Hardy–Littlewood inequality

From layer cake representation we have:[1][2]

${\displaystyle f(x)=\int _{0}^{\infty }\chi _{f(x)>r}\,dr}$

${\displaystyle g(x)=\int _{0}^{\infty }\chi _{g(x)>s}\,ds}$

where ${\displaystyle \chi _{f(x)>r}}$ denotes the indicator function of the subset E _f given by

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

Analogously, ${\displaystyle \chi _{g(x)>s}}$ denotes the indicator function of the subset E _g given by

${\displaystyle E_{g}=\left\{x\in X:g(x)>s\right\}}$

${\displaystyle \int _{\mathbb {R} ^{n}}f(x)g(x)\,dx=\displaystyle \int _{\mathbb {R} ^{n}}\int _{0}^{\infty }\int _{0}^{\infty }\chi _{f(x)>r}\chi _{g(x)>s}\,dr\,ds\,dx}$

${\displaystyle =\int _{0}^{\infty }\int _{0}^{\infty }\int _{\mathbb {R} ^{n}}\chi _{f(x)>r\cap g(x)>s}\,dx\,dr\,ds}$

${\displaystyle =\int _{0}^{\infty }\int _{0}^{\infty }\mu \left(\left\{f(x)>r\right\}\cap \left\{g(x)>s\right\}\right)\,dr\,ds}$

${\displaystyle \leq \int _{0}^{\infty }\int _{0}^{\infty }\min \left(\mu \left(f(x)>r\right);\mu \left(g(x)>s\right)\right)\,dr\,ds}$

${\displaystyle =\int _{0}^{\infty }\int _{0}^{\infty }\min \left(\mu \left(f^{*}(x)>r\right);\mu \left(g^{*}(x)>s\right)\right)\,dr\,ds}$

${\displaystyle =\int _{0}^{\infty }\int _{0}^{\infty }\mu \left(\left\{f^{\ast }(x)>r\right\}\cap \left\{g^{\ast }(x)>s\right\}\right)\,dr\,ds}$

${\displaystyle =\int _{\mathbb {R} ^{n}}f^{*}(x)g^{*}(x)\,dx}$

• Rearrangement inequality
• Chebyshev's sum inequality
• Lorentz space