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Words of analytic paraproducts on Hardy and weighted Bergman spaces

Aleman, Alexandru LU ; Cascante, Carme ; Fàbrega, Joan ; Pascuas, Daniel and Peláez, José Ángel (2024) In Journal des Mathematiques Pures et Appliquees 188. p.179-214
Abstract

For a fixed analytic function g on the unit disc, we consider the analytic paraproducts induced by g, which are formally defined by Tgf(z)=∫0zf(ζ)g′(ζ)dζ, Sgf(z)=∫0zf′(ζ)g(ζ)dζ, and Mgf(z)=g(z)f(z). We are concerned with the study of the boundedness of operators in the algebra Ag generated by the above operators acting on Hardy, or standard weighted Bergman spaces on the disc. The general question is certainly very challenging, since operators in Ag are finite linear combinations of finite products (words) of Tg,Sg,Mg which may involve a large amount of cancellations to be understood.... (More)

For a fixed analytic function g on the unit disc, we consider the analytic paraproducts induced by g, which are formally defined by Tgf(z)=∫0zf(ζ)g′(ζ)dζ, Sgf(z)=∫0zf′(ζ)g(ζ)dζ, and Mgf(z)=g(z)f(z). We are concerned with the study of the boundedness of operators in the algebra Ag generated by the above operators acting on Hardy, or standard weighted Bergman spaces on the disc. The general question is certainly very challenging, since operators in Ag are finite linear combinations of finite products (words) of Tg,Sg,Mg which may involve a large amount of cancellations to be understood. The results in [1] show that boundedness of operators in a fairly large subclass of Ag can be characterized by one of the conditions g∈H∞, or gn belongs to BMOA or the Bloch space, for some integer n>0. However, it is also proved that there are many operators, even single words in Ag whose boundedness cannot be described in terms of these conditions. The present paper provides a considerable progress in this direction. Our main result provides a complete quantitative characterization of the boundedness of an arbitrary word in Ag in terms of a “fractional power” of the symbol g, that only depends on the number of appearances of each of the letters Tg,Sg,Mg in the given word.

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author
; ; ; and
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Analytic paraproduct, Bloch space, BMOA, Hardy spaces, Iterated composition of operators, Weighted Bergman spaces
in
Journal des Mathematiques Pures et Appliquees
volume
188
pages
36 pages
publisher
Elsevier
external identifiers
  • scopus:85195477325
ISSN
0021-7824
DOI
10.1016/j.matpur.2024.05.002
language
English
LU publication?
yes
id
2a1ff8ea-83ec-495e-a239-fa775169dc34
date added to LUP
2024-08-23 16:03:40
date last changed
2024-08-23 16:03:56
@article{2a1ff8ea-83ec-495e-a239-fa775169dc34,
  abstract     = {{<p>For a fixed analytic function g on the unit disc, we consider the analytic paraproducts induced by g, which are formally 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)=g(z)f(z). We are concerned with the study of the boundedness of operators in the algebra A<sub>g</sub> generated by the above operators acting on Hardy, or standard weighted Bergman spaces on the disc. The general question is certainly very challenging, since operators in A<sub>g</sub> are finite linear combinations of finite products (words) of T<sub>g</sub>,S<sub>g</sub>,M<sub>g</sub> which may involve a large amount of cancellations to be understood. The results in [1] show that boundedness of operators in a fairly large subclass of A<sub>g</sub> can be characterized by one of the conditions g∈H<sup>∞</sup>, or g<sup>n</sup> belongs to BMOA or the Bloch space, for some integer n&gt;0. However, it is also proved that there are many operators, even single words in A<sub>g</sub> whose boundedness cannot be described in terms of these conditions. The present paper provides a considerable progress in this direction. Our main result provides a complete quantitative characterization of the boundedness of an arbitrary word in A<sub>g</sub> in terms of a “fractional power” of the symbol g, that only depends on the number of appearances of each of the letters T<sub>g</sub>,S<sub>g</sub>,M<sub>g</sub> in the given word.</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; Hardy spaces; Iterated composition of operators; Weighted Bergman spaces}},
  language     = {{eng}},
  pages        = {{179--214}},
  publisher    = {{Elsevier}},
  series       = {{Journal des Mathematiques Pures et Appliquees}},
  title        = {{Words of analytic paraproducts on Hardy and weighted Bergman spaces}},
  url          = {{http://dx.doi.org/10.1016/j.matpur.2024.05.002}},
  doi          = {{10.1016/j.matpur.2024.05.002}},
  volume       = {{188}},
  year         = {{2024}},
}