Revista Matemática Iberoamericana

Integration Operators on Bergman Spaces with exponential weight

Milutin R. Dostanić

Full-text: Open access

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


Export citation

References

  • Aleman, A. and Siskakis, A. G.: An integral operator on $H^p$. Complex Variables Theory Appl. 28 (1995), no. 2, 149-158.
  • Aleman, A. and Siskakis, A. G.: Integration Operators on Bergman Spaces. Indiana Univ. Math. J. 46 (1997), no. 2, 337-356.
  • Aleman, A. and Cima, J. A.: An integral operator on $H^p$ and Hardy's inequality. J. Anal. Math. 85 (2001), 157-176.
  • Dzhrbashyan, M. M.: Integral Transforms and Representation of Function in Complex Domain. Izdat. ``Nauka'', Moscow, 1966.
  • Fedoryuk, M. V.: Asymptotics Integrals and Series (Russian). Nauka, Moscow, 1987.
  • Hedenmalm, H., Korenblum, B. and Zhu, K.: Theory of Bergman Spaces. Graduate Texts in Mathematics 199. Springer, New York, 2000.
  • Oleinik, V. L.: Imbedding theorems for weighted classes of harmonic and analytic functions. Investigations on linear operators and the theory of functions, V. Zap.Nauč. Sem. Lenningrad. Otdel. Math. Inst. Steklov. (LOMI) 47 (1974), 120-137 (in Russian).
  • Pommerenke, Ch.: Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation. Comment. Math. Helv. 52 (1977), no. 4, 591-602.
  • Postnikov, A. G.: Introduction to the analytic number theory. Izdat. ``Nauka'', Moscow, 1971.
  • Ingham, A. E.: A Tauberian theorem for partitions. Ann. of Math. (2) 42 (1941), 1075-1090.