Continuity of composition operators in Sobolev spaces

Abstract

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