Open Access

On the -Boundedness of Nonisotropic Spherical Riesz Potentials

Journal of Inequalities and Applications20072007:036503

DOI: 10.1155/2007/36503

Received: 20 November 2006

Accepted: 1 March 2007

Published: 19 April 2007


We introduced the concept of nonisotropic spherical Riesz potential operators generated by the -distance of variable order on -sphere and its -boundedness were investigated.


Authors’ Affiliations

Department of Mathematics, Faculty of Science and Arts, Kocatepe University


  1. Besov OV, Il'in VP, Lizorkin PI: The -estimates of a certain class of non-isotropically singular integrals. Doklady Akademii Nauk SSSR 1966, 169: 1250–1253.MathSciNetGoogle Scholar
  2. Çınar İ: The Hardy-Littlewood-Sobolev inequality for non-isotropic Riesz potentials. Turkish Journal of Mathematics 1997,21(2):153–157.MathSciNetMATHGoogle Scholar
  3. Çınar İ, Duru H: The Hardy-Littlewood-Sobolev inequality for -distance Riesz potentials. Applied Mathematics and Computation 2004,153(3):757–762. 10.1016/S0096-3003(03)00671-4MathSciNetView ArticleMATHGoogle Scholar
  4. Gadjiev AD, Dogru O: On combination of Riesz potentials with non-isotropic kernels. Indian Journal of Pure and Applied Mathematics 1999,30(6):545–556.MathSciNetMATHGoogle Scholar
  5. Sarikaya MZ, Yıldırım H: The restriction and the continuity properties of potentials depending on -distance. Turkish Journal of Mathematics 2006,30(3):263–275.MathSciNetMATHGoogle Scholar
  6. Sarikaya MZ, Yıldırım H: On the -spherical Riesz potential generated by the -distance. International Journal of Contemporary Mathematical Sciences 2006,1(2):85–89.MathSciNetMATHGoogle Scholar
  7. Sarikaya MZ, Yıldırım H: On the non-isotropic fractional integrals generated by the -distance. Selçuk Journal of Applied Mathematics 2006,7(1):17–23.MATHGoogle Scholar
  8. Sarikaya MZ, Yıldırım H: On the Hardy type inequality with non-isotropic kernels. Lobachevskii Journal of Mathematics 2006, 22: 47–57.MathSciNetMATHGoogle Scholar
  9. Sarikaya MZ, Yıldırım H, Ozkan UM: Norm inequalities with non-isotropic kernels. International Journal of Pure and Applied Mathematics 2006,31(3):337–344.MathSciNetMATHGoogle Scholar
  10. Stein EM: Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, no. 30. Princeton University Press, Princeton, NJ, USA; 1970:xiv+290.Google Scholar
  11. Zhou Z, Hong Y, Zhou CZ: The-boundedness of Riesz potential operators of variable order on a sphere. Journal of South China Normal University 1999, (2):20–24.Google Scholar
  12. Yıldırım H: On generalization of the quasi homogeneous Riesz potential. Turkish Journal of Mathematics 2005,29(4):381–387.MathSciNetMATHGoogle Scholar
  13. Sadosky C: Interpolation of Operators and Singular Integrals, Monographs and Textbooks in Pure and Applied Math.. Volume 53. Marcel Dekker, New York, NY, USA; 1979:xii+375.Google Scholar


© M. Z. Sarikaya and H. Yildirim 2007

This article is published under license to BioMed Central Ltd. 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.