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Hardy-type Operators

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

Abstract

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,
$$ \int_0^\infty {{F^p}} (x){x^{p(\varepsilon - 1)}}dx \leqslant C\int_0^\infty {{f^p}} (x){x^{\varepsilon p}}dx $$
(2.1.1)
for some constant C > 0 independent of f. If ε > 1/p′, the inequality takes the form
$$ \int_0^\infty {{G^p}} (x){x^{p(\varepsilon - 1)}}dx \leqslant C\int_0^\infty {{f^p}(x){x^{\varepsilon p}}} dx $$
(2.1.2)
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
$$ C = {\left| {\varepsilon - 1/p'} \right|^{ - p}}. $$
(2.1.3)

Keywords

Approximation Number Hardy Operator Entropy Number Finite Degree Polygonal Path 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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