Weighted norm inequalities for derivatives on Bergman spaces

Peláez, José Ángel; Rättyä, Jouni

doi:10.5802/aif.3632

Weighted norm inequalities for derivatives on Bergman spaces
[Inégalités de normes pondérées pour les dérivées dans les espaces de Bergman]

Peláez, José Ángel ¹ ; Rättyä, Jouni ²

¹ Departamento de Análisis Matemático, Universidad de Málaga, Campus de Teatinos, 29071 Málaga, (Spain)
² University of Eastern Finland, P.O.Box 111, 80101 Joensuu (Finland)

Annales de l'Institut Fourier, Online first, 24 p.

Résumé
Abstract

Nous construisons une norme équivalente, définie à l’aide des dérivées supérieures, dans un espace de Bergman pondéré $A_{ω}^{p}$ où $ω$ appartient à une large classe des poids non radiaux. Nous analysons aussi autres inégalités de Littlewood–Paley. Avant de démontrer les résultats principaux nous caractérisons les $q$ -mesures de Carleson sur les espaces $A_{ω}^{p}$ et montrons que la fonction maximale de Hörmander est bornée. En utilisant nos résultats nous pouvons décrire l’ensemble résolvant de l’opérateur intégral $T_{g} (f) (z) = \int_{0}^{z} g^{'} (ζ) f (ζ) d ζ$ agissant sur $A_{ω}^{p}$ .

An equivalent norm in the weighted Bergman space $A_{ω}^{p}$ , induced by an $ω$ in a certain large class of non-radial weights, is established in terms of higher order derivatives. Other Littlewood–Paley inequalities are also considered. On the way to the proofs, we characterize the $q$ -Carleson measures for the weighted Bergman space $A_{ω}^{p}$ and the boundedness of a Hörmander-type maximal function. Results obtained are further applied to describe the resolvent set of the integral operators $T_{g} (f) (z) = \int_{0}^{z} g^{'} (ζ) f (ζ) d ζ$ acting on $A_{ω}^{p}$ .

Reçu le : 2020-07-03
Révisé le : 2022-07-11
Accepté le : 2022-11-17
Première publication : 2024-05-17

DOI : 10.5802/aif.3632

Classification : 30H20, 47G10
Keywords: Bergman space, Carleson measure, integral operator, Littlewood–Paley inequality, Hörmander-type maximal function, resolvent set.
Mot clés : Espace de Bergman, mesure de Carleson, opérateur intégral, inégalité de Littlewood–Paley, fonction maximale de Hörmander, ensemble résolvant.

Affiliations des auteurs :

Peláez, José Ángel ¹ ; Rättyä, Jouni ²

¹ Departamento de Análisis Matemático, Universidad de Málaga, Campus de Teatinos, 29071 Málaga, (Spain)
² University of Eastern Finland, P.O.Box 111, 80101 Joensuu (Finland)

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     author = {Pel\'aez, Jos\'e \'Angel and R\"atty\"a, Jouni},
     title = {Weighted norm inequalities for derivatives on {Bergman} spaces},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2024},
     doi = {10.5802/aif.3632},
     language = {en},
     note = {Online first},
}

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Peláez, José Ángel; Rättyä, Jouni. Weighted norm inequalities for derivatives on Bergman spaces. Annales de l'Institut Fourier, Online first, 24 p.

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