Question: What can you conclude about a tempered distribution G\ \in\ S'(R^n) that is concentrated in some k-dimensional manifold $M\ \subset\ R^n$ (for k < n)? More specifically, is there a result analogous to the following n=1 result?
$n=1$ result (hope I remember it correctly):
Let $\ S(R)\ $ be the set of Schwartz functions ($C^\infty$ functions $f:\ R\ \to\ C\ $ s.t. $\ f^{(n)}$ goes to 0 at infinity faster than any inverse power of x (for n=0, 1, ...)). Let \ S'(R)\ be the set of tempered distributions. G\ \in\ S'(R) is said to be concentrated in a set $A\ \subset\ R\ $ iff $\forall\ \phi\ \in\ S(R)$ that vanishes on some open set $B\ \supset\ A\ $, $G(\phi)\ =\ 0$.
Suppose $\ G\ $ is concentrated in {$\ x\ $}, for some $x\ \in\ R$. Then $\exists\ c_0,\ ...\ c_L\ \in\ C\ $ s.t. $\ G\ $ = $\sum_{j=0}^L\ c_j\ \delta_x^{(j)}$.
[where $\delta_x^{(j)}\ (\phi)\ \equiv\ (-1)^j\ \phi^{(j)}(x)\ $]