Regularly bounded functions and Hardy’s inequality

  • Tatyana Ostrogorski
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


We consider Hardy’s inequality
$$\int\limits_{0}^{\infty } {{{{\left( {\frac{1}{t}\int\limits_{0}^{t} F (s)ds} \right)}}^{p}}} W(t)dt \leqslant C\int\limits_{0}^{\infty } {{{F}^{P}}} (t)W(t)dt $$
where W, a positive function, is the weight and 1 ≤ p < ∞. In [6, Th. 330] this inequality is given with the weight W(t) = tα, for 0 < α < p - 1.


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  1. [1]
    Aljancic, A. and Arandjelovic, D. O-regularly varying functions, Publ. Inst. Math. (Beograd), 22 (36) 1977, 5–22.Google Scholar
  2. [2]
    Andersen, K. Weighted generalized Hardy inequalities for non-increasing functions, Can. J. Math. 43 1991, 1121–1135.MATHCrossRefGoogle Scholar
  3. [3]
    Arino, M.M. and Muckenhoupt, B. Maximal functions on classical Lorentz spaces and Hardy’s inequality with weights for nonincreasing functions, Trans. Amer. Math. Soc. 320 1990, 727–735.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Bingham, N.H., Göldie, C.M. and J.L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987.MATHGoogle Scholar
  5. [5]
    Geluk, J.L. and De Haan, L., Regular Variation, Extensions and Tauberian Theorems, CWI Tract, 40, Stichting Mathematish Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1987.MATHGoogle Scholar
  6. [6]
    Hardy, G., Littlewood, J. and Polya, G., Inequalities, Cambridge University Press, 1952.MATHGoogle Scholar
  7. [7]
    Muckenhoupt, B., Hardy’s inequality with weights, Studia Math. 44 1972, 31–38.MathSciNetMATHGoogle Scholar
  8. [8]
    Sawyer, E. Weighted inequalities for the two-dimensional Hardy operator, Studia Math. 82 1985, 1–16.MathSciNetMATHGoogle Scholar
  9. [9]
    Seneta, E., Regularly Varying Functions, Lecture Notes in Mathematics 508, Springer, 1976.MATHCrossRefGoogle Scholar

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© Birkhäuser Boston 1999

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  • Tatyana Ostrogorski

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