Weakly multiplicative distributions and weighted Dirichlet spaces

TL;DR

This paper characterizes distributions with a weak multiplicative property, showing they are point-supported, and applies this to classify weighted Dirichlet spaces as de Branges-Rovnyak spaces with superharmonic weights.

Contribution

It proves a unique support property for certain distributions and completes the classification of weighted Dirichlet spaces as de Branges-Rovnyak spaces with superharmonic weights.

Findings

Distributions satisfying the multiplicative condition are supported at a single point.

Weighted Dirichlet spaces that are de Branges-Rovnyak spaces have superharmonic weights.

The classification of such spaces is now complete.

Abstract

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.