Composition operators with closed range for one-dimensional smooth symbols

We give a sufficient condition for a smooth not necessarily injective function ψ : R → R ensuring that the composition operator C ψ : C ∞ ( R ) → C ∞ ( R ) , F ↦ F ∘ ψ has closed range. This generalizes in a special case the result of Kenessey and Wengenroth which gave a description of composition operators C ψ with closed range for smooth injective symbols ψ : R → R d .