous function f from I to Y is a derivative provided that F has the (H) property. Some application of this theorem to the existence of solutions of differential inclusions f′(x) ∈ F(x, f(x)) is given.     

On vector lattices of elementary Carathéodory functions

In this paper we deal with the vector lattice C(B) of all elementary Carathéodory functions corresponding to a generalized Boolean algebra B.This work was supported by grant VEGA 2/1131/21.     

Hardy–Sobolev Type Inequalities with Sharp Constants in Carnot–Carathéodory Spaces

We prove a generalization with sharp constants of a classical inequality due to Hardy to Carnot groups of arbitrary step, or more general Carnot–Carathéodory spaces associated with a system of vector fields of Hörmander type. Under a suitable additional assumption (see Eq. 1.6 below) we are able to extend such result to the nonlinear case \(p\not= 2\). We also obtain a sharp inequality of Hardy–Sobolev type.     

A remark on Carathéodory balls

Member of URA CNRS D 1322 Groupes de Lie et Géométrie     

The Grunsky function and Carathéodory metric of Teichmüller spaces

A new proof of the equality of all non-expanding invariant metrics on the universal Teichmüller space is given. The arguments extend to Teichmüller space of the punctured disc.     

Carnot–Carathéodory metrics in unbounded subdomains of {{\mathbb{C}}^2}

We introduce a new class of unbounded model subdomains of \({\mathbb{C}^2}\) for the \({\Box_b}\) problem. Unlike previous finite type models, these domains need not be bounded by algebraic varieties. In this paper we obtain precise global estimates for the Carnot–Carathéodory metric induced on the boundary of such domains by the real and imaginary parts of the CR vector field.     

Non-Commutative Carathéodory Interpolation

We prove a Carathéodory–Fejér type interpolation theorem for certain matrix convex sets in mathbbC^d{mathbb{C}^d} using the Blecher–Ruan–Sinclair characterization of abstract operator algebras. Our results generalize the work of Dmitry S. Kalyuzhnyĭ-Verbovetzkiĭ for the d-dimensional non-commutative polydisc.     

On the Carathéodory–Fejér Interpolation Problem for Generalized Schur Functions

The solutions of the Carathéodory–Fejér interpolation problem for generalized Schur functions can be parametrized via a linear fractional transformation over the class of classical Schur functions. The linear fractional transformation of some of these functions may have a pole (simple or multiple) in one or more of the interpolation points or not satisfy one or more interpolation conditions, hence not all Schur functions can serve as a parameter. The set of excluded parameters is characterized in terms of the related Pick matrix.Research was supported by the Summer Research Grant from the College of William and MarySubmitted: June 26, 2002 Revised: January 31, 2003