A Poincare inequality on fractional Sobolev space

Question

Let $\Omega$ be a bounded smooth domain. Does the following inequality hold for all $u \in H^s_0(\Omega)$:

$$\lVert u \rVert_{L^2(\Omega)} \leq C|u|_{H^s_0(\Omega)}$$ where the right hand side is the $H^s_0(\Omega)$ seminorm.

$H^s_0$ is defined as an interpolaton space of $H^1_0$ and $L^2$.

Answer 1

It will also depend on the dimension of $\Omega$. Roughly speaking, the embedding theorem works the same for fraction ordered Sobolev space. Please check Theorem 6.14 from this book, and some related theorems from there as well.