# Interpolated Sobolev norm inequality

Let $$\Omega \subset \mathbb{R}^n$$ be a bounded Lipschitz set, and let $$W^{k,p}(\Omega)$$ denote the usual Sobolev space with $$k \in \mathbb{N}$$ being the order of the derivatives and $$p \in [1, \infty)$$ the rate of integrability. I know that there exists a result of this type: fix $$\varepsilon >0$$, then $$\Vert f\Vert_{W^{k-1,p}(\Omega)} \leq \varepsilon \Vert f\Vert_{W^{k,p}(\Omega)} + c(\varepsilon) \Vert f\Vert_{L^1(\Omega)}$$ for any function $$f \in W^{k-1,p}(\Omega)$$, where the constant $$c(\varepsilon)>0$$ depends on $$\varepsilon$$ but not on $$f \in W^{k-1,p}(\Omega)$$.

Do you know how to prove it, or have any reference to a book/paper where I can find the proof of this result?

This can be proved by contradiction. Let $$\epsilon > 0$$ be given, and $$(f_j \mid j \in \mathbf{N})$$ be a sequence of functions in $$W^{k,p}(\Omega)$$ with $$\lvert f_j \rvert_{k-1,p} \geq \epsilon \lvert f_j \rvert_{k,p} + j \lvert f_j \rvert_{0,1}$$. Rescale these functions to have $$\lvert f_j \rvert_{k-1,p} = 1$$ for all $$j$$. Then $$\lvert f_j \rvert_{k,p} \leq \epsilon^{-1}$$ and one can extract a subsequence that converges weakly in $$W^{k,p}(\Omega)$$ and strongly in $$W^{k-1,p}(\Omega)$$. Let $$f \in W^{k,p}(\Omega)$$ be their limit. At the same time $$\lvert f_j \rvert_{0,1} \leq j^{-1}$$, so that $$f_j \to 0$$ in $$L^1(\Omega)$$. Therefore $$f = 0$$, but this is absurd because the convergence in $$W^{k-1,p}(\Omega)$$ means that $$\lvert f \rvert_{k-1,p} = 1$$.
• Just a remark. You are supposed to contradict : $\forall \epsilon >0, \exists C$ such that blah blah. And you start with $\forall \epsilon>0$ : ) (it is fine since you are proving more, of course). Apr 30, 2021 at 11:30