Continuity of composition operators in Sobolev spaces

We prove that all the composition operators Tf(g):=f∘g, which take the Adams-Frazier space Wpm∩W˙mp1(Rn) to itself, are continuous mappings from Wpm∩W˙mp1(Rn) to itself, for every integer m≥2 and every real number 1≤p<∞. The same automatic continuity property holds for Sobolev spaces Wpm(Rn) for m≥2 and 1≤p<∞.