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 $C^1$-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.
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.
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.
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.
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 $C^1$-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.
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.
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.
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.
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