Could we extend the functional $\xi \in C_0^\infty \left( \Omega \right) \to \xi \left( 0 \right) \in \mathbb R$ to a cont functional in Sobolev sp

Question

Consider the space of $C_0^\infty \left( \Omega \right)$ functions where $\Omega =(-1,1)$. Define the functional

$$\phi :\xi \in C_0^\infty \left( \Omega \right) \to \xi \left( 0 \right) \in \mathbb R$$

could we extend this functional to a continuous linear functional on the Sobolev space $H_0^0$ or $H_0^1$ respectively ?

Answer 1

Yes, we can extend it to $H^1$, because Sobolev’s lemma says $$ |u(0)| \leq c\|u\|_{H^1} \qquad\textrm{for}\quad u\in C^\infty_0(\mathbb{R}), $$ where $c$ is a constant independent of $u$. However, note that this is only true in 1 dimension.

For $L^2$ it doesn’t work in any dimension.