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Boundedness of Fractional Integral with Variable Kernel and Their Commutators on Variable Exponent Herz Spaces
College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China
College of Education, Shendi University, Shendi, Sudan
College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China
Faculty of Science, University of Dalanj, Dalanj, Sudan
- 1 College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China
- 2 College of Education, Shendi University, Shendi, Sudan
- 3 College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China
- 4 Faculty of Science, University of Dalanj, Dalanj, Sudan
Applied Mathematics·Volume 07 (2016)·Pages 1165–1182·Published 7 June 2016·DOI10.4236/am.2016.710104
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Abstract
In this paper, we study the boundedness of the fractional integral operator and their commutator on Herz spaecs with two variable exponents . By using the properties of the variable exponents Lebesgue spaces, the boundedness of the fractional integral operator and their commutator generated by Lipschitz function is obtained on those Herz spaces.
KeywordsFractional IntegralVariable KernelCommutatorVariable ExponentLipschitz SpaceHerz Spaces
- Cruz-Uribe, D. and Fiorenza, A. (2013) Variable Lebesgue Spaces. Foundations and Harmonic Analysis. Applied and Numerical Harmonic Analysis, Springer, New York.
- Kenig, C. (1994) Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems. American Mathematical Society, Providence. http://dx.doi.org/10.1090/cbms/083
- Calderón, A. and Zygmund, A. (1955) On a Problem of Mihilim. Transations of the American Mathematical Society, 78, 209-224. http://dx.doi.org/10.2307/1992955
- Calderón, A. and Zygmund, A. (1978) On Singular Integral with Variable Kernels. Journal of Applied Analysis, 7, 221-238. http://dx.doi.org/10.1080/00036817808839193
- Christ, M., Duoandikoetxea, J. and Rubio de Francia, J. (1986) Maximal Operators Related to the Radon Transform and the Calderóon-Zygmund Method of Rotations. Duke Mathematical Journal, 53, 189-209. http://dx.doi.org/10.1215/S0012-7094-86-05313-5
- Muckenhoupt, B. and Wheeden, R. (1971) Weighted Norm Inequalities for Singular and Fractional Integrals. Transations of the American Mathematical Society, 161, 249-258. http://dx.doi.org/10.1090/S0002-9947-1971-0285938-7
- Kovácik, O. and Rákosník, J. (1991) On Spaces and . Czechoslovak Matematical Journal, 41, 592-618.
- Izuki, M. (2010) Boundedness of Commutators on Herz Spaces with Variable Exponent. Rendiconti del Circolo Matematico di Palermo, 59, 199-213. http://dx.doi.org/10.1007/s12215-010-0015-1
- Izuki, M. (2010) Fractional Integrals on Herz-Morrey Spaces with Variable Exponent. Hiroshima Mathematical Journal, 40, 343-355.
- Wang, L. and Tao, S. (2014) Boundedness of Littlewood-Paley Operators and Their Commutators on Herz-Morrey Spaces with Variable Exponent. Journal of Inequalities and Applications, 227, 1-17. http://dx.doi.org/10.1186/1029-242x-2014-227
- Wang, L. and Tao, S. (2015) Parameterized Littlewood-Paley Operators and Their Commutators on Lebegue Spaces with variable Exponent. Analysis in Theory and Applications, 31, 13-24.
- Izuki, M. (2009) Herz and Amalgam Spaces with Variable Exponent, the Haar Wavelets and Greediness of the Wavelet System. East Journal on Approximations, 15, 87-109.
- Tan, J. and Liu, Z. (2015) Some Boundedness of Homogeneous Fractional Integrals on Variable Exponent Function Spaces. ACTA Mathematics Science (Chinese Series), 58, 310-320.