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Weighted conformal invariance of Banach spaces of analytic functions

Aleman, Alexandru LU and Mas, Alejandro LU (2021) In Journal of Functional Analysis 280(9).
Abstract

We consider Banach spaces of analytic functions in the unit disc which satisfy a weighted conformal invariance property, that is, for a fixed α>0 and every conformal automorphism φ of the disc, f→f∘φ(φ′)α defines a bounded linear operator on the space in question, and the family of all such operators is uniformly bounded in operator norm. Many common examples of Banach spaces of analytic functions like Korenblum growth classes, Hardy spaces, standard weighted Bergman and certain Besov spaces satisfy this condition. The aim of the paper is to develop a general approach to the study of such spaces based on this property alone. We consider polynomial approximation, duality and complex interpolation, we identify the... (More)

We consider Banach spaces of analytic functions in the unit disc which satisfy a weighted conformal invariance property, that is, for a fixed α>0 and every conformal automorphism φ of the disc, f→f∘φ(φ′)α defines a bounded linear operator on the space in question, and the family of all such operators is uniformly bounded in operator norm. Many common examples of Banach spaces of analytic functions like Korenblum growth classes, Hardy spaces, standard weighted Bergman and certain Besov spaces satisfy this condition. The aim of the paper is to develop a general approach to the study of such spaces based on this property alone. We consider polynomial approximation, duality and complex interpolation, we identify the largest and the smallest as well as the “unique” Hilbert space satisfying this property for a given α>0. We investigate the weighted conformal invariance of the space of derivatives, or anti-derivatives with the induced norm, and arrive at the surprising conclusion that they depend entirely on the properties of the (modified) Cesàro operator acting on the original space. Finally, we prove that this last result implies a John-Nirenberg type estimate for analytic functions g with the property that the integration operator f→∫0zf(t)g′(t)dt is bounded on a Banach space satisfying the weighted conformal invariance property.

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publishing date
type
Contribution to journal
publication status
published
subject
keywords
Banach space, Integration operator, Weighted composition operator, Weighted conformal invariance
in
Journal of Functional Analysis
volume
280
issue
9
article number
108946
publisher
Elsevier
external identifiers
  • scopus:85100492791
ISSN
0022-1236
DOI
10.1016/j.jfa.2021.108946
language
English
LU publication?
yes
id
ec583426-bafb-4849-b4d3-e4bf2afd7444
date added to LUP
2021-02-16 11:07:49
date last changed
2022-04-27 00:17:05
@article{ec583426-bafb-4849-b4d3-e4bf2afd7444,
  abstract     = {{<p>We consider Banach spaces of analytic functions in the unit disc which satisfy a weighted conformal invariance property, that is, for a fixed α&gt;0 and every conformal automorphism φ of the disc, f→f∘φ(φ<sup>′</sup>)<sup>α</sup> defines a bounded linear operator on the space in question, and the family of all such operators is uniformly bounded in operator norm. Many common examples of Banach spaces of analytic functions like Korenblum growth classes, Hardy spaces, standard weighted Bergman and certain Besov spaces satisfy this condition. The aim of the paper is to develop a general approach to the study of such spaces based on this property alone. We consider polynomial approximation, duality and complex interpolation, we identify the largest and the smallest as well as the “unique” Hilbert space satisfying this property for a given α&gt;0. We investigate the weighted conformal invariance of the space of derivatives, or anti-derivatives with the induced norm, and arrive at the surprising conclusion that they depend entirely on the properties of the (modified) Cesàro operator acting on the original space. Finally, we prove that this last result implies a John-Nirenberg type estimate for analytic functions g with the property that the integration operator f→∫<sub>0</sub><sup>z</sup>f(t)g<sup>′</sup>(t)dt is bounded on a Banach space satisfying the weighted conformal invariance property.</p>}},
  author       = {{Aleman, Alexandru and Mas, Alejandro}},
  issn         = {{0022-1236}},
  keywords     = {{Banach space; Integration operator; Weighted composition operator; Weighted conformal invariance}},
  language     = {{eng}},
  number       = {{9}},
  publisher    = {{Elsevier}},
  series       = {{Journal of Functional Analysis}},
  title        = {{Weighted conformal invariance of Banach spaces of analytic functions}},
  url          = {{http://dx.doi.org/10.1016/j.jfa.2021.108946}},
  doi          = {{10.1016/j.jfa.2021.108946}},
  volume       = {{280}},
  year         = {{2021}},
}