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 (m ...
, 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 Geometry#Length, area, and volume, geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly ...
real functions vanishing at
infinity
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
that are defined on
-
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
al
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, 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 ...
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 In mathematics, the layer cake representation of a non- negative, real-valued measurable function f defined on a measure space (\Omega,\mathcal,\mu) is the formula
:f(x) = \int_0^\infty 1_ (x) \, \mathrmt,
for all x \in \Omega, where 1_E denotes t ...
[ 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 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
:
The technique that is used to obtain the above property of the Normal distribution can be utilized for other unimodal distributions.
See also
* Rearrangement inequality In mathematics, the rearrangement inequality states that
x_n y_1 + \cdots + x_1 y_n
\leq x_ y_1 + \cdots + x_ y_n
\leq x_1 y_1 + \cdots + x_n y_n
for every choice of real numbers
x_1 \leq \cdots \leq x_n \quad \text \quad y_1 \leq \cdots \leq y_n ...
* Chebyshev's sum inequality
In mathematics, Chebyshev's sum inequality, named after Pafnuty Chebyshev, states that if
:a_1 \geq a_2 \geq \cdots \geq a_n \quad and \quad b_1 \geq b_2 \geq \cdots \geq b_n,
then
: \sum_^n a_k b_k \geq \left(\sum_^n a_k\right)\!\!\left(\sum_^n ...
* Lorentz space In mathematical analysis, Lorentz spaces, introduced by George G. Lorentz in the 1950s,G. Lorentz, "On the theory of spaces Λ", ''Pacific Journal of Mathematics'' 1 (1951), pp. 411-429. are generalisations of the more familiar L^ spaces.
The Lor ...
References
{{DEFAULTSORT:Hardy-Littlewood inequality
Inequalities
Articles containing proofs