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arXiv · 2509.19714

Higher order weighted Dirichlet type spaces with poly-superharmonic weights and Dirichlet type operators of finite order

Ashish Kujur·Md. Ramiz Reza
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Abstract

We study higher-order weighted Dirichlet-type spaces on the unit disc associated with a class of poly-superharmonic weights. A higher-order Littlewood Paley formula is established enabling the computation of higher-order weighted Dirichlet integrals and allowing us to relate iterates of the Laplacian of the weight to higher-order defect operators of the shift operator on these spaces. This leads to the introduction of Dirichlet-type operators of finite order, a class containing $m$-isometries as well as completely hyperexpansive and completely hypercontractive operators of finite order. We prove that every cyclic operator in this class admits a functional model as the shift on a suitable higher-order weighted Dirichlet-type space, thereby providing a unified extension of the model theories for cyclic completely hyperexpansive operators and cyclic $m$-isometries.

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Ashish Kujur, Md. Ramiz Reza. 2025-09-24. Higher order weighted Dirichlet type spaces with poly-superharmonic weights and Dirichlet type operators of finite order. https://arxiv.org/abs/2509.19714

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