This is the statement that I seek to find a reference for:
Let $n,m\in\mathbb{N}$ with $n>m$ and let $u\in H^{\frac{m-n}{2}}(\mathbb{R}^n)$ be supported only on a smooth manifold of dimension $m$ which is embedded in $\mathbb{R}^n$. Then $u=0$.
To me, it seems like a basic result, but my advisor insisted that I cite or prove it, and I on a tight schedule and don't really have time to have to write up the proof. I'd much rather cite it, but I've been looking around in the library and I'm not sure how to find it.
If anyone could point me to a reference for this fact, I would be very grateful.
Note that I don't actually need the general result. I just need to know it for a cylinder, a line, or a point embedded in $\mathbb{R}^3$
In various places (e.g., Avner Friedman's book on PDE) one can find the theorem that a distribution supported on a nicely-imbedded submanifold is (at least locally) the composition of application of normal derivatives with a distribution on the submanifold. After a simplifying coordinate change (e.g., to make the submanifold the span of a subset of the canonical basis vectors) Fourier transform shows that for $u\in H^s(\mathrm{sub})$ the composition with restriction of codimension $m$ is in $H^{s-{m\over 2}-\epsilon}$ for every $\epsilon>0$. This is a nice extension of the fact that Dirac delta is in $H^{-{n\over 2}-\epsilon}(\mathbb R^n)$.