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 W\limits^{˙}_{mp}^{1}(\mathbb{R}^{n}) to itself, are continuous mappings from W_{p}^{m} \cap W\limits^{˙}_{mp}^{1}(\mathbb{R}^{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 $W_{p}^{m}(\mathbb{R}^{n})$ for $m \geq 2$ and $1 \leq p < \infty$.