A local Douglas formula for higher order weighted Dirichlet-type integrals

TL;DR

This paper establishes a local Douglas formula for higher order weighted Dirichlet-type integrals, explores their algebraic and invariant subspace properties, and distinguishes these spaces from de Branges-Rovnyak spaces.

Contribution

It introduces a local Douglas formula for higher order weighted Dirichlet spaces and analyzes their algebraic structure and invariant subspaces, providing new insights into their functional properties.

Findings

Higher order weighted Dirichlet spaces form an algebra under pointwise multiplication for m ≥ 3.

Non-zero closed M_z-invariant subspaces have codimension 1 under certain conditions.

These spaces are not equivalent to any de Branges-Rovnyak space.

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 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.