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In mathematical analysis , the Hardy–Littlewood inequality , named after G. H. Hardy and John Edensor Littlewood , states that if f and g are nonnegative measurable real functions vanishing at infinity that are defined on n -dimensional Euclidean space R n
∫
R
n
f
(
x
)
g
(
x
)
d
x
≤
∫
R
n
f
∗
(
x
)
g
∗
(
x
)
d
x
{\displaystyle \int _{\mathbb {R} ^{n}}f(x)g(x)\,dx\leq \int _{\mathbb {R} ^{n}}f^{*}(x)g^{*}(x)\,dx}
where f * and g * are the symmetric decreasing rearrangements of f (x ) and g (x ), respectively.[ 1] [ 2]
Proof
From layer cake representation we have:[ 1] [ 2]
f
(
x
)
=
∫
0
∞
χ
f
(
x
)
>
r
d
r
{\displaystyle f(x)=\int _{0}^{\infty }\chi _{f(x)>r}\,dr}
g
(
x
)
=
∫
0
∞
χ
g
(
x
)
>
s
d
s
{\displaystyle g(x)=\int _{0}^{\infty }\chi _{g(x)>s}\,ds}
where
χ
f
(
x
)
>
r
{\displaystyle \chi _{f(x)>r}}
indicator function of the subset E f given by
E
f
=
{
x
∈
X
:
f
(
x
)
>
r
}
{\displaystyle E_{f}=\left\{x\in X:f(x)>r\right\}}
Analogously,
χ
g
(
x
)
>
s
{\displaystyle \chi _{g(x)>s}}
E g given by
E
g
=
{
x
∈
X
:
g
(
x
)
>
s
}
{\displaystyle E_{g}=\left\{x\in X:g(x)>s\right\}}
∫
R
n
f
(
x
)
g
(
x
)
d
x
=
∫
R
n
∫
0
∞
∫
0
∞
χ
f
(
x
)
>
r
χ
g
(
x
)
>
s
d
r
d
s
d
x
{\displaystyle \int _{\mathbb {R} ^{n}}f(x)g(x)\,dx=\displaystyle \int _{\mathbb {R} ^{n}}\int _{0}^{\infty }\int _{0}^{\infty }\chi _{f(x)>r}\chi _{g(x)>s}\,dr\,ds\,dx}
=
∫
0
∞
∫
0
∞
∫
R
n
χ
f
(
x
)
>
r
∩
g
(
x
)
>
s
d
x
d
r
d
s
{\displaystyle =\int _{0}^{\infty }\int _{0}^{\infty }\int _{\mathbb {R} ^{n}}\chi _{f(x)>r\cap g(x)>s}\,dx\,dr\,ds}
=
∫
0
∞
∫
0
∞
μ
(
{
f
(
x
)
>
r
}
∩
{
g
(
x
)
>
s
}
)
d
r
d
s
{\displaystyle =\int _{0}^{\infty }\int _{0}^{\infty }\mu \left(\left\{f(x)>r\right\}\cap \left\{g(x)>s\right\}\right)\,dr\,ds}
≤
∫
0
∞
∫
0
∞
min
(
μ
(
f
(
x
)
>
r
)
;
μ
(
g
(
x
)
>
s
)
)
d
r
d
s
{\displaystyle \leq \int _{0}^{\infty }\int _{0}^{\infty }\min \left(\mu \left(f(x)>r\right);\mu \left(g(x)>s\right)\right)\,dr\,ds}
=
∫
0
∞
∫
0
∞
min
(
μ
(
f
∗
(
x
)
>
r
)
;
μ
(
g
∗
(
x
)
>
s
)
)
d
r
d
s
{\displaystyle =\int _{0}^{\infty }\int _{0}^{\infty }\min \left(\mu \left(f^{*}(x)>r\right);\mu \left(g^{*}(x)>s\right)\right)\,dr\,ds}
=
∫
0
∞
∫
0
∞
μ
(
{
f
∗
(
x
)
>
r
}
∩
{
g
∗
(
x
)
>
s
}
)
d
r
d
s
{\displaystyle =\int _{0}^{\infty }\int _{0}^{\infty }\mu \left(\left\{f^{\ast }(x)>r\right\}\cap \left\{g^{\ast }(x)>s\right\}\right)\,dr\,ds}
=
∫
R
n
f
∗
(
x
)
g
∗
(
x
)
d
x
{\displaystyle =\int _{\mathbb {R} ^{n}}f^{*}(x)g^{*}(x)\,dx}
See also
References
