Riesz rearrangement inequality

In mathematics, the Riesz rearrangement inequality (sometimes called Riesz-Sobolev inequality)states that for any three non-negative functions f,g,h, the integral

$I(f,g,h)=\iint _{\mathbb {R} ^{n}\times \mathbb {R} ^{n}}f(x)g(x-y)h(y)\,dxdy$

satisfies the inequality

$I(f,g,h)\leq I(f^{*},g^{*},h^{*})$

where $f^{*},g^{*},h^{*}$ are the symmetric decreasing rearrangements of the functions f,g, and h, respectively.

The inequality was first proved by Frigyes Riesz in 1930, and independently reproved by S.L.Sobolev in 1938. It can be generalized to arbitrarily (but finitely) many functions acting on arbitrarily many variables. 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.

Sources

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

Historical references

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

Further reading

• Hardy, G. H.; Littlewood, J. E.; Polya, G. (1952), Inequalities, Cambridge: Cambridge University Press, ISBN 0-521-35880-9