Exercise 41, § 5

Let $n\geqslant 1$ be an integer. Denote by Δ the Laplacian on $\mathbf{R}^n$, and denote by $u$ the restriction of Δ to the subspace $\mathscr{D}(\mathbf{R}^n-\{0\})$.

(a) The domain of $u$ is dense in the Hilbert space $E_q$ associated with the partial form $q$ associated with Δ if and only if $n\geqslant 2$. (Identify the dual of $E_q$ with a space of tempered distributions; if $g\in E'_q$ is zero on dom($u$), now prove that $g$ is a distribution with support contained in $\{0\}$; conclude using exercise 24 of IV, p. 338.)

(b) The domain of $u$ is dense in the Hilbert space $E_{\Delta}$ if and only if $n\geqslant 4$.