Weakly multiplicative distributions and weighted Dirichlet spaces

We show that if $u$ is a compactly supported distribution on the complex plane such that, for every pair of entire functions $f,g$, $$\langle u,f\overline{g}\rangle=\langle u,f\rangle\langle u,\overline{g}\rangle,$$ then $u$ is supported at a single point. As an application, we complete the classification of all weighted Dirichlet spaces on the unit disk that are de Branges-Rovnyak spaces by showing that, for such spaces, the weight is necessarily a superharmonic function.