1

Let $\mathbb{R}^n$ be $n$-dimensional Euclidean space and $S^n$ be the $n$-sphere.

If we fix a point $x \in S$, the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ can be characterized as the subspace of $C^\infty(S^n)$ whose elements have vanishing derivatives of all orders at $x$, according to my previous post.

Now, let $\mathcal{S}'(\mathbb{R}^n)$ be the space of tempered distributions and $\mathcal{D}'(S^n)$ be the dual space of $C^\infty(S^n)$. Then my question is

What would be some explicit description of $\mathcal{S}'(\mathbb{R}^n)$ as a space of distributions on $S^n$?

Could anyone please provide an answer or any relevant reference?

Keith
  • 8,359
  • It is not clear to me how you can see $\mathcal{S}'(\mathbb{R}^n)$ as a subspace of $\mathcal{D}'(S^n)$. Such an injection should usually come as the dual of an injection of type $C^\infty(S^n) \rightarrow \mathcal{S}(\mathbb{R}^n)$ but I don't see how you do this injection since elements of $C^\infty(S^n)$ don't necessarily vanish at the point corresponding to $\mathbb R^n$ 's infinity. – alex440 Jul 11 '24 at 18:31
  • I edited my question as the tempered distributions do not seem to be a subspace of $D'(S^n)$. – Keith Jul 11 '24 at 19:33
  • 2
    Now posted to MO, https://mathoverflow.net/questions/474878/an-equivalent-characterization-of-mathcals-as-distributions-on-the-sphere – Gerry Myerson Jul 14 '24 at 00:56
  • @GerryMyerson Thank you. I forgot to mention this here... – Keith Jul 14 '24 at 16:32

0 Answers0