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.

Proof

From layer cake representation we have:

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

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

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

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

${\begin{aligned}\int _{\mathbb {R} ^{n}}f(x)g(x)\,dx&=\displaystyle \int _{\mathbb {R} ^{n}}\int _{0}^{\infty }\int _{0}^{\infty }\chi _{f(x)>r}\chi _{g(x)>s}\,dr\,ds\,dx\\[8pt]&=\int _{\mathbb {R} ^{n}}\int _{0}^{\infty }\int _{\mathbb {R} ^{n}}\chi _{f(x)>r\cap g(x)>s}\,dx\,dr\,ds\\[8pt]&=\int _{0}^{\infty }\int _{0}^{\infty }\mu \left(\left\{\chi _{f(x)>r\cap g(x)>s}\right\}\right)\,dr\,ds\\[8pt]&\leq \int _{0}^{\infty }\int _{0}^{\infty }\min \left(\mu \left(f(x)>r\right);\mu \left(g(x)>s\right)\right)\,dr\,ds\\[8pt]&=\int _{0}^{\infty }\int _{0}^{\infty }\min \left(\mu \left(f^{*}(x)>r\right);\mu \left(g^{*}(x)>s\right)\right)\,dr\,ds\\[8pt]&=\int _{0}^{\infty }\int _{0}^{\infty }\mu \left(\left\{\chi _{f^{*}(x)>r\cap g^{*}(x)>s}\right\}\right)\,dr\,ds\\[8pt]&=\int _{\mathbb {R} ^{n}}f^{*}(x)g^{*}(x)\,dx\end{aligned}}$

See also

• Rearrangement inequality
• Chebyshev's sum inequality