According to this pdf, the "Schwartz representation theorem" says that for any tempered distribution $u \in \mathcal{S}^\prime(\mathbb{R}^n)$ there exists a finite collection $u_{\alpha\beta}: \mathbb{R}^n \to \mathbb{C}$ of bounded continuous functions, $|\alpha|+|\beta| \leq k$, such that
$$ u = \sum_{|\alpha|+|\beta| \leq k} x^\beta D^\alpha_x u_{\alpha\beta}. $$
My question is: is there some straightforward criterion for knowing when this representation reduces to its trivial form, and $u$ can be simply represented by a bounded continuous function, without any derivatives? Are there any "hallmarks" of such distributions? In my specific context I know that the distributions obey a linear first order PDE, if that helps!