Superposition in homogeneous and vector valued Sobolev spaces

Abstract:

We give a sufficient condition on a function $f:\mathbb {R}^{k}\rightarrow \mathbb {R}$ so that it takes by superposition the homogeneous vector valued space $\dot {W}^{m}_{p}\cap \dot {W}^{1}_{mp}(\mathbb {R}^n, \mathbb {R}^k)$ into the corresponding real valued space, for integers $m,n,k$ such that $m\geq 2$, $k,n\geq 1$, and $p\in [1,+\infty [$. In case $k=1$, this condition also turns out to be necessary. For $k>1$, it is not proved to be necessary, but it is weaker than the conditions used till now, such as the continuity and boundedness of all derivatives up to order $m$.