Hardy–Littlewood inequality

In mathematical analysis, the Hardy–Littlewood inequality, named after G. H. Hardy and John Edensor Littlewood, states that if and are nonnegative measurable real functions vanishing at infinity that are defined on -dimensional Euclidean space , then

where and are the symmetric decreasing rearrangements of and , respectively.[1][2]

The decreasing rearrangement of is defined via the property that for all the two super-level sets

and

have the same volume (-dimensional Lebesgue measure) and is a ball in centered at , i.e. it has maximal symmetry.

Proof

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The layer cake representation[1][2] allows us to write the general functions   and   in the form

  and  

where   equals   for   and   otherwise. Analogously,   equals   for   and   otherwise.

Now the proof can be obtained by first using Fubini's theorem to interchange the order of integration. When integrating with respect to   the conditions   and   the indicator functions   and   appear with the superlevel sets   and   as introduced above:

 
 

Denoting by   the  -dimensional Lebesgue measure we continue by estimating the volume of the intersection by the minimum of the volumes of the two sets. Then, we can use the equality of the volumes of the superlevel sets for the rearrangements:

 
 
 

Now, we use that the superlevel sets   and   are balls in   centered at  , which implies that   is exactly the smaller one of the two balls:

 
 

The last identity follows by reversing the initial five steps that even work for general functions. This finishes the proof.

An application

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Let random variable   is Normally distributed with mean   and finite non-zero variance  , then using the Hardy–Littlewood inequality, it can be proved that for   the   reciprocal moment for the absolute value of   is

 [3]


The technique that is used to obtain the above property of the Normal distribution can be utilized for other unimodal distributions.

See also

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References

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  1. ^ a b Lieb, Elliott; Loss, Michael (2001). Analysis. Graduate Studies in Mathematics. Vol. 14 (2nd ed.). American Mathematical Society. ISBN 978-0821827833.
  2. ^ a b Burchard, Almut. A Short Course on Rearrangement Inequalities (PDF).
  3. ^ Pal, Subhadip; Khare, Kshitij (2014). "Geometric ergodicity for Bayesian shrinkage models". Electronic Journal of Statistics. 8 (1): 604–645. doi:10.1214/14-EJS896. ISSN 1935-7524.