Weakly multiplicative distributions and weighted Dirichlet spaces (Q2682385)

The multiplicativity property is important for various applications of distribution theory. In this article the authors generalize and develop the classic results of Sobolev-Schwartz distributions. Generalized functions were first introduced in the work of \textit{S.~Soboleff} [Rec. Math. Moscou, n.~Ser.~1, 39--71 (1936; Zbl 0014.05902)]. Here, based on results of \textit{O.~El-Fallah} et al. [Complex Anal. Oper. Theory 10, No.~1, 97--107 (2016; Zbl 1341.30046)], generalizations of RKHS theory, the following result is derived. Theorem. If \(\omega\) is a weight on \(\mathbb{D}\) such that \({\mathcal D_{\omega}}\) is a de Branges-Rovnyak space, then \(\omega\) is (almost everywhere equal to) a function that is superharmonic on \(\mathbb{D}\).