Composition of analytic paraproducts
(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
- Aleman, Alexandru LU ; Cascante, Carme ; Fàbrega, Joan ; Pascuas, Daniel and Peláez, José Ángel
- organization
- publishing date
- 2022
- 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}}, }