The Whitney problem of existence of a linear extension operator

We prove that a linear bounded extension operator exists for the trace of C ^1·ω (R ⁿ )to an arbitrary closed subset of R ⁿ .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.