The Schwarz theorem states that any function $f$ form the Schwarz class of tempered distributions $S'(\mathbb R^n)$ is a sum of derivatives from some continuous tempered distributions $g_\alpha\in С(\mathbb R^n)$:
$$ f(x) = \sum_{|\alpha|\le m}\partial^\alpha g_\alpha(x). $$
I seem to rememeber even a variant with one function $g$.
Suppose now that $\mathrm{supp} f$ is in a halfspace $\{x_n\ge 0\}$. Can one take functions $g_\alpha$ s.t. their supoprt is also in the same halfspace? Is the same true with a convex domain instead of a halfspace?
Edit: my previous answer was wrong, try with $\delta(x,y)=\delta(x)\delta(y)=(\partial_x \partial_y)^2xy 1_{x>0}1_{y >0}$ where $xy 1_{x>0}1_{y >0}$ is not equal to a polynomial on the whole of $x >0$.
For half-spaces it works due to how we can construct $g$ such that $f=(\partial_{x_1}\ldots\partial_{x_n})^m g$:
Take $\psi\in C^\infty(\Bbb{R}), \psi=0$ on $t < -1$, $\psi=1$ on $t > 0$ and let $J_l \phi(x)=\int_{-\infty}^0 \phi(x+t e_l)dt, I_l\phi(x)=J_l\phi(x+\infty e_l)\psi(x_l)-J_l\phi(x)$.
Then given $f\in S'(\Bbb{R}^n)$ of order $m-1$ define the distribution $g\in S'(\Bbb{R}^n)$ by $$\forall \phi \in S(\Bbb{R}^n) ,\qquad \langle g,\phi \rangle=\langle f,(I_1\ldots I_n)^m \phi \rangle$$