Abstract and Applied Analysis

On the Generalized Hardy Spaces

M. Fatehi

Full-text: Open access

Abstract

We introduce new spaces that are extensions of the Hardy spaces and we investigate the continuity of the point evaluations as well as the boundedness and the compactness of the composition operators on these spaces.

Article information

Source
Abstr. Appl. Anal., Volume 2010 (2010), Article ID 803230, 14 pages.

Dates
First available in Project Euclid: 2 March 2010

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1267538588

Digital Object Identifier
doi:10.1155/2010/803230

Mathematical Reviews number (MathSciNet)
MR2587612

Zentralblatt MATH identifier
1191.30021

Citation

Fatehi, M. On the Generalized Hardy Spaces. Abstr. Appl. Anal. 2010 (2010), Article ID 803230, 14 pages. doi:10.1155/2010/803230. https://projecteuclid.org/euclid.aaa/1267538588


Export citation

References

  • P. L. Duren, Theory of H$^p$ Spaces, vol. 38 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1970.
  • C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, Fla, USA, 1995.
  • J. V. Ryff, “Subordinate $H^p$ functions,” Duke Mathematical Journal, vol. 33, pp. 347–354, 1966.
  • H. J. Schwartz, Composition operators on H$^p$, Ph.D. thesis, University of Toledo, 1969.
  • J. H. Shapiro and P. D. Taylor, “Compact, nuclear, and Hilbert-Schmidt composition operators on $H^2$,” Indiana University Mathematics Journal, vol. 23, pp. 471–496, 1973.
  • J. H. Shapiro, Composition Operators and Classical Function Theory, Universitext: Tracts in Mathematics, Springer, New York, NY, USA, 1993.
  • B. R. Choe, H. Koo, and W. Smith, “Composition operators acting on holomorphic Sobolev spaces,” Transactions of the American Mathematical Society, vol. 355, no. 7, pp. 2829–2855, 2003.
  • B. R. Choe, H. Koo, and W. Smith, “Composition operators on small spaces,” Integral Equations and Operator Theory, vol. 56, no. 3, pp. 357–380, 2006.
  • D. D. Clahane and S. Stević, “Norm equivalence and composition operators between Bloch/Lipschitz spaces of the ball,” Journal of Inequalities and Applications, vol. 2006, Article ID 61018, 11 pages, 2006.
  • C. C. Cowen and B. D. MacCluer, “Linear fractional maps of the ball and their composition operators,” Acta Scientiarum Mathematicarum, vol. 66, no. 1-2, pp. 351–376, 2000.
  • X. Fu and X. Zhu, “Weighted composition operators on some weighted spaces in the unit ball,” Abstract and Applied Analysis, vol. 2008, Article ID 605807, 8 pages, 2008.
  • S. Li and S. Stević, “Weighted composition operators from Bergman-type spaces into Bloch spaces,” Proceedings of the Indian Academy of Sciences. Mathematical Sciences, vol. 117, no. 3, pp. 371–385, 2007.
  • S. Li and S. Stević, “Weighted composition operators from $\alpha $-Bloch space to $H^\infty $ on the polydisc,” Numerical Functional Analysis and Optimization, vol. 28, no. 7-8, pp. 911–925, 2007.
  • S. Li and S. Stević, “Weighted composition operators from $H^\infty $ to the Bloch space on the polydisc,” Abstract and Applied Analysis, vol. 2007, Article ID 48478, 13 pages, 2007.
  • S. Li and S. Stević, “Weighted composition operators from Zygmund spaces into Bloch spaces,” Applied Mathematics and Computation, vol. 206, no. 2, pp. 825–831, 2008.
  • B. D. MacCluer and J. H. Shapiro, “Angular derivatives and compact composition operators on the Hardy and Bergman spaces,” Canadian Journal of Mathematics, vol. 38, no. 4, pp. 878–906, 1986.
  • S. Stević, “On generalized weighted Bergman spaces,” Complex Variables. Theory and Application, vol. 49, no. 2, pp. 109–124, 2004.
  • S. Stević, “Composition operators between $H^\infty $ and $\alpha $-Bloch spaces on the polydisc,” Zeitschrift für Analysis und ihre Anwendungen, vol. 25, no. 4, pp. 457–466, 2006.
  • S. Stević, “Weighted composition operators between mixed norm spaces and $H_\alpha ^\infty $ spaces in the unit ball,” Journal of Inequalities and Applications, vol. 2007, Article ID 28629, 9 pages, 2007.
  • S. Stević, “Essential norms of weighted composition operators from the $\alpha $-Bloch space to a weighted-type space on the unit ball,” Abstract and Applied Analysis, vol. 2008, Article ID 279691, 11 pages, 2008.
  • S. Stević, “Norm of weighted composition operators from Bloch space to $H_\mu ^\infty $ on the unit ball,” Ars Combinatoria, vol. 88, pp. 125–127, 2008.
  • S. Stević, “Composition operators from the Hardy space to the Zygmund-type space on the upper half-plane,” Abstract and Applied Analysis, vol. 2009, Article ID 161528, 8 pages, 2009.
  • S. Ueki, “Composition operators on the Privalov spaces of the unit ball of $\mathbbC^n$,” Journal of the Korean Mathematical Society, vol. 42, no. 1, pp. 111–127, 2005.
  • S. Ueki, “Weighted composition operators on the Bargman-Fock space,” International Journal of Modern Mathematics, vol. 3, no. 3, pp. 231–243, 2008.
  • S. Ueki and L. Luo, “Compact weighted composition operators and multiplication operators between Hardy spaces,” Abstract and Applied Analysis, vol. 2008, Article ID 196498, 12 pages, 2008.
  • S. Ye, “Weighted composition operator between the little $\alpha $-Bloch spaces and the logarithmic Bloch,” Journal of Computational Analysis and Applications, vol. 10, no. 2, pp. 243–252, 2008.
  • K. Zhu, “Compact composition operators on Bergman spaces of the unit ball,” Houston Journal of Mathematics, vol. 33, no. 1, pp. 273–283, 2007.
  • X. Zhu, “Weighted composition operators from logarithmic Bloch spaces to a class of weighted-type spaces in the unit ball,” Ars Combinatoria, vol. 91, pp. 87–95, 2009.
  • K. Zhu, Operator Theory in Function Spaces, vol. 139 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1990.
  • B. D. MacCluer, “Compact composition operators on $H^p(B_N)$,” The Michigan Mathematical Journal, vol. 32, no. 2, pp. 237–248, 1985.
  • W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, NY, USA, 3rd edition, 1987.
  • R. G. Bartle, The Elements of Real Analysis, John Wiley & Sons, New York, NY, USA, 2nd edition, 1976.
  • P. R. Halmos, Measure Theory, Springer, New York, NY, USA, 1974.