Support of polynomial distributions Assume $u\in\mathcal{S}'(\mathbb{R}^n)$ is a tempered distribution such that $\widehat{u}$ is compactly supported and $u^k$ defines a distribution for each $k=1,\cdots,m$. Let   $p_1,\cdots,p_m$ be polynomials in $\mathbb{R}^n.$ 
How are supports of $\widehat{p_1u}+\cdots+\widehat{p_mu^m}$ and $\widehat{u}$ compared?
Does the following inclusion hold?
 supp$(\widehat{p_1u}+\cdots+\widehat{p_mu^m})\subseteq$ supp$(\widehat{u})$ 
 A: I am not completely sure at the moment how the supports of $\widehat{p_1 u} + \dots + \widehat{p_m u^m}$ and of $\widehat{u}$ could be connected, but if you consider
$$
\widehat{u} = \delta_x,
$$
i.e. $u = e^{2\pi i \langle x, \cdot\rangle}$, then $u^2 = e^{2\pi i \langle 2x, \cdot\rangle}$, i.e.
$$
\widehat{u^2} = \delta_{2x},
$$
so that even ${\rm supp}(\widehat{u^2}) \subset {\rm supp}(\widehat{u})$ is false in general (because $\{2x\} \nsubseteq \{x\}$ for $x \neq 0$).
EDIT: I would expect that ${\rm supp}(\widehat{u^k}) \subset {\rm supp}(\widehat{u}) + \dots + {\rm supp}(\widehat{u})$ (with $k$ summands) holds, because for ordinary Schwartz functions, we would have (by the convolution theorem)
$$
\widehat{u^k} = \widehat{u} \ast \cdots \ast \widehat{u},
$$
and we have ${\rm supp}(f \ast g) \subset {\rm supp}(f) + {\rm supp}(g)$. But I am not completely sure how to lift this to general distributions at the moment.
EDIT 2: Finally, (and this is even easy to see for general tempered distributions), you have $\widehat{x^\alpha u} = \partial^\alpha \widehat{u}$ (up to multiplication by powers of $\pm 2 \pi i$ or something like that) and ${\rm supp}(\partial^\alpha v) \subset {\rm supp}(v)$. Hence, the multplication with polynomials is ok to do (will not increase the support).
A: (rather long to be a comment)Theorem:( J.L-. Lions 1951) If $u,v\in\mathcal{S}'({\mathbb{R}})$ with compact support then co.supp$(u\ast v)=$co.supp$(u)$+co.supp$(v)$, where  "co" denotes the convex hull.
