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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 $\mathbb{R}^3$

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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 $H^{-{n\over 2}-\epsilon}(\mathbb R^n)$.

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  • $\begingroup$ I was looking in Freedman's book entitled "Partial Differential Equations," and I found no mention of distributions, the delta function, or manifolds anywhere in the book. Can you direct me more precisely? Is it possibly a different one of his books? $\endgroup$ Commented Aug 12, 2017 at 23:10
  • $\begingroup$ Hm, maybe, sorry for the misdirection! Unfortunately, I don't have those books nearby. A similar argument can be found at math.umn.edu/~garrett/m/fun/dists_on_subs.pdf $\endgroup$ Commented Aug 12, 2017 at 23:32
  • $\begingroup$ ... and I guess a useful google phrase would be "distributions supported on submanifolds"... $\endgroup$ Commented Aug 12, 2017 at 23:33
  • $\begingroup$ Also, this may seem like a very basic and silly question to you, but how do you define restriction for a general distribution? The trace theorems I have seen only hold for u in H^s where s>1/2. I'm an applied mathematician... I only know the very basic elements of distribution theory. $\endgroup$ Commented Aug 13, 2017 at 0:00
  • $\begingroup$ Ah, I don't think we can define restriction of distributions nicely in general (as you note), but we can do a much easier thing: define a distribution $U$ on a larger space as $U(\varphi)=(u\circ r)(\varphi)=u(r\varphi)$ where $u$ is a distribution on a sub, and $r$ is the restriction of test functions from the bigger space to (nice) smaller. Sorry to not be clear. $\endgroup$ Commented Aug 13, 2017 at 0:08

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