Hardy–Littlewood inequality explained

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

Rⁿ

, then

\int
	Rⁿ

f(x)g(x)dx\leq

\int
	Rⁿ

f^*(x)g^*(x)dx

where

f^*

and

g^*

are the symmetric decreasing rearrangements of

and

, respectively.^[1] ^[2]

The decreasing rearrangement

f^*

is defined via the property that for all

r>0

the two super-level sets

E_{f(r)=\left\{x\in}X:f(x)>r\right\}

and

E
	f^*

(r)=\left\{x\inX:f^{*(x)>r\right\}}

have the same volume (

-dimensional Lebesgue measure) and

E
	f^*

(r)

is a ball in

Rⁿ

centered at

x=0

, i.e. it has maximal symmetry.

Proof

The layer cake representation^[1] ^[2] allows us to write the general functions

and

in the form

f(x)=

	infty
\int
	0

\chi_f(x)>rdr

and

g(x)=

	infty
\int
	0

\chi_g(x)>sds

where

r\mapsto\chi_f(x)>r

equals

for

r<f(x)

and

otherwise. Analogously,

s\mapsto\chi_g(x)>s

equals

for

s<g(x)

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

x\inRⁿ

the conditions

f(x)>r

and

g(x)>s

the indicator functions

x\mapsto

\chi
	E_f(r)

(x)

and

x\mapsto

\chi
	E_g(s)

(x)

appear with the superlevel sets

E_f(r)

and

E_g(s)

as introduced above:

\int
	Rⁿ

f(x)g(x)dx=

\displaystyle\int
	Rⁿ

	infty
\int
	0

\chi_f(x)>rdr

	infty
\int
	0

\chi_g(x)>sdsdx=

\int
	Rⁿ

	infty
\int
	0

	infty
\int
	0

\chi_f(x)>r \chi_g(x)>sdrdsdx

	infty
\int
	0

	infty
\int
	0

\int
	Rⁿ

\chi
	E_f(r)

(x)

\chi
	E_g(s)

(x)dxdrds =

	infty
\int
	0

	infty
\int
	0

\int
	Rⁿ

\chi
	E_f(r)\capE_g(s)

(x)dxdrds.

Denoting by

\mu

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:

	infty
\int
	0

	infty
\int
	0

\mu\left(E_f(r)\capE_g(s)\right)drds

\leq

	infty
\int
	0

	infty
\int
	0

min\left\{\mu(E_f(r)),\mu(E_g(s))\right\}drds

	infty
\int
	0

	infty
\int
	0

min\left\{\mu(E
	f^*

(r)),

\mu(E
	g^*

(s))\right\}drds.

Now, we use that the superlevel sets

E
	f^*

(r)

and

E
	g^*

(s)

are balls in

Rⁿ

centered at

x=0

, which implies that

E
	f^*

(r)\cap

E
	g^*

(s)

is exactly the smaller one of the two balls:

	infty
\int
	0

	infty
\int
	0

\mu\left(

E
	f^*

(r)\cap

E
	g^*

(s)\right)drds

\int
	Rⁿ

f^*(x)g^*(x)dx

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

\mu

and finite non-zero variance

\sigma²

, then using the Hardy–Littlewood inequality, it can be proved that for

0<\delta<1

the

\delta^th

reciprocal moment for the absolute value of

\begin{align} \operatorname{E}\left[

	1
	\vertX\vert^\delta

\right]&\leq

	(1-\delta)
	2

\Gamma\left(	1-\delta	\right)
	2

\sigma^\delta\sqrt{2\pi

}irrespectiveofthevalueof\mu\inR.\end{align}

^[3]

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

References

Book: Lieb. Elliott. Elliott H. Lieb. Loss. Michael. Michael Loss. Analysis. 2001. 2nd. American Mathematical Society. Graduate Studies in Mathematics. 14. 978-0821827833.
Book: Burchard, Almut. A Short Course on Rearrangement Inequalities. Almut Burchard.
Pal . Subhadip . Khare . Kshitij . Geometric ergodicity for Bayesian shrinkage models . Electronic Journal of Statistics . 2014 . 8 . 1 . 604–645 . 10.1214/14-EJS896 . 1935-7524. free .

Hardy–Littlewood inequality explained

Proof

An application

See also

References