In mathematical analysis, the Hardy–Littlewood inequality, named after G. H. Hardy and John Edensor Littlewood, states that if
are nonnegative measurable real functions vanishing at infinity that are defined on
-dimensional Euclidean space
are the symmetric decreasing rearrangements of
, respectively.
[1][2] The decreasing rearrangement
is defined via the property that for all
the two super-level sets have the same volume (
-dimensional Lebesgue measure) and
, i.e. it has maximal symmetry.
The layer cake representation[1][2] allows us to write the general functions
equals
Analogously,
equals
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
the indicator functions
appear with the superlevel sets
as introduced above: Denoting by
-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
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.
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
reciprocal moment for the absolute value of
The technique that is used to obtain the above property of the Normal distribution can be utilized for other unimodal distributions.