Table of Contents Author Guidelines Submit a Manuscript
Journal of Function Spaces and Applications
Volume 2012, Article ID 682960, 28 pages
http://dx.doi.org/10.1155/2012/682960
Research Article

Lebesgue's Differentiation Theorems in R.I. Quasi-Banach Spaces and Lorentz Spaces Γ𝑝,𝑀

1Institute of Mathematics, PoznaΕ„ University of Technology, Piotrowo 3a, 60-965 PoznaΕ„, Poland
2Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, USA

Received 10 October 2011; Accepted 20 January 2012

Academic Editor: Lech Maligranda

Copyright Β© 2012 Maciej Ciesielski and Anna KamiΕ„ska. 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. Mazzone and H. Cuenya, β€œMaximal inequalities and Lebesgue's differentiation theorem for best approximant by constant over balls,” Journal of Approximation Theory, vol. 110, no. 2, pp. 171–179, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  2. C. Bennett and R. Sharpley, Interpolation of Operators, vol. 129 of Pure and Applied Mathematics, Academic Press, Boston, Mass, USA, 1988.
  3. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, NJ, USA, 1970.
  4. J. Bastero, M. Milman, and F. J. Ruiz, β€œRearrangement of Hardy-Littlewood maximal functions in Lorentz spaces,” Proceedings of the American Mathematical Society, vol. 128, no. 1, pp. 65–74, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  5. F. E. Levis, H. H. Cuenya, and A. N. Priori, β€œBest constant approximants in Orlicz-Lorentz spaces,” Commentationes Mathematicae, vol. 48, no. 1, pp. 59–73, 2008. View at Google Scholar Β· View at Zentralblatt MATH
  6. F. E. Levis, β€œWeak inequalities for maximal functions in Orlicz-Lorentz spaces and applications,” Journal of Approximation Theory, vol. 162, no. 2, pp. 239–251, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  7. M. Ciesielski and A. KamiΕ„ska, β€œThe best constant approximant operators in Lorentz spaces Ξ“p,w and their applications,” Journal of Approximation Theory, vol. 162, no. 9, pp. 1518–1544, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  8. S. G. KreΔ­n, Yu. Δͺ. PetunΔ«n, and E. M. SemΓ«nov, Interpolation of Linear Operators, vol. 54 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 1982.
  9. A. KamiΕ„ska and L. Maligranda, β€œOrder convexity and concavity of Lorentz spaces Ξ›p,w, 0<p<∞,” Studia Mathematica, vol. 160, no. 3, pp. 267–286, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  10. G. G. Lorentz, β€œOn the theory of spaces Ξ›,” Pacific Journal of Mathematics, vol. 1, pp. 411–429, 1951. View at Google Scholar Β· View at Zentralblatt MATH
  11. M. A. AriΓ±o and B. Muckenhoupt, β€œMaximal functions on classical Lorentz spaces and Hardy's inequality with weights for nonincreasing functions,” Transactions of the American Mathematical Society, vol. 320, no. 2, pp. 727–735, 1990. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  12. E. Sawyer, β€œBoundedness of classical operators on classical Lorentz spaces,” Studia Mathematica, vol. 96, no. 2, pp. 145–158, 1990. View at Google Scholar Β· View at Zentralblatt MATH
  13. V. D. Stepanov, β€œThe weighted Hardy's inequality for nonincreasing functions,” Transactions of the American Mathematical Society, vol. 338, no. 1, pp. 173–186, 1993. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  14. H. L. Royden, Real Analysis, Macmillan Publishing Company, New York, NY, USA, 3rd edition, 1988.
  15. N. L. Carothers, R. Haydon, and P.-K. Lin, β€œOn the isometries of the Lorentz function spaces,” Israel Journal of Mathematics, vol. 84, no. 1-2, pp. 265–287, 1993. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  16. J. V. Ryff, β€œMeasure preserving transformations and rearrangements,” Journal of Mathematical Analysis and Applications, vol. 31, pp. 449–458, 1970. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  17. R. E. Megginson, An Introduction to Banach Space Theory, vol. 183 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1998.
  18. E. W. Cheney, Introduction to Approximation Theory, AMS Chelsea Publishing, Providence, RI, USA, 1998.
  19. A. M. Pinkus, On L1-Approximation, vol. 93 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, UK, 1989.
  20. M. Ciesielski, A. KamiΕ„ska, and R. PΕ‚uciennik, β€œGateaux derivatives and their applications to approximation in Lorentz spaces Ξ“p,w,” Mathematische Nachrichten, vol. 282, no. 9, pp. 1242–1264, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  21. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. II. Function Spaces, vol. 97 of Results in Mathematics and Related Areas, Springer, Berlin, Germany, 1979.
  22. M. J. Carro, J. A. Raposo, and J. Soria, β€œRecent developments in the theory of Lorentz spaces and weighted inequalities,” Memoirs of the American Mathematical Society, vol. 187, no. 877, 2007. View at Google Scholar Β· View at Zentralblatt MATH
  23. N. J. Kalton, N. T. Peck, and J. W. Roberts, An F-Space Sampler, vol. 89 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, UK, 1984.
  24. A. KamiΕ„ska and L. Maligranda, β€œOn Lorentz spaces Ξ“p,w,” Israel Journal of Mathematics, vol. 140, pp. 285–318, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  25. J. Bastero and F. J. Ruiz, β€œElementary reverse HΓΆlder type inequalities with application to operator interpolation theory,” Proceedings of the American Mathematical Society, vol. 124, no. 10, pp. 3183–3192, 1996. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH