Gagliardo-Nirenberg inequality for fractional Sobolev spaces Wikipedia states two versions of the Gagliardo-Nirenberg inequality for nonfractional Sobolev spaces. I'm interested in generalizations to fractional (Slobodeckij) Sobolev spaces. 
Such a generalization of the version for functions on $\mathbb{R}^n$ can be found e.g. here.  
Unfortunately, I don't find such a generalization of the version for functions defined on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^n$. I'm pretty sure that the inequality still holds if one replaces the terms $\|D^j u\|_{L^p}$ and $\|D^m u\|_{L^r}$ by the corresponding Gagliardo semi-norms. 
Does anyone know an article/book where such a generalization can be found?

It is straightforward. You have $u$ defined on $\Omega$ and then the extension $\tilde u$ defined on $\mathbb{R}^n$. Now the Gagliardo semi-norm of $u$ can be estimated trivially by the Gagliardo semi-norm of $\tilde u$ (just because it's an extension). Since $\tilde u$ is defined on $\mathbb{R}^n$ you can now use the Gagliardo-Nirenberg inequality for such functions. Finally, use Theorem 5.4 of Hitchhiker's guide to bound the appearing norms of $\tilde u$ by norms of $u$.
