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Journal of Function Spaces and Applications
Volume 2013, Article ID 809704, 12 pages
http://dx.doi.org/10.1155/2013/809704
Research Article

Weak Type Inequalities for Some Integral Operators on Generalized Nonhomogeneous Morrey Spaces

1Department of Mathematics, Bandung Institute of Technology, Bandung 41032, Indonesia
2Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji, Tokyo 192-0397, Japan
3Department of Mathematics, Jenderal Soedirman University, Purwokerto 53122, Indonesia

Received 9 September 2013; Accepted 20 November 2013

Academic Editor: Vagif Guliyev

Copyright © 2013 Hendra Gunawan et al. 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.

Linked References

  1. F. Nazarov, S. Treil, and A. Volberg, “Cauchy integral and Calderón-Zygmund operators on nonhomogeneous spaces,” International Mathematics Research Notices, vol. 1997, no. 15, pp. 703–726, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  2. F. Nazarov, S. Treil, and A. Volberg, “Weak type estimates and Cotlar inequalities for Calderón-Zygmund operators on nonhomogeneous spaces,” International Mathematics Research Notices, vol. 1998, no. 9, pp. 463–487, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  3. F. Nazarov, S. Treil, and A. Volberg, “The Tb-theorem on non-homogeneous spaces,” Acta Mathematica, vol. 190, no. 2, pp. 151–239, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. J. Verdera, “The fall of the doubling condition in Calderón-Zygmund theory,” in Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations, pp. 275–292, Publicacions Matematiques, 2002. View at MathSciNet
  5. D. E. Edmunds, V. Kokilashvili, and A. Meskhi, Bounded and Compact Integral Operators, vol. 543 of Mathematics and Its Applications, Kluwer Academic Publishers, London, UK, 2002. View at MathSciNet
  6. G. H. Hardy and J. E. Littlewood, “Some properties of fractional integrals. I,” Mathematische Zeitschrift, vol. 27, no. 1, pp. 565–606, 1928. View at Publisher · View at Google Scholar · View at Scopus
  7. S. L. Sobolev, “On a theorem in functional analysis,” Matematicheskiĭ Sbornik. Novaya Seriya 4, vol. 46, pp. 471–497, 1938, English translation in American Mathematical Society Translations Series 2, vol. 34, pp. 39–68, 1963. View at Google Scholar
  8. V. Kokilashvili, “Weighted estimates for classical integral operators,” in Nonlinear Analysis, Function Spaces and Applications, vol. 4 of Teubner-Texte zur Mathematik, pp. 86–103, Teubner, Leipzig, Germany, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  9. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series no. 30, Princeton University Press, Princeton, NJ, USA, 1970. View at MathSciNet
  10. Eridani, “On the boundedness of a generalized fractional integral on generalized Morrey spaces,” Tamkang Journal of Mathematics, vol. 33, no. 4, pp. 335–340, 2002. View at Google Scholar · View at MathSciNet
  11. H. Gunawan, “A note on the generalized fractional integral operators,” Journal of the Indonesian Mathematical Society, vol. 9, no. 1, pp. 39–43, 2003. View at Google Scholar · View at MathSciNet
  12. E. Nakai, “On generalized fractional integrals,” Taiwanese Journal of Mathematics, vol. 5, no. 3, pp. 587–602, 2001. View at Google Scholar · View at MathSciNet · View at Scopus
  13. Y. Mizuta, E. Nakai, T. Ohno, and T. Shimomura, “Boundedness of fractional integral operators on Morrey spaces and Sobolev embeddings for generalized Riesz potentials,” Journal of the Mathematical Society of Japan, vol. 62, no. 3, pp. 707–744, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. V. S. Guliyev and Y. Sawano, “Linear and sublinear operators on generalized Morrey spaces with non-doubling measures,” Publicationes Mathematicae Debrecen, vol. 83, no. 3, pp. 1–17, 2013. View at Google Scholar
  15. E. Nakai, “Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces,” Mathematische Nachrichten, vol. 166, no. 1, pp. 95–103, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  16. D. I. Hakim and H. Gunawan, “Weak (p, q) inequalities for fractional integral operators on generalized Morrey spaces of non-homogeneous type,” Mathematica Aeterna, vol. 3, no. 3, pp. 161–168, 2013. View at Google Scholar
  17. J. García-Cuerva and A. E. Gatto, “Boundedness properties of fractional integral operators associated to non-doubling measures,” Studia Mathematica, vol. 162, no. 3, pp. 245–261, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. J. García-Cuerva and J. M. Martell, “Two-weight norm inequalities for maximal operators and fractional integrals on non-homogeneous spaces,” Indiana University Mathematics Journal, vol. 50, no. 3, pp. 1241–1280, 2001. View at Google Scholar · View at MathSciNet · View at Scopus
  19. I. Sihwaningrum, S. Maryani, and H. Gunawan, “Weak type inequalities for fractional integral operators on generalized non-homogeneous Morrey spaces,” Analysis Theory Applications, vol. 28, no. 1, pp. 65–72, 2012. View at Google Scholar
  20. H. L. Royden and P. M. Fitzpatrick, Real Analysis, Pearson, London, UK, 4th edition, 2010.
  21. V. S. Guliyev, “Generalized weighted Morrey spaces and higher order commutators of sublinear operators,” Eurasian Mathematical Journal, vol. 3, no. 3, pp. 33–61, 2012. View at Google Scholar · View at MathSciNet
  22. N. Aronszajn and K. T. Smith, “Theory of Bessel potentials. part I,” Annales de l'Institut Fourier, vol. 11, pp. 385–475, 1961. View at Google Scholar · View at MathSciNet
  23. S. Nagayasu and H. Wadade, “Characterization of the critical Sobolev space on the optimal singularity at the origin,” Journal of Functional Analysis, vol. 258, no. 11, pp. 3725–3757, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. Y. Sawano and H. Wadade, “On the Gagliardo-Nirenberg type inequality in the critical Sobolev-Morrey space,” The Journal of Fourier Analysis and Applications, vol. 19, no. 1, pp. 20–47, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  25. Eridani, H. Gunawan, E. Nakai, and Y. Sawano, “Characterizations for the generalized fractional integral operators on Morrey spaces,” Mathematical Inequalities & Applications.
  26. Eridani and Y. Sawano, “Fractional integral operators in generalized Morrey spaces defined on metric measure spaces,” Proceedings of A. Razmadze Mathematical Institute, vol. 158, pp. 13–24, 2012. View at Google Scholar · View at MathSciNet
  27. Y. Sawano and T. Shimomura, “Sobolev's inequality for Riesz potentials of functions in generalized Morrey spaces with variable exponent attaining the value 1 over non-doubling measure spaces,” Journal of Inequalities and Applications, vol. 2013, article 12, pp. 1–19, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  28. Y. Sawano and T. Shimomura, “Sobolev embeddings for Riesz potentials of functions in non-doubling Morrey spaces of variable exponents,” Collectanea Mathematica, vol. 64, no. 3, pp. 313–350, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  29. H. Gunawan, E. Nakai, Y. Sawano, and H. Tanaka, “Generalized stummel class and Morrey spaces,” Publications de l'Institut Mathematique, vol. 92, no. 106, pp. 127–138, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  30. C. Pérez, “Two weighted inequalities for potential and fractional type maximal operators,” Indiana University Mathematics Journal, vol. 43, no. 2, pp. 663–683, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  31. D. I. Hakim, Weak type inequalities for fractional integral operators on generalized non-homogeneous Morrey spaces [M.S. thesis], Institut Teknologi Bandung, Bandung, Indonesia, 2013.
  32. Y. Sawano, “Generalized Morrey spaces for non-doubling measures,” Nonlinear Differential Equations and Applications, vol. 15, no. 4-5, pp. 413–425, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. E. Nakai, “Generalized fractional integrals on generalized Morrey spaces,” Mathematische Nachrichten, 2013. View at Publisher · View at Google Scholar
  34. Y. Sawano, S. Sugano, and H. Tanaka, “Generalized fractional integral operators and fractional maximal operators in the framework of morrey spaces,” Transactions of the American Mathematical Society, vol. 363, no. 12, pp. 6481–6503, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  35. A. Akbulut, V. Guliyev, and R. Mustafayev, “On the boundedness of the maximal operator and singular integral operators in generalized Morrey spaces,” Mathematica Bohemica, vol. 137, no. 1, pp. 27–43, 2012. View at Google Scholar · View at MathSciNet