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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)\ $]

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That's a wrong definition of "concentrated". For example, you would like $\delta'$ to be concentrated on $\{0\}$, but there are functions (e.g. $\phi(x) = x \exp(-x^2)$) with $\phi(0) = 0$ but $\delta'(\phi) = \phi'(0) \ne 0$. –  Robert Israel Mar 23 '11 at 22:22
    
Instead, you want to say that $\phi$ is supported in $R \backslash A$, i.e. it is 0 on a neighbourhood of $A$. –  Robert Israel Mar 23 '11 at 22:33

1 Answer 1

Yes, at least locally, every distribution (temperedness becomes irrelevant if we are talking about local things) supported on a submanifold is (locally) a finite linear combination $f\rightarrow \sum_i u_i(\nu_i f)$ of distributions $u_i$ supported on the submanifold applied to (iterated) normal derivatives $\nu_i$ of a test function $f$. The proof is not so different from the proof in case the submanifold is a point... A sample argument is here .

Edit: Yes, as in Mariano S.-A.'s comment, by this is meant that the local pieces can then be glued back together by a (smooth) partition of unity. Then the temperedness would/could come into play, constraining the orders of the local pieces. But to address that precisely would certainly require further details about the imbedding of the submanifold, since temperedness on the ambient space, meaning relative to a metric, can obviously have a range of translations to the submanifold. Spiral imbeddings of $\mathbb R^1$ in $\mathbb R^2$ already illustrate how the metrics can be disparate. I don't know anything clear and systematic to say except the caution about this.

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Why only locally? One can use a partition of unity to glue, no? –  Mariano Suárez-Alvarez Jul 24 '11 at 0:23

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