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 ${\mathcal{H}}_{\mu },$ induced by an $m$-tuple $\mu =({\mu }_1,\dots ,{\mu }_m)$ 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 $m⩾3,$ forms an algebra under pointwise product. We also prove that every non-zero closed $M_z$-invariant subspace of ${\mathcal{H}}_{\mu },$ has codimension $1$ property if $m⩾3$ or ${\mu }_2$ is finitely supported. As another application of local Douglas formula obtained in this article, it is shown that for any $m⩾2,$ weighted Dirichlet-type space of order $m$ does not coincide with any de Branges-Rovnyak space $\mathcal{H}(b)$ with equivalence of norms.

21 pages