Continuity of composition operators in Sobolev spaces

Gérard Bourdaud, Madani Moussai

Published 2018-12-01Version 1

We prove that all the composition operators $T_f(g):= f\circ g$, which take the Adams-Frazier space $W^{m}_{p}\cap \dot{W}^{1}_{mp}(\re^n)$ to itself, are continuous mappings from $W^{m}_{p}\cap \dot{W}^{1}_{mp}(\re^n)$ to itself, for every integer $m\geq 2$ and every real number $1\leq p<+\infty$. The same automatic continuity property holds for Sobolev spaces $W^m_p(\R)$ for $m\geq 2$ and $1\leq p<+\infty$.