2
$\begingroup$

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 $f({H^{s+k}})\subseteq H^{s+k}$?

$\endgroup$
1
  • 2
    $\begingroup$ There are whole books on this. Give a try at Runst and Sickel, "Sobolev spaces of fractional order etc" $\endgroup$ Sep 10 '10 at 18:16
1
$\begingroup$

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 $L^\infty(\mathbb R^n)$.

Bazin.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.