In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
, the Hardy–Littlewood inequality, named after
G. H. Hardy
Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
and
John Edensor Littlewood
John Edensor Littlewood (9 June 1885 – 6 September 1977) was a British mathematician. He worked on topics relating to analysis, number theory, and differential equations and had lengthy collaborations with G. H. Hardy, Srinivasa Ramanu ...
, states that if
and
are nonnegative
measurable
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts hav ...
real functions vanishing at
infinity
Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol.
From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophic ...
that are defined on
-
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
al
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
, then
:
where
and
are the
symmetric decreasing rearrangement In mathematics, the symmetric decreasing rearrangement of a function is a function which is symmetric and decreasing, and whose level sets are of the same size as those of the original function.
Definition for sets
Given a measurable set, A, in \R ...
s of
and
, respectively.
[
]
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
The
layer cake representation[ 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
Let be a normally-distributed random variable with mean and finite non-zero variance . Using the Hardy–Littlewood inequality, it can be proved that for the reciprocal moment for the absolute value of is bounded above as
:
The technique used to obtain the above property of the normal distribution can be applied to other unimodal distributions.
See also
* Rearrangement inequality
* Chebyshev's sum inequality
* Lorentz space
References
{{DEFAULTSORT:Hardy-Littlewood inequality
Inequalities (mathematics)
Articles containing proofs