Integration Operators on Bergman Spaces with exponential weight

  • Milutin R. Dostanić

    University of Belgrade, Beograd, Serbia


We study operators of the form Tgf(z)=0zf(ξ)g(ξ)d(ξ)T_{g}f\left( z\right) =\int\nolimits_{0}^{z}f\left( \xi \right) \,g^{\prime }\left( \xi \right) \,d\left( \xi \right) (gg is an analytic function unity disc) on weighted Bergman spaces Lap(w)L_{a}^{p}\left( w\right) of the unit disc where symbol gg is analytic function on the disc. For the case of

w(r)=exp(a(1r)β)(a>0,0<β1)w(r) =\exp \Big( \frac{-a}{( 1-r)^{\beta }}\Big)\qquad \left( a>0, 0<\beta \leq 1\right)

it is shown that operator TgT_{g} is bounded (compact) on La2(w)L_{a}^{2}\left( w\right) if and only if (1z)β+1g(z)=O(1)(=o(1))\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 z1\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.].

Cite this article

Milutin R. Dostanić, Integration Operators on Bergman Spaces with exponential weight. Rev. Mat. Iberoam. 23 (2007), no. 2, pp. 421–436

DOI 10.4171/RMI/501