Riesz rearrangement inequality information

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

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

Riesz rearrangement inequality information

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