# 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?

• I could solve my original problem (that I didn't mention here) without using the Gagliardo-Nirenberg inequality for functions defined on bounded Lipschitz domains. I combined the Gagliardo-Nirenberg inequality for functions on $\mathbb{R}^n$ with Lemma 5.1 and Lemma 5.3 from Hitchhiker's guide. – shipwater Apr 14 at 8:38
• If it helps someone: The Gagliardo-Nirenberg inequality for functions defined on bounded Lipschitz domains $\Omega \subset \mathbb{R}^n$ follows from the version for functions on $\mathbb{R}^n$ and Theorem 5.4 from Hitchhiker's guide. Only the following must be taken into account here: From the proof of Theorem 5.4, it is clear that the extension $\tilde u$ satisfies not only $\|\tilde u\|_{W^{s,p}(\mathbb{R}^n)} \leq C \|u\|_{W^{s,p}(\Omega)}$ but also $\|\tilde u\|_{L^2(\mathbb{R}^n)} \leq C_2 \|u\|_{L^2(\Omega)}$. – shipwater Apr 15 at 8:28