- Proof
- See also
- References
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
- Rearrangement inequality
- Chebyshev's sum inequality
- Lorentz space
References
1. ^1 {{cite book|last1=Lieb|first1=Elliott|authorlink1=Elliott H. Lieb|last2=Loss|first2=Michael|author2-link=Michael Loss|title=Analysis|year=2001|edition=2nd|publisher=American Mathematical Society|series=Graduate Studies in Mathematics|volume=14|isbn=978-0821827833}}
2. ^1 {{cite book|title=A Short Course on Rearrangement Inequalities|first=Almut|last=Burchard|url=http://www.math.toronto.edu/almut/rearrange.pdf}}
{{DEFAULTSORT:Hardy-Littlewood inequality}}2. ^1 {{cite book|title=A Short Course on Rearrangement Inequalities|first=Almut|last=Burchard|url=http://www.math.toronto.edu/almut/rearrange.pdf}}
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