DOI QR코드

DOI QR Code

SOME PROOFS OF THE CLASSICAL INTEGRAL HARDY INEQUALITY

  • Received : 2014.05.05
  • Accepted : 2014.07.25
  • Published : 2014.09.30

Abstract

We present some proofs of the classical integral Hardy inequality. Our approach makes use of continuous functions with compact support in (0, ${\infty}$), homogeneity of the norm and Schur's criterion for integral operators.

Keywords

References

  1. S. Abramovich, K. Krulic, J. Pecaric and L.-E. Persson, Some new refined Hardy type inequalities with general kernels and measures, Aequat. Math. 79 (2010), 157-172. https://doi.org/10.1007/s00010-010-0011-8
  2. G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed., John Willey and Sons, Inc., New York, 1999.
  3. C. Gasquet and P. Witomski, Fourier Analysis and Applications: Filtering, Numerical Computation, Wavelets, Springer-Verlag New York, Inc., 1999.
  4. G. H. Hardy, Note on a theorem of Hilbert, Math. Z. 6 (1920), 314-317. https://doi.org/10.1007/BF01199965
  5. G. H. Hardy, J. E. Littlewood and G.Polya, Inequalities, 2nd ed., Cambridge University Press, Cambridge, 1952.
  6. S. Kaijser, L. Nikolova, L.- E. Persson and A. Wedestig, Hardy-Type Inequalities Via Convexity, J. Math. Ineq. Appl., 8 (3) (2005), 403-417.
  7. A. Kufner, L. Maligranda and L.- E. Persson, The Hardy inequality: About its History and Some Related Results, Vydavatelsky Sevis Publishing House, Pilsen, 2007.
  8. A. Kufner, L. Maligranda and L.-E. Persson, The Prehistory of the Hardy Inequality, Amer. Math. Monthly, 113 (8) (2006), 715-732. https://doi.org/10.2307/27642033
  9. J. A. Oguntuase and L.- E. Persson, Refinement of Hardy inequalities via superquadratic and subquadratic functions, J. Math. Anal. Appl., 339 (2) (2008), 1305-1312. https://doi.org/10.1016/j.jmaa.2007.08.007
  10. W. Rudin, Real and Complex Analysis, 2nd edition, McGraw-Hill Inc, 1974.
  11. T. Tao, Analysis I, Hindustan Book Agency, Volume I, 2006.
  12. K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Springer, NewYork, 2005.