Table of Contents Author Guidelines Submit a Manuscript
Journal of Function Spaces
Volume 2015, Article ID 160635, 19 pages
http://dx.doi.org/10.1155/2015/160635
Research Article

Herz-Morrey-Hardy Spaces with Variable Exponents and Their Applications

Department of Mathematics, Hainan Normal University, Haikou 571158, China

Received 5 July 2014; Accepted 20 December 2014

Academic Editor: Józef Banaś

Copyright © 2015 Jingshi Xu and Xiaodi Yang. 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. A. Beurling, “Construction and analysis of some convolution algebras,” Annales de L'Institut Fourier Grenoble, vol. 14, pp. 1–32, 1964. View at Google Scholar
  2. C. S. Herz, “Lipschitz spaces and Bernstein's theorem on absolutely convergent Fourier transforms,” vol. 18, pp. 283–324, 1968. View at Google Scholar · View at MathSciNet
  3. A. Baernstein II and E. Sawyer, “Embedding and multiplier theorems for Hp(R2),” Memoirs of the American Mathematical Society, vol. 59, no. 318, 1985. View at Google Scholar
  4. Y. Z. Chen and K.-S. Lau, “Some new classes of Hardy spaces,” Journal of Functional Analysis, vol. 84, no. 2, pp. 255–278, 1989. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. J. García-Cuerva, “Hardy spaces and Beurling algebras,” Journal of the London Mathematical Society, vol. 39, no. 3, pp. 499–513, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  6. S. Lu, D. Yang, and G. Hu, Herz Type Spaces and Their Applications, Science Press, Beijing, China, 2008.
  7. M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, vol. 1748 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2000.
  8. P. Harjulehto, P. Hästö, Ú. V. Lê, and M. Nuortio, “Overview of differential equations with non-standard growth,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 12, pp. 4551–4574, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. Y. Chen, S. Levine, and M. Rao, “Variable exponent, linear growth functionals in image restoration,” SIAM Journal on Applied Mathematics, vol. 66, no. 4, pp. 1383–1406, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. F. Li, Z. Li, and L. Pi, “Variable exponent functionals in image restoration,” Applied Mathematics and Computation, vol. 216, no. 3, pp. 870–882, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. P. Harjulehto, P. Hästö, V. Latvala, and O. Toivanen, “Critical variable exponent functionals in image restoration,” Applied Mathematics Letters, vol. 26, no. 1, pp. 56–60, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. 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
  13. 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 · View at Scopus
  14. L. Diening, P. Hästö, and S. Roudenko, “Function spaces of variable smoothness and integrability,” Journal of Functional Analysis, vol. 256, no. 6, pp. 1731–1768, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. B. Dong and J. Xu, “New Herz type Besov and Triebel-Lizorkin spaces with variable exponents,” Journal of Function Spaces and Applications, vol. 2012, Article ID 384593, 27 pages, 2012. View at Publisher · View at Google Scholar · View at Scopus
  16. X. Fan, “Variable exponent Morrey and Campanato spaces,” Nonlinear Analysis, vol. 72, no. 11, pp. 4148–4161, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. J.-J. Fu and J.-S. 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
  18. P. Gurka, P. Harjulehto, and A. Nekvinda, “Bessel potential spaces with variable exponent,” Mathematical Inequalities & Applications, vol. 10, no. 3, pp. 661–676, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  19. P. A. Hästö, “Local-to-global results in variable exponent spaces,” Mathematical Research Letters, vol. 16, no. 2, pp. 263–278, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. H. Kempka, “2-microlocal Besov and Triebel-Lizorkin spaces of variable integrability,” Revista Matematica Complutense, vol. 22, no. 1, pp. 227–251, 2009. View at Google Scholar
  21. 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
  22. 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
  23. 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
  24. J.-S. Xu, “An atomic decomposition of variable Besov and Triebel-Lizorkin spaces,” Armenian Journal of Mathematics, vol. 2, no. 1, pp. 1–12, 2009. View at Google Scholar · View at MathSciNet
  25. T. Noi, “Duality of variable exponent Triebel-Lizorkin and Besov spaces,” Journal of Function Spaces and Applications, vol. 2012, Article ID 361807, 19 pages, 2012. View at Publisher · View at Google Scholar · View at Scopus
  26. Y. Sawano, “Atomic decompositions of Hardy spaces with variable exponents and its application to bounded linear operators,” Integral Equations and Operator Theory, vol. 77, no. 1, pp. 123–148, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. 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
  28. 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
  29. 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
  30. B. Dong and J. Xu, “New Herz type Besov and Triebel-Lizorkin spaces with variable exponents,” Journal of Function Spaces and Applications, vol. 2012, Article ID 384593, 27 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  31. 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
  32. Z.-H. Xuan and L.-S. Shu, “Boundedness of higher order commutators on Herz-Morrey spaces with variable exponent,” Journal of Nanjing University Mathematical Biquarterly, vol. 30, no. 2, pp. 188–196, 2013. View at Google Scholar · View at MathSciNet
  33. Y. Lu and Y. P. Zhu, “Boundedness of multilinear Calderón-Zygmund singular operators on Morrey-Herz spaces with variable exponents,” Acta Mathematica Sinica (English Series), vol. 30, no. 7, pp. 1180–1194, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  34. C. Tang, Q. Wu, and J. Xu, “Commutators of multilinear Calderón-Zygmund operator and BMO functions in Herz-Morrey spaces with variable exponents,” Journal of Function Spaces, vol. 2014, Article ID 162518, 12 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  35. W. Gao, Y. Jiang, and X. Gong, “Boundedness of linear commutators on Herz-Morrey spaces on spaces of homogeneous type,” Chinese Quarterly Journal of Mathematics, vol. 25, no. 2, pp. 172–181, 2010. View at Google Scholar
  36. X. Q. Zhao and W. H. Gao, “Boundedness of sublinear operators on Herz-Morrey spaces,” Applied Mathematics. A Journal of Chinese Universities. Series A, vol. 20, no. 1, pp. 55–62, 2005. View at Google Scholar · View at MathSciNet
  37. D. V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces, Applied and Numerical Harmonic Analysis, Birkhäuser, Heidelberg, Germany, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  38. D. Cruz-Uribe, A. Fiorenza, C. Martell, and C. Perez, “The Boundedness of classical operators on variable Lp spaces,” Annales Academiae Scientiarum Fennicae—Mathematica, vol. 31, pp. 239–264, 2006. View at Google Scholar
  39. D. Cruz-Uribe, A. Fiorenza, and C. J. Neugebauer, “The maximal function on variable Lp spaces,” Annales Academiæ Scientiarum Fennicæ Mathematica, vol. 28, no. 1, pp. 223–238, 2003. View at Google Scholar · View at MathSciNet
  40. L. Diening, “Maximal function on generalized lebesgue spaces LP(x),” Mathematical Inequalities and Applications, vol. 7, no. 2, pp. 245–253, 2004. View at Google Scholar · View at Scopus
  41. L. Diening, P. Harjulehto, P. Hasto, and M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, Springer, Berlin, Germany, 2011.
  42. A. Nekvinda, “Hardy-littlewood maxell operator on LP(x),” Mathematical Inequalities and Applications, vol. 7, no. 2, pp. 255–265, 2004. View at Google Scholar · View at Scopus
  43. O. Kovacik and J. Rakosnik, “On spaces Lpx and Wk,px,” Czechoslovak Mathematical Journal, vol. 41, no. 4, pp. 592–681, 1991. View at Google Scholar
  44. C. Fefferman and E. M. Stein, “HP spaces of several variables,” Acta Mathematica, vol. 129, no. 1, pp. 137–193, 1972. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  45. L. Grafakos, Classical Fourier Analysis, Springer, New York, NY, USA, 2008.
  46. E. M. Stein, Harmonic Analysis, Princeton University Press, Princeton, NJ, USA, 1993. View at MathSciNet
  47. A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Academic Press, New York, NY, USA, 1986.
  48. S. Lu and D. Yang, “The local versions of Hp (Rn) spaces at the origin,” Studia Mathematica, vol. 116, pp. 103–131, 1995. View at Google Scholar
  49. L. Grafakos, Modern Fourier Analysis, Springer, New York, NY, USA, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  50. M. H. Taibleson and G. Weiss, “The molecular characterization of certain Hardy spaces,” Astérisque, vol. 77, pp. 67–149, 1980. View at Google Scholar
  51. P. Rocha and M. Urciuolo, “Fractional type integral operators on variable Hardy spaces,” Acta Mathematica Hungarica, vol. 143, no. 2, pp. 502–514, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus