Extending differentiable functions and quasiconformal mappings on Carnot groups (Q1380247)

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 \({\mathcal D}\subset G\) under which a continuous extension operator \(\text{ext}: BMO ({\mathcal D}) \to BMO\) exists. A sufficient condition for the existence of a linear bounded extension operator \(\text{ext}: W^1_p ({\mathcal D})\to W^1_p(G)\) \((\text{ext}: L^1_p ({\mathcal D})\to L^1_p(G))\), \(p\in [1, \infty]\), \(W^1_p ({\mathcal D})\) 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 \(f:{\mathcal D}\to {\mathcal D}'\) is a \(q\)-quasiconformal homeomorphism of domains in Carnot groups, then the operator \(\varphi: v\to v\circ f^{-1}\) is a bijective isomorphism of \(BMO\) spaces and \(\| v\circ f^{-1} \|_*\leq \| v\|_*\), where \(\| \cdot \|_*\) denotes the \(BMO\)-norm.