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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 function|measurable]] [[real functions]] vanishing at [[infinity]] that are defined on ''n''-[[dimension]]al [[Euclidean space]] '''R'''<sup>''n''</sup> then
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| :<math>\int_{\mathbb{R}^n} f(x)g(x) \, dx \leq \int_{\mathbb{R}^n} f^*(x)g^*(x) \, dx</math>
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| where ''f''<sup>*</sup> and ''g''<sup>*</sup> are the [[symmetric decreasing rearrangement]]s of ''f''(''x'') and ''g''(''x''), respectively.<ref name=liebloss>{{cite book | author=Lieb, Elliott H., & Loss, Michael | title=Analysis | edition=Second | publisher=American Mathematical Society | location=Providence, RI | year=2001 | isbn=0-8218-2783-9 }}
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| </ref><ref name=burchard>{{cite book|title=A Short Course on Rearrangement Inequalities|first=Almut|last=Burchard|url=http://www.math.toronto.edu/almut/rearrange.pdf}}</ref> | |
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| ==Proof==
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| From [[layer cake representation]] we have:<ref name=liebloss/><ref name=burchard/>
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| :<math>f(x)= \int_0^\infty \chi_{f(x)>r} \, dr</math>
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| :<math>g(x)= \int_0^\infty \chi_{g(x)>s} \, ds</math> | |
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| where <math>\chi_{f(x)>r}</math> denotes the [[indicator function]] of the subset ''E''<sub> ''f''</sub> given by
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| :<math>E_f=\left\{x\in X: f(x)>r\right\} \, </math> | |
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| Analogously, <math>\chi_{g(x)>s}</math> denotes the indicator function of the subset ''E''<sub> ''g''</sub> given by
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| :<math>E_g=\left\{x\in X: g(x)>s\right\} \, </math>
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| :<math>\begin{align}
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| \int_{\mathbb{R}^n} f(x)g(x) \, dx &= \displaystyle\int_{\mathbb{R}^n}\int_0^\infty \int_0^\infty \chi_{f(x)>r}\chi_{g(x)>s} \, dr \, ds \, dx \\[8pt]
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| &= \int_0^\infty \int_0^\infty \int_{\mathbb{R}^n}\chi_{f(x)>r\cap g(x)>s} \, dx \, dr \, ds \\[8pt]
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| &= \int_0^\infty \int_0^\infty \mu\left(\left\{f(x)>r\right\}\cap\left\{ g(x)>s\right\}\right) \, dr \, ds\\[8pt]
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| &\leq \int_0^\infty \int_0^\infty \min\left(\mu\left(f(x)>r\right);\mu\left(g(x)>s\right)\right) \, dr \, ds\\[8pt]
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| &= \int_0^\infty \int_0^\infty \min\left(\mu\left(f^*(x)>r\right);\mu\left(g^*(x)>s\right)\right) \, dr \, ds\\[8pt]
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| &= \int_0^\infty \int_0^\infty \mu\left(\left\{f^\ast(x)>r\right\}\cap\left\{ g^\ast(x)>s\right\}\right) \, dr \, ds\\[8pt]
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| &= \int_{\mathbb{R}^n} f^*(x)g^*(x) \, dx
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| \end{align}
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| </math>
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| ==See also==
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| * [[Rearrangement inequality]]
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| * [[Chebyshev's sum inequality]]
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| ==References==
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| <references/>
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| {{DEFAULTSORT:Hardy-Littlewood inequality}}
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| [[Category:Inequalities]]
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| [[Category:Articles containing proofs]]
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