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.].