Hardy–Littlewood inequality

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

\int _{\mathbb {R} ^{n}}f(x)g(x)\,dx\leq \int _{\mathbb {R} ^{n}}f^{*}(x)g^{*}(x)\,dx

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

Proof

f(x)=\int _{0}^{\infty }\chi _{f(x)>r}\,dr

g(x)=\int _{0}^{\infty }\chi _{g(x)>s}\,ds

where $\chi _{f(x)>r}$ denotes the indicator function of the subset E_f given by

E_{f}=\left\{x\in X:f(x)>r\right\}

Analogously, $\chi _{g(x)>s}$ denotes the indicator function of the subset E_g given by

E_{g}=\left\{x\in X:g(x)>s\right\}

\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

=\int _{0}^{\infty }\int _{0}^{\infty }\int _{\mathbb {R} ^{n}}\chi _{f(x)>r\cap g(x)>s}\,dx\,dr\,ds

=\int _{0}^{\infty }\int _{0}^{\infty }\mu \left(\left\{f(x)>r\right\}\cap \left\{g(x)>s\right\}\right)\,dr\,ds

\leq \int _{0}^{\infty }\int _{0}^{\infty }\min \left(\mu \left(f(x)>r\right);\mu \left(g(x)>s\right)\right)\,dr\,ds

=\int _{0}^{\infty }\int _{0}^{\infty }\min \left(\mu \left(f^{*}(x)>r\right);\mu \left(g^{*}(x)>s\right)\right)\,dr\,ds

=\int _{0}^{\infty }\int _{0}^{\infty }\mu \left(\left\{f^{\ast }(x)>r\right\}\cap \left\{g^{\ast }(x)>s\right\}\right)\,dr\,ds

=\int _{\mathbb {R} ^{n}}f^{*}(x)g^{*}(x)\,dx

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

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