Let $\Omega$ be open bounded set in $\mathbb R^n$,$0<s<1$, $1\leq p<\infty$. Does the following inequality holds, $$||u||_p\leq c [u]_{s,p}, \forall u\in W^{s,p}_0(\Omega)$$

Where $[u]_{s,p}$ is the Gagliardo (semi) norm of $u$ in fractional sobolev space $W^{s,p}(\Omega)$.

  • $\begingroup$ Did you try to transfer the proof of the Poincaré inequality to this case of fractional spaces? $\endgroup$
    – gerw
    Feb 12, 2018 at 19:57
  • $\begingroup$ It is not true for $W^{s,p}(\Omega)$ that I can see but in $W^{s,p}_0(\Omega)$ I didn't get it. $\endgroup$
    – MathRock
    Feb 12, 2018 at 20:07

1 Answer 1


For $s<\frac{1}{2}$, $W^{s, p} (D) = W^{s, p}_0 (D)$. So, we can find a sequence of functions $u_k\in W^{s, p}_0 (D)$ such that $u_k\to \bar{1}$ in $W^{s, p} (D)$. If the inequality in the question is to hold, then that would mean $1<0$.

Hence, the answer is negative.


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .