Hardy–Littlewood inequality

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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[edit]

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[edit]

References[edit]

  1. ^ a b Lieb, Elliott; Loss, Michael (2001). Analysis. Graduate Studies in Mathematics. 14 (2nd ed.). American Mathematical Society. ISBN 978-0821827833. 
  2. ^ a b Burchard, Almut. A Short Course on Rearrangement Inequalities (PDF).