Riesz rearrangement inequality explained
In mathematics, the Riesz rearrangement inequality, sometimes called Riesz–Sobolev inequality, states that any three non-negative functions
,
and
satisfy the inequality
f(x)g(x-y)h(y)dxdy
\le
f*(x)g*(x-y)h*(y)dxdy,
where
,
and
are the
symmetric decreasing rearrangements of the functions
,
and
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
,
and
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]
Notes and References
- Riesz . Frigyes . Frigyes Riesz. 1930. Sur une inégalité intégrale. Journal of the London Mathematical Society. 5. 3 . 162–168. 10.1112/jlms/s1-5.3.162. 1574064.
- Brascamp. H.J.. Lieb. Elliott H.. Elliott H. Lieb. Luttinger. J.M.. 1974. A general rearrangement inequality for multiple integrals. Journal of Functional Analysis. 17. 227–237. 0346109.
- Book: G. H. . Hardy. G. H. Hardy . Littlewood. J. E.. J. E. Littlewood. Polya. G.. G. Polya. Inequalities. registration . 1952 . Cambridge University Press. Cambridge . 978-0-521-35880-4.
- Book: Lieb. Elliott. Elliott H. Lieb. Loss. Michael. Michael Loss. Analysis. 2001. 2nd. American Mathematical Society. Graduate Studies in Mathematics. 14. 978-0821827833.
- Burchard. Almut. Almut Burchard. 1996. Cases of Equality in the Riesz Rearrangement Inequality. 2118534. Annals of Mathematics. 143. 3. 499–527. 10.2307/2118534. 10.1.1.55.3241.
