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Journal of Function Spaces and Applications
Volume 2013, Article ID 859402, 21 pages
http://dx.doi.org/10.1155/2013/859402
Research Article

Applications of Littlewood-Paley Theory for -Morrey Spaces to the Boundedness of Integral Operators

1School of High Technology for Human Welfare, Tokai University, 317 Nishino Numazu, Shizuoka 410-0395, Japan
2College of Economics, Nihon University, 1-3-2 Misaki-cho, Chiyoda-ku, Tokyo 101-8360, Japan
3Department of Mathematics, Ibaraki University, Mito, Ibaraki 310-8512, Japan
4Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1 Minami-Ohsawa, Hachioji, Tokyo 192-0397, Japan

Received 18 March 2012; Accepted 27 November 2012

Academic Editor: Alberto Fiorenza

Copyright © 2013 Yasuo Komori-Furuya 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. A. L. Mazzucato, “Decomposition of Besov-Morrey spaces,” in Harmonic Analysis at Mount Holyoke, vol. 320 of Contemporary Mathematics, pp. 279–294, American Mathematical Society, Providence, RI, USA, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. S. Zhang and X. Tao, “Boundedness of Littlewood-Paley operators in generalized Orlicz-Campanato spaces,” The Rocky Mountain Journal of Mathematics, vol. 40, no. 4, pp. 1355–1375, 2010. View at Google Scholar
  3. A. Beurling, “Construction and analysis of some convolution algebras,” Annales de l'Institut Fourier, vol. 14, no. 2, pp. 1–32, 1964. View at Google Scholar · View at Zentralblatt MATH
  4. N. Wiener, “Generalized harmonic analysis,” Acta Mathematica, vol. 55, no. 1, pp. 117–258, 1930. View at Publisher · View at Google Scholar
  5. N. Wiener, “Tauberian theorems,” Annals of Mathematics, vol. 33, no. 1, pp. 1–100, 1932. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. H. G. Feichtinger, “An elementary approach to Wiener's third Tauberian theorem for the Euclidean n-space,” in Symposia Mathematica, vol. 29, pp. 267–301, Academic Press, New York, NY, USA, 1987. View at Google Scholar
  7. C. S. Herz, “Lipschitz spaces and Bernstein's theorem on absolutely convergent Fourier transforms,” Journal of Mathematics and Mechanics, vol. 18, pp. 283–323, 1969. View at Google Scholar
  8. S. Z. Lu and D. C. Yang, “The Littlewood-Paley function and Φ-transform characterizations of a new Hardy space HK2 associated with the Herz space,” Studia Mathematica, vol. 101, no. 3, pp. 285–298, 1992. View at Google Scholar
  9. S. Lu and D. Yang, “The central BMO spaces and Littlewood-Paley operators,” Approximation Theory and its Applications, vol. 11, no. 3, pp. 72–94, 1995. View at Google Scholar · View at Zentralblatt MATH
  10. K. Matsuoka and E. Nakai, “Fractional integral operators on Bp,λ with Morrey-Campanato norms,” in Function Spaces IX, vol. 92, pp. 249–264, Institute of Mathematics Polish Academy of Sciences, Warsaw, Poland, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. V. I. Burenkov and H. V. Guliyev, “Necessary and sufficient conditions for boundedness of the maximal operator in local Morrey-type spaces,” Studia Mathematica, vol. 163, no. 2, pp. 157–176, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. Y. Komori-Furuya, K. Matsuoka, E. Nakai, and Y. Sawano, “Integral operators on B˙σ-Morrey-Campanato spaces,” Revista Mathematica Complutense, vol. 26, no. 1, pp. 1–32, 2013. View at Google Scholar
  13. H. Triebel, Theory of Function Spaces, vol. 78 of Monographs in Mathematics, Birkhäuser, Basel, Switzerland, 1983. View at Publisher · View at Google Scholar
  14. Y. Liang, Y. Sawano, T. Ullrich, D. Yang, and W. Yuan, “New characterizations of Besov-Triebel-Lizorkin-Hausdorff spaces including coorbits and wavelets,” The Journal of Fourier Analysis and Applications, vol. 18, no. 5, pp. 1067–1111, 2012. View at Publisher · View at Google Scholar
  15. Y. Sawano, D. Yang, and W. Yuan, “New applications of Besov-type and Triebel-Lizorkin-type spaces,” Journal of Mathematical Analysis and Applications, vol. 363, no. 1, pp. 73–85, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. W. Yuan, W. Sickel, and D. Yang, Morrey and Campanato Meet Besov, Lizorkin and Triebel, vol. 2005 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2010. View at Publisher · View at Google Scholar
  17. D. Yang and W. Yuan, “A new class of function spaces connecting Triebel-Lizorkin spaces and Q spaces,” Journal of Functional Analysis, vol. 255, no. 10, pp. 2760–2809, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. D. Yang and W. Yuan, “New Besov-type spaces and Triebel-Lizorkin-type spaces including Q spaces,” Mathematische Zeitschrift, vol. 265, no. 2, pp. 451–480, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. D. Yang and W. Yuan, “Characterizations of Besov-type and Triebel-Lizorkin-type spaces via maximal functions and local means,” Nonlinear Analysis. Theory, Methods & Applications, vol. 73, no. 12, pp. 3805–3820, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. D. Yang and W. Yuan, “Dual properties of Triebel-Lizorkin-type spaces and their applications,” Zeitschrift für Analysis und ihre Anwendungen, vol. 30, no. 1, pp. 29–58, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. D. Yang, W. Yuan, and C. Zhuo, “Fourier multipliers on Triebel-Lizorkin-type spaces,” Journal of Function Spaces and Applications, vol. 2012, Article ID 431016, 37 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. A. Uchiyama, “A constructive proof of the Fefferman-Stein decomposition of BMO BMO(Rn),” Acta Mathematica, vol. 148, pp. 215–241, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. C. Fefferman and E. M. Stein, “Some maximal inequalities,” American Journal of Mathematics, vol. 93, pp. 107–115, 1971. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. F. Chiarenza and M. Frasca, “Morrey spaces and Hardy-Littlewood maximal function,” Rendiconti di Matematica e delle sue Applicazioni, vol. 7, no. 3-4, pp. 273–279, 1987. View at Google Scholar · View at Zentralblatt MATH
  25. V. I. Burenkov and V. S. Guliyev, “Necessary and sufficient conditions for the boundedness of the Riesz potential in local Morrey-type spaces,” Potential Analysis, vol. 30, no. 3, pp. 211–249, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. V. I. Burenkov, A. Gogatishvili, V. S. Guliyev, and R. Ch. Mustafayev, “Boundedness of the fractional maximal operator in local Morrey-type spaces,” Complex Variables and Elliptic Equations, vol. 55, no. 8–10, pp. 739–758, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. V. I. Burenkov, A. Gogatishvili, V. S. Guliyev, and R. Ch. Mustafayev, “Boundedness of the Riesz potential in local Morrey-type spaces,” Potential Analysis, vol. 35, no. 1, pp. 67–87, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. J. García-Cuerva and María-J.L. Herrero, “A theory of Hardy spaces associated to the Herz spaces,” Proceedings of the London Mathematical Society, vol. 69, no. 3, pp. 605–628, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. K. Morii, T. Sato, Y. Sawano, and H. Wadade, “Sharp constants of Brézis-Gallouët-Wainger type inequalities with a double logarithmic term on bounded domains in Besov and Triebel-Lizorkin spaces,” Boundary Value Problems, vol. 2010, Article ID 584521, 38 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  30. L. Grafakos, Classical Fourier Analysis, vol. 249 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2nd edition, 2008.
  31. D. S. Kurtz, “Littlewood-Paley and multiplier theorems on weighted Lp spaces,” Transactions of the American Mathematical Society, vol. 259, no. 1, pp. 235–254, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  32. N. Kikuchi, E. Nakai, N. Tomita, K. Yabuta, and T. Yoneda, “Calderón-Zygmund operators on amalgam spaces and in the discrete case,” Journal of Mathematical Analysis and Applications, vol. 335, no. 1, pp. 198–212, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  33. J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, vol. 116, North-Holland Publishing Co., Amsterdam, The Netherlands, 1985.
  34. Y. Mizuta, E. Nakai, Y. Sawano, and T. Shimomura, “Gagliardo–Nirenberg inequality for generalized Riesz potentials of functions in Musielak-Orlicz spaces,” Journal of the Mathematical Society of Japan, vol. 98, no. 3, pp. 253–263, 2012. View at Publisher · View at Google Scholar
  35. J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, vol. 223 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 1976.
  36. M. Bownik and K. P. Ho, “Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces,” Transactions of the American Mathematical Society, vol. 358, no. 4, pp. 1469–1510, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH