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Let  G : ℝ → ℝ be a continuous function. Denote by  T G   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 likeis true. Here  F   s    p, q   denotes a space of Triebel-Lizorkin type and  W   m    p   denotes a Sobolev space, respectively. Necessary and sufficient conditions for such an inclusion to hold will be given in cases  G ( t ) =  t   k  ,  k  ∈ ℕ,  G ( t ) = | t | μ ,  G ( t ) =  t | t | μ - 1 , μ &gt; 1,  G  ∈  C  0  ∞ , and  G  a periodic  C  ∞ -function.

Conditions on composition operators which map a space of Triebel-Lizorkin type into a Sobolev space. The case 1 < s < n / p . II