# Proper sobolev spaces invariant under no-linearities

Let $f:H^s\to H^s$ at least continuous and not necesarily linear. Is there some kind of criterion or condition over $f$ that lets to ensure that $f({H^{s+k}})\subseteq H^{s+k}$?

• There are whole books on this. Give a try at Runst and Sickel, "Sobolev spaces of fractional order etc" Sep 10 '10 at 18:16

Let $f:\mathbb C\rightarrow\mathbb C$ be a $C^\infty$ function such that $f(0)=0$. Let $1\le n\in \mathbb N$ and $s>n/2$. Then $$\forall u\in H^s(\mathbb R^n),\quad f(u)\in H^s(\mathbb R^n).$$ In particular, for this range of values, $H^s(\mathbb R^n)$ is an algebra included in $L^\infty(\mathbb R^n)$.