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

From layer cake representation we have:^[1][2]

$f(x)= \int_0^\infty \chi_{f(x)>r} \, dr$

$g(x)= \int_0^\infty \chi_{g(x)>s} \, ds$

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

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

$\int_{\mathbb{R}^n} f(x)g(x) \, dx = \displaystyle\int_{\mathbb{R}^n}\int_0^\infty \int_0^\infty \chi_{f(x)>r}\chi_{g(x)>s} \, dr \, ds \, dx$

$= \int_0^\infty \int_0^\infty \int_{\mathbb{R}^n}\chi_{f(x)>r\cap g(x)>s} \, dx \, dr \, ds$

$= \int_0^\infty \int_0^\infty \mu\left(\left\{f(x)>r\right\}\cap\left\{ g(x)>s\right\}\right) \, dr \, ds$

$\leq \int_0^\infty \int_0^\infty \min\left(\mu\left(f(x)>r\right);\mu\left(g(x)>s\right)\right) \, dr \, ds$

$= \int_0^\infty \int_0^\infty \min\left(\mu\left(f^*(x)>r\right);\mu\left(g^*(x)>s\right)\right) \, dr \, ds$

$= \int_0^\infty \int_0^\infty \mu\left(\left\{f^\ast(x)>r\right\}\cap\left\{ g^\ast(x)>s\right\}\right) \, dr \, ds$

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

See also

• Rearrangement inequality
• Chebyshev's sum inequality
• Lorentz space

References

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