Using interpolation in Sobolev Space

Question

good afternoon. I'm studying an article and the author uses a notation that I don't know and this is making it difficult for me to understand an inequality.

Consider a function $\eta$ in $H^{1}(0,1)$, that is, $\eta \in L^{2}(0,1)$ and $\eta_{x} \in L^{2} (0.1)$. So, the author claims that by interpolation, we have $$ \|\eta\|_{L^{2}(0,1)} \leq C\|\eta\|_{-1,2}\|\eta\|_{1,2} $$ where $C >0$.

I don't know what the meaning of $\| \bullet\|_{-1,2}$. Does anyone know a reference that addresses this notation? And what about inequality? Is it some classical theorem?

Answer 1

I am following Adams and Fournier's Sobolev Spaces.

In general, $W^{-k,p}(\Omega)=(W_{0}^{-k,p}(\Omega))'$ (well, this is not the precise definition, but you can think this space in this form because they are isometrically isomorphic, see Theorem 3.12 from the book that I mentioned or Theorem 11.62 of Leoni's Sobolev Spaces 2nd ed), where $'$ is used to denote the dual space. In your case, $H^{-1,2}(0,1)=(H_{0}^{1,2}(0,1))'$.

• What is the norm here? and there make sense to compute $\|\eta\|_{-1,2}$ for $\eta \in L^{2}$?: for each $v \in L^2(0,1)$ define $$L_{v}(u)=\langle u,v\rangle=\int_{\Omega}u(x)v(x)dx,$$ where $u \in H_{0}^{1,2}(0,1)$ Note that $L_{v}$ makes sense as an element of $H^{-1,2}$ because $$|L_{v}(u)|=|\langle u,v \rangle| \underbrace{\leq}_{\text{Holder}} \|u\|_{L^{2}(0,1)} \|v\|_{L^{2}(0,1)} \leq \|u\|_{H^{1,2}(0,1)} \|v\|_{L^{2}(0,1)}<\infty.$$ Now, define the norm of $v$ in this space as the operator norm of $L_{v}$, that is, $$ \|v\|_{H^{-1,2}(0,1)}=\sup_{{u \in H_{0}^{1,2}(0,1) \\ \|u\|_{H_{0}^{1,2}(0,1)} \leq 1}} |L_{v}(u)|.$$

• How do you obtain the required inequality?: Let $u \in H_{0}^{1,2}(0,1)$, $v \in L^{2}(0,1)$ we have $$|\langle u,v \rangle|=\|u\|_{H_{0}^{1,2}(0,1)} \Bigg| \left\langle \frac{u}{\|u\|_{H_{0}^{1,2}(0,1)}},v \right\rangle \Bigg| \leq \|u\|_{H_{0}^{1,2}(0,1)} \|v\|_{H^{-1,2}(0,1)}.$$ So, taking $u=v=\eta$, you obtain the desire inequality.

Finally, let me comment that this inequality is "well-known" (for people that works on PDE), and sometimes is called "Sobolev duality inequality" or "generalized Hölder inequality".