AbstractWe consider a real-valued function r = M(t) on the real axis, such that M(t) < 0 for t < 0. Under appropriate assumptions on M, the pull-back operator M* gives rise to a transform of Sobolev spaces Ws.p (-∞, 0) that restricts to a transform of Ws.p(-∞, ∞). We construct a bounded linear extension operator Ws.p(-∞, 0) → Ws.p(−∞, ∞), commuting with this transform.