Continuity of composition operators in Sobolev spaces

We prove that all the composition operators T-f (g) := f o g, which take the Adams-Frazier space W-m(p) boolean AND (W)over dot(mp)(1) (R-n) to itself, are continuous mappings from W-p(m) boolean AND (W)over dot(mp)(1) (R-n) to itself, for every integer m >= 2 and every real number 1 <= p < infinity. The same automatic continuity property holds for Sobolev spaces W-p(m) (R-n) for m >= 2 and 1 <= p < infinity. (C) 2019 Elsevier Masson SAS. All rights reserved.