Hardy–Littlewood inequality

In mathematical analysis, the Hardy–Littlewood inequality, named after G. H. Hardy and John Edensor Littlewood, states that if ${\displaystyle f}$ and ${\displaystyle g}$ are nonnegative measurable real functions vanishing at infinity that are defined on ${\displaystyle n}$-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{n}}$, then

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

where ${\displaystyle f^{*}}$ and ${\displaystyle g^{*}}$ are the symmetric decreasing rearrangements of ${\displaystyle f}$ and ${\displaystyle g}$, respectively.

The decreasing rearrangement ${\displaystyle f^{*}}$ of ${\displaystyle f}$ is defined via the property that for all ${\displaystyle r>0}$ the two super-level sets

${\displaystyle E_{f}(r)=\left\{x\in X:f(x)>r\right\}\quad }$ and ${\displaystyle \quad E_{f^{*}}(r)=\left\{x\in X:f^{*}(x)>r\right\}}$

have the same volume (${\displaystyle n}$-dimensional Lebesgue measure) and ${\displaystyle E_{f^{*}}(r)}$ is a ball in ${\displaystyle \mathbb {R} ^{n}}$ centered at ${\displaystyle x=0}$, i.e. it has maximal symmetry.