Table of Contents Author Guidelines Submit a Manuscript
Journal of Function Spaces
Volume 2017, Article ID 3764142, 9 pages
https://doi.org/10.1155/2017/3764142
Research Article

Precompact Sets, Boundedness, and Compactness of Commutators for Singular Integrals in Variable Morrey Spaces

1School of Sciences, Central South University of Forestry and Technology, Changsha 410004, China
2Department of Mathematics, Hainan Normal University, Haikou 571158, China

Correspondence should be addressed to Jingshi Xu; moc.621@uxihsgnij

Received 10 May 2017; Accepted 18 June 2017; Published 20 July 2017

Academic Editor: Dashan Fan

Copyright © 2017 Wei Wang and Jingshi Xu. 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. R. R. Coifman, R. Rochberg, and G. Weiss, “Factorization theorems for Hardy spaces in several variables,” Annals of Mathematics, vol. 103, no. 3, pp. 611–635, 1976. View at Publisher · View at Google Scholar · View at MathSciNet
  2. A. Uchiyama, “On the compactness of operators of Hankel type,” The Tohoku Mathematical Journal. Second Series, vol. 30, no. 1, pp. 163–171, 1978. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. S. Janson, “Mean oscillation and commutators of singular integral operators,” Arkiv för Matematik, vol. 16, no. 2, pp. 263–270, 1978. View at Publisher · View at Google Scholar · View at MathSciNet
  4. R. Coifman, P. Lions, Y. Meyer, and S. Semmes, “Compensated compactness and Hardy spaces,” Journal de Mathématiques Pures et Appliquées, vol. 72, pp. 247–286, 1993. View at Google Scholar · View at MathSciNet
  5. F. Beatrous and S.-Y. Li, “On the boundedness and compactness of operators of Hankel type,” Journal of Functional Analysis, vol. 111, no. 2, pp. 350–379, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. Y. Chen and Y. Ding, “Compactness of the commutators of parabolic singular integrals,” Science China. Mathematics, vol. 53, no. 10, pp. 2633–2648, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. Y. Chen, Y. Ding, and X. Wang, “Compactness of commutators of Riesz potential on Morrey spaces,” Potential Analysis, vol. 30, no. 4, pp. 301–313, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  8. Y. Chen, Y. Ding, and X. Wang, “Compactness of commutators for singular integrals on Morrey spaces,” Canadian Journal of Mathematics. Journal Canadien de Math\'ematiques, vol. 64, no. 2, pp. 257–281, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. C. Morrey, “On the solutions of quasi-linear elliptic partial differential equations,” Transactions of the American Mathematical Society, vol. 43, no. 1, pp. 126–166, 1938. View at Publisher · View at Google Scholar · View at MathSciNet
  10. T. Kato, “Strong solutions of the Navier-Stokes equation in Morrey spaces,” Boletim da Sociedade Brasileira de Matemática, vol. 22, no. 2, pp. 127–155, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  11. Z. Shen, “The periodic schrödinger operators with potentials in the morrey class,” Journal of Functional Analysis, vol. 193, no. 2, pp. 314–345, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  12. D. R. Adams and J. Xiao, “Nonlinear potential analysis on Morrey spaces and their capacities,” Indiana University Mathematics Journal, vol. 53, no. 6, pp. 1629–1663, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. D. R. Adams and J. Xiao, “Regularity of Morrey commutators,” Transactions of the American Mathematical Society, vol. 364, no. 9, pp. 4801–4818, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. D. R. Adams and J. Xiao, “Restrictions of Riesz-Morrey potentials,” Arkiv för Matematik, vol. 54, no. 2, pp. 201–231, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  15. A. Almeida, J. Hasanov, and S. Samko, “Maximal and potential operators in variable exponent Morrey spaces,” Georgian Mathematical Journal, vol. 15, no. 2, pp. 195–208, 2008. View at Google Scholar · View at MathSciNet
  16. A. Almeida and D. Drihem, “Maximal, potential and singular type operators on Herz spaces with variable exponents,” Journal of Mathematical Analysis and Applications, vol. 394, no. 2, pp. 781–795, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. A. Almeida and P. Hästö, “Besov spaces with variable smoothness and integrability,” Journal of Functional Analysis, vol. 258, no. 5, pp. 1628–1655, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  18. D. Cruz-Uribe, A. Fiorenza, J. M. Martell, and C. Pérez, “The boundedness of classical operators on variable Lp spaces,” Annales Academiæ Scientiarum Fennicæ Mathematica, vol. 31, no. 1, pp. 239–264, 2006. View at Google Scholar · View at MathSciNet
  19. L. Diening, P. Harjulehto, P. Hästö, and M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, vol. 2017 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  20. B. Dong and J. Xu, “Herz-Morrey type Besov and Triebel-Lizorkin spaces with variable exponents,” Banach Journal of Mathematical Analysis, vol. 9, no. 1, pp. 75–101, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. J. Fu and J. Xu, “Characterizations of Morrey type Besov and Triebel-Lizorkin spaces with variable exponents,” Journal of Mathematical Analysis and Applications, vol. 381, no. 1, pp. 280–298, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. M. Izuki, “Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization,” Analysis Mathematica, vol. 36, no. 1, pp. 33–50, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. M. Izuki, “Boundedness of commutators on Herz spaces with variable exponent,” Rendiconti del Circolo Matematico di Palermo. Second Series, vol. 59, no. 2, pp. 199–213, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. M. Izuki and Y. Sawano, “Variable Lebesgue norm estimates for BMO functions,” Czechoslovak Mathematical Journal, vol. 62(137), no. 3, pp. 717–727, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  25. H. Kempka, “2-microlocal Besov and Triebel-Lizorkin spaces of variable integrability,” Revista Matemática Complutense, vol. 22, no. 1, pp. 227–251, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  26. H. Kempka, “Atomic, molecular and wavelet decomposition of generalized 2-microlocal Besov spaces,” Journal of Function Spaces and Applications, vol. 8, no. 2, pp. 129–165, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. H. Kempka and J. Vybíral, “Lorentz spaces with variable exponents,” Mathematische Nachrichten, vol. 287, no. 8-9, pp. 938–954, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  28. O. Kovácik and J. Rákosník, “On spaces Lp(x) and Wk,p(x),” Czechoslovak Mathematical Journal, vol. 41, no. 116, pp. 592–618, 1991. View at Google Scholar · View at MathSciNet
  29. E. Nakai and Y. Sawano, “Hardy spaces with variable exponents and generalized Campanato spaces,” Journal of Functional Analysis, vol. 262, no. 9, pp. 3665–3748, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  30. T. Noi, “Fourier multiplier theorems for Besov and Triebel-Lizorkin spaces with variable exponents,” Mathematical Inequalities & Applications, vol. 17, no. 1, pp. 49–74, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  31. T. Noi, “Trace and extension operators for Besov spaces and Triebel-Lizorkin spaces with variable exponents,” Revista Matemática Complutense, vol. 29, no. 2, pp. 341–404, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  32. T. Noi and Y. Sawano, “Complex interpolation of Besov spaces and Triebel-Lizorkin spaces with variable exponents,” Journal of Mathematical Analysis and Applications, vol. 387, no. 2, pp. 676–690, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. T. Noi and Y. Sawano, “Weighted variable modulation spaces,” Scientiae Mathematicae Japonicae, vol. 78, pp. 29–43, 2015. View at Google Scholar · View at MathSciNet
  34. H. Rafeiro and S. Samko, “Variable exponent Campanato spaces,” Journal of Mathematical Sciences (New York), vol. 172, no. 1, pp. 143–164, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  35. S. Samko, “Variable exponent Herz spaces,” Mediterranean Journal of Mathematics, vol. 10, no. 4, pp. 2007–2025, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  36. C. Shi and J. Xu, “Herz type Besov and Triebel-Lizorkin spaces with variable exponent,” Frontiers of Mathematics in China, vol. 8, no. 4, pp. 907–921, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  37. H. Wang and Z. Liu, “The Herz-type HARdy spaces with variable exponent and their applications,” Taiwanese Journal of Mathematics, vol. 16, no. 4, pp. 1363–1389, 2012. View at Google Scholar · View at MathSciNet · View at Scopus
  38. J. Xu, “Variable Besov and Triebel–Lizorkin spaces,” Annales Academiæ Scientiarum Fennicæ Mathematica, vol. 33, no. 2, pp. 511–522, 2008. View at Google Scholar · View at MathSciNet
  39. J. S. Xu, “The relation between variable Bessel potential spaces and Triebel-Lizorkin spaces,” Integral Transforms and Special Functions. An International Journal, vol. 19, no. 7-8, pp. 599–605, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  40. J. Xu and X. Yang, “Herz-Morrey-Hardy spaces with variable exponents and their applications,” Journal of Function Spaces, vol. 2015, Article ID 160635, 19 pages, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  41. D. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces, Birkhäuser, Basel, Switzerland, 2013.
  42. V. S. Guliyev, J. J. Hasanov, and S. G. Samko, “Boundedness of maximal, potential type, and singular integral operators in the generalized variable exponent Morrey type spaces,” Journal of Mathematical Sciences (New York), vol. 170, no. 4, pp. 423–443, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  43. A. Y. Karlovich and A. K. Lerner, “Commutators of singular integrals on generalized Lp spaces with variable exponent,” Publicacions Matemátiques, vol. 49, no. 1, pp. 111–125, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  44. V. Kokilashvili and A. Meskhi, “Boundedness of maximal and singular operators in Morrey spaces with variable exponent,” Armenian Journal of Mathematics, vol. 1, no. 1, pp. 18–28, 2008. View at Google Scholar · View at MathSciNet
  45. H. Rafeiro, “Kolmogorov compactness criterion in variable exponent Lebesgue spaces,” Proceedings of A. Razmadze Mathematical Institute, vol. 150, pp. 105–113, 2009. View at Google Scholar · View at MathSciNet
  46. R. A. Adams and J. Fournier, Sobolev Spaces, Academic Press, Heidelberg, Germany, 2nd edition, 2006.
  47. L. Grafakos, Modern Fourier Analysis, vol. 250 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  48. C. Pérez and R. Trujillo-González, “Sharp weighted estimates for multilinear commutators,” Journal of the London Mathematical Society. Second Series, vol. 65, no. 3, pp. 672–692, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  49. S. Lu, Y. Ding, and D. Yan, Singular Integrals and Related Topics, World Scientific Publishing, Singapore, 2007. View at Scopus