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

Autor: Soumitra Ghara, Rajeev Gupta, Md. Ramiz Reza
Rok vydání: 2023
Předmět:
Zdroj: The Journal of Geometric Analysis. 33
ISSN: 1559-002X
1050-6926
DOI: 10.1007/s12220-023-01297-8
Popis: 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.
Comment: 21 pages
Databáze: OpenAIRE
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