Revista Matemática Iberoamericana
- Rev. Mat. Iberoamericana
- Volume 23, Number 2 (2007), 421-436.
Integration Operators on Bergman Spaces with exponential weight
Abstract
We study operators of the form $T_{g}f\left( z\right) =\int\nolimits_{0}^{z}f\left( \xi \right) \,g^{\prime }\left( \xi \right) \,d\left( \xi \right) $ ($g$ is an analytic function unity disc) on weighted Bergman spaces $L_{a}^{p}\left( w\right) $ of the unit disc where symbol $g$ is analytic function on the disc. For the case of $$ w(r) =\exp \Big( \frac{-a}{( 1-r)^{\beta }}\Big)\qquad \left( a>0, 0<\beta \leq 1\right) $$ it is shown that operator $T_{g}$ is bounded (compact) on $L_{a}^{2}\left( w\right) $ if and only if $\left( 1-\left\vert z\right\vert \right)^{\beta +1}\left\vert g^{\prime }\left( z\right) \right\vert =O\left( 1\right) \left( =o\left( 1\right) \right) $ as $\left\vert z\right\vert \rightarrow 1-$, thus solving a problem formulated in [Aleman, A. and Siskakis, A.G.: Integration Operators on Bergman Spaces. Indiana Univ. Math. J. 46 (1997), no. 2, 337-356.].
Article information
Source
Rev. Mat. Iberoamericana, Volume 23, Number 2 (2007), 421-436.
Dates
First available in Project Euclid: 26 September 2007
Permanent link to this document
https://projecteuclid.org/euclid.rmi/1190831217
Mathematical Reviews number (MathSciNet)
MR2371433
Zentralblatt MATH identifier
1146.47020
Subjects
Primary: 47B38: Operators on function spaces (general)
Keywords
weighted Bergman's space radial weight function tauberian theorem of Ingham
Citation
Dostanić, Milutin R. Integration Operators on Bergman Spaces with exponential weight. Rev. Mat. Iberoamericana 23 (2007), no. 2, 421--436. https://projecteuclid.org/euclid.rmi/1190831217
