Hardy-type Operators

  • David E. Edmunds
  • W. Desmond Evans
Part of the Springer Monographs in Mathematics book series (SMM)


In [118] Hardy proved the following celebrated inequality: let 1 < p < ∞ and set F(x) = ∫ o x f (t)dt, where f is a non-negative measurable function on (0, ∞). Then, if ε < 1/p′ = 1–1/p,
$$ \int_0^\infty {{F^p}} (x){x^{p(\varepsilon - 1)}}dx \leqslant C\int_0^\infty {{f^p}} (x){x^{\varepsilon p}}dx $$
for some constant C > 0 independent of f. If ε > 1/p′, the inequality takes the form
$$ \int_0^\infty {{G^p}} (x){x^{p(\varepsilon - 1)}}dx \leqslant C\int_0^\infty {{f^p}(x){x^{\varepsilon p}}} dx $$
where G(x) = x f (t)dt. The best possible constants C in (2.1.1) and (2.1.2) are equal and this common value was determined by Landau in [150] as
$$ C = {\left| {\varepsilon - 1/p'} \right|^{ - p}}. $$


Approximation Number Hardy Operator Entropy Number Finite Degree Polygonal Path 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • David E. Edmunds
    • 1
  • W. Desmond Evans
    • 2
  1. 1.Department of MathematicsSussex UniversityBrightonUK
  2. 2.School of MathematicsCardiff UniversityCardiffUK

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