Riesz rearrangement inequality
In mathematics, the Riesz rearrangement inequality, sometimes called Riesz–Sobolev inequality, states that any three non-negative functions [math]\displaystyle{ f : \mathbb{R}^n \to \mathbb{R}^+ }[/math], [math]\displaystyle{ g : \mathbb{R}^n \to \mathbb{R}^+ }[/math] and [math]\displaystyle{ h : \mathbb{R}^n \to \mathbb{R}^+ }[/math] satisfy the inequality
- [math]\displaystyle{ \iint_{\mathbb{R}^n\times \mathbb{R}^n} f(x) g(x-y) h(y) \, dx\,dy \le \iint_{\mathbb{R}^n\times \mathbb{R}^n} f^*(x) g^*(x-y) h^*(y) \, dx\,dy, }[/math]
where [math]\displaystyle{ f^* : \mathbb{R}^n \to \mathbb{R}^+ }[/math], [math]\displaystyle{ g^* : \mathbb{R}^n \to \mathbb{R}^+ }[/math] and [math]\displaystyle{ h^* : \mathbb{R}^n \to \mathbb{R}^+ }[/math] are the symmetric decreasing rearrangements of the functions [math]\displaystyle{ f }[/math], [math]\displaystyle{ g }[/math] and [math]\displaystyle{ h }[/math] 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 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 [math]\displaystyle{ f }[/math], [math]\displaystyle{ g }[/math] and [math]\displaystyle{ h }[/math] 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]
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.
- ↑ Brascamp, H.J.; Lieb, Elliott H.; Luttinger, J.M. (1974). "A general rearrangement inequality for multiple integrals". Journal of Functional Analysis 17: 227–237.
- ↑ Hardy, G. H.; Littlewood, J. E.; Polya, G. (1952). Inequalities. Cambridge: Cambridge University Press. ISBN 978-0-521-35880-4. https://archive.org/details/inequalities0000hard.
- ↑ Lieb, Elliott; Loss (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. doi:10.2307/2118534.
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