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

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 $\mathcal H_{\pmbμ},$ induced by an $m$-tuple $\pmb μ=(μ_1,\ldots,μ_{m})$ 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 $\mathcal H_{\pmbμ},$ has codimension $1$ property if $m\geqslant 3$ or $μ_2$ 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.

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