Revista Matemática Iberoamericana

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Volume 23, Issue 2, 2007, pp. 421–436
DOI: 10.4171/RMI/501

Published online: 2007-08-31

Integration Operators on Bergman Spaces with exponential weight

Milutin R. Dostanić[1]

(1) University of Belgrade, Beograd, Serbia

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.].

Keywords: Weighted Bergman’s space, radial weight function, Tauberian theorem of Ingham

Dostanić Milutin: Integration Operators on Bergman Spaces with exponential weight. Rev. Mat. Iberoam. 23 (2007), 421-436. doi: 10.4171/RMI/501