If by 'finitely supported' you mean supported on a set made by a finite number of points, the answer is no. As a simple example, let $D=\partial_t-\partial_{xx}$ be the heat operator, $E=\{(t,x):t>0,x\in R\}$ and $u(t,x)=t^{-1/2}e^{-|x-y|^2/4t}$ the fundamental solution. Then $D(u\chi_E)$ is the delta at $(0,0)$.
Piero D'Ancona
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