Integral-type operators acting on a Hilbert-Bergman space of logarithmic weights on the unit ball

Stevo Stević; Sei-Ichiro Ueki; Stevo Stević; Sei-Ichiro Ueki

doi:10.3934/math.2026368

AIMS Mathematics

2026, Volume 11, Issue 4: 8926-8944. doi: 10.3934/math.2026368

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Integral-type operators acting on a Hilbert-Bergman space of logarithmic weights on the unit ball

Stevo Stević ^{1,2,3
,
,},
Sei-Ichiro Ueki ⁴

1.
Mathematical Institute of the Serbian Academy of Sciences and Arts, Knez Mihailova 36/III, 11000 Beograd, Serbia
2.
Serbian Academy of Sciences and Arts, Knez Mihailova 35, 11000 Beograd, Serbia
3.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
4.
Department of Mathematical Science, Faculty of Engineering, Yokohama National University, Hodogaya, Yokohama, 240–8501, Japan

Received: 03 February 2026 Revised: 16 March 2026 Accepted: 25 March 2026 Published: 01 April 2026
MSC : Primary 47B38; Secondary 47B32, 30H20

For a given holomorphic function $ g $ on the open unit ball in $ \mathbb{C}^{N} $, we consider the following integral operators
$ T_{g}f(z) = \int_{0}^{1}f(tz)\mathcal{R}g(tz)\dfrac{dt}{t} \quad \text{and} \quad I_{g}f(z) = \int_{0}^{1}\mathcal{R}f(tz)g(tz)\dfrac{dt}{t} $
on a Hilbert-Bergman space of logarithmic weights. By using an estimate for the norm of the Hilbert-Bergman space in terms of the radial derivative, we describe necessary and sufficient conditions for the boundedness of the operators. We also estimate the essential norm of these operators via the boundary behavior of some quantities that involve a symbol function $ g $.
- Bounded operator,
- essential norm,
- holomorphic function,
- open unit ball
Citation: Stevo Stević, Sei-Ichiro Ueki. Integral-type operators acting on a Hilbert-Bergman space of logarithmic weights on the unit ball[J]. AIMS Mathematics, 2026, 11(4): 8926-8944. doi: 10.3934/math.2026368

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Abstract

For a given holomorphic function $ g $ on the open unit ball in $ \mathbb{C}^{N} $, we consider the following integral operators

$ T_{g}f(z) = \int_{0}^{1}f(tz)\mathcal{R}g(tz)\dfrac{dt}{t} \quad \text{and} \quad I_{g}f(z) = \int_{0}^{1}\mathcal{R}f(tz)g(tz)\dfrac{dt}{t} $

on a Hilbert-Bergman space of logarithmic weights. By using an estimate for the norm of the Hilbert-Bergman space in terms of the radial derivative, we describe necessary and sufficient conditions for the boundedness of the operators. We also estimate the essential norm of these operators via the boundary behavior of some quantities that involve a symbol function $ g $.

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