Dirac-Delta distribution lies in $H^{-1}(B_1(0))$ only for $n = 1$

Question

Show that the delta distribution $\delta_0$ is in $H^{-1}(B_1(0))$ ($B_1$ denotes the unit ball in $\mathbb{R}^n$) if and only if $n = 1$.

I've successfully proven that for $n = 1$ we have $\delta_0 \in H^{-1}(B_1(0))$. However, I'm facing difficulties in establishing the reverse implication. Can anyone provide guidance or tips on how to prove it?

Answer 1

A sketch too long for a comment: Leverage the fact that a) on $\mathbb R^n$ $H^{-1}(\mathbb R^n)$ has an easy description in terms of Fourier transforms and b) The support of $\delta$ (defined here just as the distribution defined by integrating a function against the probability measure at 0) as a distribution is $0$, meaning precisely that if $\phi\in H^1$ and $\psi\in \mathcal{S}(\mathbb R^n)$ has $\text{supp}(\psi)\subset B_\delta(0)$ for some $\delta>0$, then $\delta(\psi \phi)=\delta(\phi)$.

So, since $\hat{\delta}=1$ up to some normalization, $\delta$ is not in $H^{-1}(\mathbb R^n)$ if and only if $n>1$, and there is a sequence of $\phi_k\in H^1(\mathbb{R}^n)$ with $\phi_k\to \phi$ in $H^1(\mathbb R^n)$ but $\delta(\phi_k)\not\to \delta(\phi)$. Now, take a smooth cut off $\eta$ and consider $\eta\cdot \phi_k$.

Answer 2

In addition to @operatorerror's good answer from the viewpoint of Fourier transform characterization of $L^2$ Sobolev spaces, ... one might dually ask for examples of functions in $H^1$ but not in $C^o$, in larger dimensions. The Sobolev imbedding theorem does give $H^1\subset C^o$ in one dimension, but this fails sharply in dimensions $\ge 3$, and does fail in dimension $2$. (This is "dual" to the fact that $\delta\in H^{-{n\over 2}-\epsilon}$ for every $\epsilon>0$, on $\mathbb R^n$.)

Granting ourselves the (true!) fact that $H^1$ is also (characterizable as) the collection of $L^2$ functions whose distributional gradient is (given by a distribution) in $L^2$, we want such a function that is not continuous. Although things are more delicate for $n=2$, for $n\ge 3$ there is the nice example of $\log r$, where $r$ is of course radius. The verification of $L^2$-ness is easy. :)