Riesz rearrangement inequality

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

History

The inequality was first proved by Frigyes Riesz in 1930,[1] and independently reproved by S.L.Sobolev in 1938. 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 $f$ , $g$ and $h$ 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]

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References

Riesz, Frigyes (1930). "Sur une inégalité intégrale". Journal of the London Mathematical Society. 5 (3): 162–168. doi:10.1112/jlms/s1-5.3.162. MR 1574064.
Brascamp, H.J.; Lieb, Elliott H.; Luttinger, J.M. (1974). "A general rearrangement inequality for multiple integrals". Journal of Functional Analysis. 17: 227–237. MR 0346109.
Hardy, G. H.; Littlewood, J. E.; Polya, G. (1952). Inequalities. Cambridge: Cambridge University Press. ISBN 978-0-521-35880-4.
Lieb, Elliott; Loss, Michael (2001). Analysis. Graduate Studies in Mathematics. 14 (2nd ed.). American Mathematical Society. ISBN 978-0821827833.
Burchard, Almut (1996). "Cases of Equality in the Riesz Rearrangement Inequality". Annals of Mathematics. 143 (3): 499–527. CiteSeerX 10.1.1.55.3241. doi:10.2307/2118534. JSTOR 2118534.

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[1] Riesz, Frigyes (1930). "Sur une inégalité intégrale". Journal of the London Mathematical Society. 5 (3): 162–168. doi:10.1112/jlms/s1-5.3.162. MR 1574064.

[2] Brascamp, H.J.; Lieb, Elliott H.; Luttinger, J.M. (1974). "A general rearrangement inequality for multiple integrals". Journal of Functional Analysis. 17: 227–237. MR 0346109.

[3] Hardy, G. H.; Littlewood, J. E.; Polya, G. (1952). Inequalities. Cambridge: Cambridge University Press. ISBN 978-0-521-35880-4.

[4] Lieb, Elliott; Loss, Michael (2001). Analysis. Graduate Studies in Mathematics. 14 (2nd ed.). American Mathematical Society. ISBN 978-0821827833.

[5] Burchard, Almut (1996). "Cases of Equality in the Riesz Rearrangement Inequality". Annals of Mathematics. 143 (3): 499–527. CiteSeerX 10.1.1.55.3241. doi:10.2307/2118534. JSTOR 2118534.