Let G: ℝ → ℝ be a continuous function. Denote by TG the corresponding composition operator which sends ƒ to G(ƒ). Then we investigate necessary and sufficient conditions on the parameters s, p, q, r and on the function G such that an inclusion like
is true. Here Fsp, q denotes a space of Triebel-Lizorkin type and Wmp denotes a Sobolev space, respectively. Necessary and sufficient conditions for such an inclusion to hold will be given in cases G(t) = tk, k ∈ ℕ, G(t) = |t|μ, G(t) = t|t|μ - 1, μ > 1, G ∈ C0∞, and G a periodic C∞-function.