The Whitney problem of existence of a linear extension operator

We prove that a linear bounded extension operator exists for the trace of C1·ω (Rn)to an arbitrary closed subset ofRn.The similar result is obtained for some other spaces of multivariate smooth functions. We also show that unlike the one-dimensional case treated by Whitney, for some trace spaces of multivariate smooth functions a linear bounded extension operator does not exist. The proofs are based on a relation between the problem under consideration and a similar problem for Lipschitz spaces defined on hyperbolic Riemannian manifolds.