A Local Douglas formula for Higher Order Weighted Dirichlet-Type Integrals

Abstract

We prove a local Douglas formula for higher order weighted Dirichlet-type integrals. With the help of this formula, we study the multiplier algebra of the associated higher order weighted Dirichlet-type spaces $${\mathcal {H}}_{\pmb \mu },$$ induced by an m-tuple $$\pmb \mu =(\mu _1,\ldots ,\mu _{m})$$ of finite non-negative Borel measures on the unit circle. In particular, it is shown that any weighted Dirichlet-type space of order m, for $$m\geqslant 3,$$ forms an algebra under pointwise product. We also prove that every non-zero closed $$M_z$$ -invariant subspace of $${\mathcal {H}}_{\pmb \mu },$$ has codimension 1 property if $$m\geqslant 3$$ or $$\mu _2$$ is finitely supported. As another application of this local Douglas formula obtained in this article, it is shown that for any $$m\geqslant 2,$$ weighted Dirichlet-type space of order m does not coincide with any de Branges–Rovnyak space $${\mathcal {H}}(b)$$ with equivalence of norms.