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Composition of analytic paraproducts

Aleman, Alexandru LU ; Cascante, Carme ; Fàbrega, Joan ; Pascuas, Daniel and Peláez, José Ángel (2022) In Journal des Mathematiques Pures et Appliquees 158. p.293-319
Abstract

For a fixed analytic function g on the unit disc D, we consider the analytic paraproducts induced by g, which are defined by Tgf(z)=∫0zf(ζ)g′(ζ)dζ, Sgf(z)=∫0zf′(ζ)g(ζ)dζ, and Mgf(z)=f(z)g(z). The boundedness of these operators on various spaces of analytic functions on D is well understood. The original motivation for this work is to understand the boundedness of compositions of two of these operators, for example Tg2,TgSg,MgTg, etc. Our methods yield a characterization of the boundedness of a large class of operators contained in the algebra generated by these analytic... (More)

For a fixed analytic function g on the unit disc D, we consider the analytic paraproducts induced by g, which are defined by Tgf(z)=∫0zf(ζ)g′(ζ)dζ, Sgf(z)=∫0zf′(ζ)g(ζ)dζ, and Mgf(z)=f(z)g(z). The boundedness of these operators on various spaces of analytic functions on D is well understood. The original motivation for this work is to understand the boundedness of compositions of two of these operators, for example Tg2,TgSg,MgTg, etc. Our methods yield a characterization of the boundedness of a large class of operators contained in the algebra generated by these analytic paraproducts acting on the classical weighted Bergman and Hardy spaces in terms of the symbol g. In some cases it turns out that this property is not affected by cancellation, while in others it requires stronger and more subtle restrictions on the oscillation of the symbol g than the case of a single paraproduct.

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author
; ; ; and
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Analytic paraproduct, Bloch space, BMOA space, Hardy spaces, Weighted Bergman spaces
in
Journal des Mathematiques Pures et Appliquees
volume
158
pages
293 - 319
publisher
Elsevier
external identifiers
  • scopus:85120667301
ISSN
0021-7824
DOI
10.1016/j.matpur.2021.11.007
language
English
LU publication?
yes
id
04be51b7-2dc9-42d6-81ab-2997d91de9c8
date added to LUP
2022-01-18 13:50:21
date last changed
2022-06-29 21:04:57
@article{04be51b7-2dc9-42d6-81ab-2997d91de9c8,
  abstract     = {{<p>For a fixed analytic function g on the unit disc D, we consider the analytic paraproducts induced by g, which are defined by T<sub>g</sub>f(z)=∫<sub>0</sub><sup>z</sup>f(ζ)g<sup>′</sup>(ζ)dζ, S<sub>g</sub>f(z)=∫<sub>0</sub><sup>z</sup>f<sup>′</sup>(ζ)g(ζ)dζ, and M<sub>g</sub>f(z)=f(z)g(z). The boundedness of these operators on various spaces of analytic functions on D is well understood. The original motivation for this work is to understand the boundedness of compositions of two of these operators, for example T<sub>g</sub><sup>2</sup>,T<sub>g</sub>S<sub>g</sub>,M<sub>g</sub>T<sub>g</sub>, etc. Our methods yield a characterization of the boundedness of a large class of operators contained in the algebra generated by these analytic paraproducts acting on the classical weighted Bergman and Hardy spaces in terms of the symbol g. In some cases it turns out that this property is not affected by cancellation, while in others it requires stronger and more subtle restrictions on the oscillation of the symbol g than the case of a single paraproduct.</p>}},
  author       = {{Aleman, Alexandru and Cascante, Carme and Fàbrega, Joan and Pascuas, Daniel and Peláez, José Ángel}},
  issn         = {{0021-7824}},
  keywords     = {{Analytic paraproduct; Bloch space; BMOA space; Hardy spaces; Weighted Bergman spaces}},
  language     = {{eng}},
  pages        = {{293--319}},
  publisher    = {{Elsevier}},
  series       = {{Journal des Mathematiques Pures et Appliquees}},
  title        = {{Composition of analytic paraproducts}},
  url          = {{http://dx.doi.org/10.1016/j.matpur.2021.11.007}},
  doi          = {{10.1016/j.matpur.2021.11.007}},
  volume       = {{158}},
  year         = {{2022}},
}