Abstract and Applied Analysis

Necessary and Sufficient Conditions for Boundedness of Commutators of the General Fractional Integral Operators on Weighted Morrey Spaces

Abstract

We prove that $b$ is in $\text{L}\text{i}{\text{p}}_{\beta }(\omega )$ if and only if the commutator $[b,{L}^{-\alpha /2}]$ of the multiplication operator by $b$ and the general fractional integral operator ${L}^{-\alpha /2}$ is bounded from the weighted Morrey space ${L}^{p,k}(\omega )$ to ${L}^{q,kq/p}({\omega }^{1-(1-\alpha /n)q},\omega )$, where $0<\beta <1$, $0<\alpha +\beta , $1/q=1/p-(\alpha +\beta )/n$, $0\le k, ${\omega }^{q/p}\in {A}_{1,}$ and ${r}_{\omega }>(1-\mathrm{k)}/\mathrm{(p}/(q-k))$, and here ${r}_{\omega }$ denotes the critical index of $\omega$ for the reverse Hölder condition.

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 929381, 14 pages.

Dates
First available in Project Euclid: 14 December 2012

https://projecteuclid.org/euclid.aaa/1355495851

Digital Object Identifier
doi:10.1155/2012/929381

Mathematical Reviews number (MathSciNet)
MR2965481

Zentralblatt MATH identifier
1256.42024

Citation

Si, Zengyan; Zhao, Fayou. Necessary and Sufficient Conditions for Boundedness of Commutators of the General Fractional Integral Operators on Weighted Morrey Spaces. Abstr. Appl. Anal. 2012 (2012), Article ID 929381, 14 pages. doi:10.1155/2012/929381. https://projecteuclid.org/euclid.aaa/1355495851

References

• X. T. Duong and L. X. Yan, “On commutators of fractional integrals,” Proceedings of the American Mathematical Society, vol. 132, no. 12, pp. 3549–3557, 2004.
• M. Paluszyński, “Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss,” Indiana University Mathematics Journal, vol. 44, no. 1, pp. 1–17, 1995.
• S. Shirai, “Necessary and sufficient conditions for boundedness of commutators of fractional integral operators on classical Morrey spaces,” Hokkaido Mathematical Journal, vol. 35, no. 3, pp. 683–696, 2006.
• H. Wang, “On some commutator theorems for fractional integral operators on the weighted morrey spacesčommentComment on ref. [12?]: Please update the information of this reference, if possible.,” http://128.84.158.119/abs/1010.2638v1.
• Y. Komori and S. Shirai, “Weighted Morrey spaces and a singular integral operator,” Mathematische Nachrichten, vol. 282, no. 2, pp. 219–231, 2009.
• H. Wang, “Some estimates for the commutators of fractional integrals associated to operators with Gaussian kenerl boundsčommentComment on ref. [13?]: Please update the information of this reference, if possible.,” http://xxx.tau.ac.il/abs/1102.4380v1.
• J. M. Martell, “Sharp maximal functions associated with approximations of the identity in spaces of homogeneous type and applications,” Studia Mathematica, vol. 161, no. 2, pp. 113–145, 2004.
• J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, vol. 116 of North-Holland Mathematics Studies, North-Holland Publishing, Amsterdam, The Netherlands, 1985.
• E. M. Stein and T. S. Murphy, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43 of Monographs in Harmonic Analysis, Princeton University Press, Princeton, NJ, USA, 1993.
• A. Torchinsky, Real-Variable Methods in Harmonic Analysis, vol. 123, Academic Press, Orlando, Fla, USA, 1986.
• J. García-Cuerva, “Weighted ${H}^{p}$ spaces,” Dissertations Math, vol. 162, pp. 1–63, 1979.
• C. Pérez, “Endpoint estimates for commutators of singular integral operators,” Journal of Functional Analysis, vol. 128, no. 1, pp. 163–185, 1995.
• S. Janson, “Mean oscillation and commutators of singular integral operators,” Arkiv för Matematik, vol. 16, no. 2, pp. 263–270, 1978.