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.

History

The inequality was first proved by Frigyes Riesz in 1930,^[1] and independently reproved by S.L.Sobolev in 1938. Brascamp, Lieb and Luttinger have shown that it can be generalized to arbitrarily (but finitely) many functions acting on arbitrarily many variables.^[2]

Applications

The Riesz rearrangement inequality can be used to prove the Pólya–Szegő inequality.

Proofs

One-dimensional case

In the one-dimensional case, the inequality is first proved when the functions ${\displaystyle f}$, ${\displaystyle g}$ and ${\displaystyle h}$ are characteristic functions of a finite unions of intervals. Then the inequality can be extended to characteristic functions of measurable sets, to measurable functions taking a finite number of values and finally to nonnegative measurable functions.^[3]

Higher-dimensional case

In order to pass from the one-dimensional case to the higher-dimensional case, the spherical rearrangement is approximated by Steiner symmetrization for which the one-dimensional argument applies directly by Fubini's theorem.^[4]

Equality cases

In the case where any one of the three functions is a strictly symmetric-decreasing function, equality holds only when the other two functions are equal, up to translation, to their symmetric-decreasing rearrangements.^[5]