Delta distribution as a bounded linear functional

Question

Let $\Omega \subset \mathbb{R}^n$ be a domain with $0 \in \Omega$. For which $n \in \mathbb{N}$ is the Dirac-delta distribution a bounded linear functional on $H_{0}^{1}(\Omega)$, that is to say, an element of $H^{-1}(\Omega)$?

I can easily show (using the Sobolev embedding theorem) that for $n=1$ the delta distribution is bounded. But what about the other possible values of $n$?

Answer 1

For $n\geq 3$ take $f_k(x)=\min\{ |x|^{-1}, k\}$. Then it's easy to see that $\| f_k\|_{H^1_0(\Omega)}$ is uniformly bounded in $k$, but $f_k(0)=k$ is unbounded.

For $n=2$ a similar argument applies to $f_k(x)=\min\{ \ln \ln(1+1/|x|), k\}$.