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Hardy–Littlewood inequality

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In mathematical analysis, the Hardy–Littlewood inequality, named after G. H. Hardy and John Edensor Littlewood, states that if f and g are nonnegative measurable real functions vanishing at infinity that are defined on n-dimensional Euclidean space Rn then

where f* and g* are the symmetric decreasing rearrangements of f(x) and g(x), respectively.[1][2]

Proof

From layer cake representation we have[1][2]:

where denotes the indicator function of the subset E f given by

Analogously, denotes the indicator function of the subset E g given by

See also

References

  1. ^ a b Lieb, Elliott H., & Loss, Michael (2001). Analysis (Second ed.). Providence, RI: American Mathematical Society. ISBN 0-8218-2783-9.{{cite book}}: CS1 maint: multiple names: authors list (link)
  2. ^ a b Burchard, Almut. A Short Course on Rearrangement Inequalities (PDF).