Counter example for analogous Poincare inequality does not hold on Fractional Sobolev spaces

Question

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 $W^{s,p}(\Omega)$.

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.