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, $$\begin{aligned} \langle u,f{\overline{g}}\rangle =\langle u,f\rangle u,{\overline{g}}\rangle , \end{aligned}$$ then at single point. As an application, we complete classification all weighted Dirichlet spaces unit disk are de Branges–Rovnyak by showing spaces, weight necessarily superharmonic function.