Given a differential operator $A$ with constant coefficients we know by Malgrange-Ehrenpreis that there exists a fundamental solution, i.e. a distribution $T$ such that $AT=\delta$ where $\delta$ is the Dirac distribution. Let $f\in C_c^\infty(\mathbb{R}^n)$, then the PDE $Au=f$ has a distributional distribution given by $u=T*f$. Now we know that $u\in C^\infty(\mathbb{R}^n)$ as $D^\alpha u = (D^\alpha T)*f$. Bit this argument shows, that the derivatives are regular distributions as-well, hence we know that the distributional derivatives are in fact weak derivatives and as the weak derivatives agree with a smooth function they are in fact classical derivatives?
I don't see where my argumentation fails, however the conclusion can't be true since I get also distributional solutions for compact distributions (that is not regular!) as data, for which we of course cannot get a classical solution...