Riemann-Stieltjes operators between Bergman-type spaces and
<mml:math id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi></mml:math>-Bloch spaces

Li, Songxiao

doi:https://doi.org/10.1155/IJMMS/2006/86259

International Journal of Mathematics and Mathematical Sciences

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Volume 2006 | Article ID 086259 | https://doi.org/10.1155/IJMMS/2006/86259

Riemann-Stieltjes operators between Bergman-type spaces and α-Bloch spaces

Songxiao Li^1,2

Received08 Dec 2005

Revised11 May 2006

Accepted28 May 2006

Published03 Oct 2006

Abstract

We study the following integral operators: Jgf(z)=∫0zf(ξ)g′(ξ)dξ; Igf(z)=∫0zf′(ξ)g(ξ)dξ, where g is an analytic function on the open unit disk in the complex plane. The boundedness and compactness of Jg, Ig between the Bergman-type spaces and the α-Bloch spaces are investigated.

References

A. Aleman and J. A. Cima, “An integral operator on $H^p$ and Hardy's inequality,” Journal d'Analyse Mathématique, vol. 85, pp. 157–176, 2001.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
A. Aleman and A. G. Siskakis, “An integral operator on $H^p$ ,” Complex Variables, vol. 28, no. 2, pp. 149–158, 1995.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
A. Aleman and A. G. Siskakis, “Integration operators on Bergman spaces,” Indiana University Mathematics Journal, vol. 46, no. 2, pp. 337–356, 1997.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. Arazy, S. D. Fisher, and J. Peetre, “Möbius invariant function spaces,” Journal für die reine und angewandte Mathematik, vol. 363, pp. 110–145, 1985.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics, CRC Press, Florida, 1995.
View at: Zentralblatt MATH | MathSciNet
G. H. Hardy and J. E. Littlewood, “Some properties of fractional integrals II,” Mathematische Zeitschrift, vol. 34, no. 1, pp. 403–439, 1932.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Z. J. Hu, “Extended Cesàro operators on mixed norm spaces,” Proceedings of the American Mathematical Society, vol. 131, no. 7, pp. 2171–2179, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
K. Madigan and A. Matheson, “Compact composition operators on the Bloch space,” Transactions of the American Mathematical Society, vol. 347, no. 7, pp. 2679–2687, 1995.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. Miao, “The Cesàro operator is bounded on $H^p$ for $0 < p <1$ ,” Proceedings of the American Mathematical Society, vol. 116, no. 4, pp. 1077–1079, 1992.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
F. Pérez-González and J. Rättyä, “Forelli-Rudin estimates, Carleson measures and $F(p,q,s)$ -functions,” Journal of Mathematical Analysis and Applications, vol. 315, no. 2, pp. 394–414, 2006.
View at: Google Scholar | MathSciNet
C. Pommerenke, “Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation,” Commentarii Mathematici Helvetici, vol. 52, no. 4, pp. 591–602, 1977.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
W. Rudin, Function Theory in the Unit Ball of $\mathbb{C}^n$ , vol. 241 of Fundamental Principles of Mathematical Science, Springer, New York, 1980.
View at: Zentralblatt MATH | MathSciNet
J.-H. Shi and G.-B. Ren, “Boundedness of the Cesàro operator on mixed norm spaces,” Proceedings of the American Mathematical Society, vol. 126, no. 12, pp. 3553–3560, 1998.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. L. Shields and D. L. Williams, “Bounded projections, duality, and multipliers in spaces of analytic functions,” Transactions of the American Mathematical Society, vol. 162, pp. 287–302, 1971.
View at: Publisher Site | Google Scholar | MathSciNet
A. G. Siskakis, “Composition semigroups and the Cesàro operator on $H^p$ ,” Journal of the London Mathematical Society. Second Series, vol. 36, no. 1, pp. 153–164, 1987.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
A. G. Siskakis, “The Cesàro operator is bounded on $H^1$ ,” Proceedings of the American Mathematical Society, vol. 110, no. 2, pp. 461–462, 1990.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. G. Siskakis and R. Zhao, “A Volterra type operator on spaces of analytic functions,” in Function Spaces (Edwardsville, IL, 1998), vol. 232 of Contemp. Math., pp. 299–311, American Mathematical Society, Rhode Island, 1999.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
S. Stević, “The generalized Cesàro operator on Dirichlet spaces,” Studia Scientiarum Mathematicarum Hungarica, vol. 40, no. 1-2, pp. 83–94, 2003.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
S. Stević, “On an integral operator on the unit ball in $\mathbb{C}^n$ ,” Journal of Inequalities and Applications, vol. 2005, no. 1, pp. 81–88, 2005.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. Xiao, “Cesàro-type operators on Hardy, BMOA and Bloch spaces,” Archiv der Mathematik, vol. 68, no. 5, pp. 398–406, 1997.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. Xiao, “Riemann-Stieltjes operators on weighted Bloch and Bergman spaces of the unit ball,” Journal of the London Mathematical Society. Second Series, vol. 70, no. 1, pp. 199–214, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. Yoneda, “Pointwise multipliers from ${\mathrm{BMOA}}^\alpha$ to ${\mathrm{BMOA}}^\beta$ ,” Complex Variables, vol. 49, no. 14, pp. 1045–1061, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
X. Zhang, J. Xiao, and Z. Hu, “The multipliers between the mixed norm spaces in $\mathbb{C}^n$ ,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 664–674, 2005.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. Zhao, “Composition operators from Bloch type spaces to Hardy and Besov spaces,” Journal of Mathematical Analysis and Applications, vol. 233, no. 2, pp. 749–766, 1999.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
K. H. Zhu, Operator Theory in Function Spaces, vol. 139 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, 1990.
View at: Zentralblatt MATH | MathSciNet

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Copyright © 2006 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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