Riesz rearrangement inequality

In mathematics, the Riesz rearrangement inequality (sometimes called Riesz-Sobolev inequality) states that for any three non-negative functions , and satisfies the inequality

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. 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]

gollark: iPhones are quite expensive, so if you value your time at $50/hour (this might be low, I'm not really sure), it would probably take a few years for the iPhone to pay off, but it could actually come out in favour if it does in fact save that much time.
gollark: I don't get anything like that on my *£120* Android phone from recently, except in Discord, in which the keyboard is occasionally ridiculously laggy due to what I assume is bad design on their end.
gollark: (very fermi estimation, but it's probably not THAT many orders of magnitude out)
gollark: If we assume you open the keyboard, I don't know, 50 times a day, and it takes 0.5 seconds each time, this is 25 seconds a day, or 144 days for it to cost an hour of time.
gollark: This seems dubious, even if we ignore the implication that there aren't reasonably fast Android phones.

References

  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.
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