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

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

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