Corona and Gleason problems for certain Banach spaces of holomorphic functions

We use a combination of the classical dead-end space method and the interpolation (by bounded holomorphic functions) method to affirmatively solve Corona and Gleason problems for familiar Banach spaces of holomorphic functions such as (Besov-)Lipschitz spaces of holomorphic functions on bounded symmetric domains (including the unit ball and polydisc) in $\mathbb{C}^n$, holomorphic mean (Besov-)Lipschitz spaces, (Besov-)Hardy-Sobolev spaces and some special classes of holomorphic mixed-norm (Bergman) spaces (including weighted Bergman spaces) as well as some special classes of holomorphic (mean) Besov-Lipschitz spaces on the unit ball in $\mathbb{C}^n$. Indeed, we extend our recent work on both problems for holomorphic Lipschitz algebras.

Authors

Publication Details

Journal
arXiv (Cornell University)
Published
2026-11-02
DOI
https://doi.org/10.48550/arxiv.2611.02949
Primary Topic
Holomorphic and Operator Theory
Type
preprint
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preprint

Corona and Gleason problems for certain Banach spaces of holomorphic functions

S. R. Patel
arXiv (Cornell University)
Holomorphic and Operator Theory
preprint

Corona and Gleason problems for certain Banach spaces of holomorphic functions

S. R. Patel
preprint en

Abstract

We use a combination of the classical dead-end space method and the interpolation (by bounded holomorphic functions) method to affirmatively solve Corona and Gleason problems for familiar Banach spaces of holomorphic functions such as (Besov-)Lipschitz spaces of holomorphic functions on bounded symmetric domains (including the unit ball and polydisc) in $\mathbb{C}^n$, holomorphic mean (Besov-)Lipschitz spaces, (Besov-)Hardy-Sobolev spaces and some special classes of holomorphic mixed-norm (Bergman) spaces (including weighted Bergman spaces) as well as some special classes of holomorphic (mean) Besov-Lipschitz spaces on the unit ball in $\mathbb{C}^n$. Indeed, we extend our recent work on both problems for holomorphic Lipschitz algebras.

arXiv (Cornell University)
Holomorphic and Operator Theory
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