Riesz rearrangement inequality

In mathematics, the Riesz rearrangement inequality, sometimes called Riesz–Sobolev inequality, states that any three non-negative functions ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{+}}$, ${\displaystyle g:\mathbb {R} ^{n}\to \mathbb {R} ^{+}}$ and ${\displaystyle h:\mathbb {R} ^{n}\to \mathbb {R} ^{+}}$ satisfy the inequality

${\displaystyle \iint _{\mathbb {R} ^{n}\times \mathbb {R} ^{n}}f(x)g(x-y)h(y)\,dx\,dy\leq \iint _{\mathbb {R} ^{n}\times \mathbb {R} ^{n}}f^{*}(x)g^{*}(x-y)h^{*}(y)\,dx\,dy,}$

where ${\displaystyle f^{*}:\mathbb {R} ^{n}\to \mathbb {R} ^{+}}$, ${\displaystyle g^{*}:\mathbb {R} ^{n}\to \mathbb {R} ^{+}}$ and ${\displaystyle h^{*}:\mathbb {R} ^{n}\to \mathbb {R} ^{+}}$ are the symmetric decreasing rearrangements of the functions ${\displaystyle f}$, ${\displaystyle g}$ and ${\displaystyle h}$ respectively.