Desigualdade de Hardy-Littlewood

Pelo representação bolo de camadas, tem-se[1][2]:

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

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

onde ${\displaystyle \chi _{f(x)>r}}$ denota a função indicadora (ou função característica) do subconjunto E _f dado por

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

Analogamente, ${\displaystyle \chi _{g(x)>s}}$ denota a função indicadora do subconjunto E _g dado por

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

${\displaystyle {\begin{aligned}\int _{\mathbb {R} ^{n}}f(x)g(x)\,dx&=\displaystyle \int _{\mathbb {R} ^{n}}\int _{0}^{\infty }\int _{0}^{\infty }\chi _{f(x)>r}\chi _{g(x)>s}\,dr\,ds\,dx\\[8pt]&=\int _{\mathbb {R} ^{n}}\int _{0}^{\infty }\int _{\mathbb {R} ^{n}}\chi _{f(x)>r\cap g(x)>s}\,dx\,dr\,ds\\[8pt]&=\int _{0}^{\infty }\int _{0}^{\infty }\mu \left(\left\{\chi _{f(x)>r\cap g(x)>s}\right\}\right)\,dr\,ds\\[8pt]&\leq \int _{0}^{\infty }\int _{0}^{\infty }\min \left(\mu \left(f(x)>r\right);\mu \left(g(x)>s\right)\right)\,dr\,ds\\[8pt]&=\int _{0}^{\infty }\int _{0}^{\infty }\min \left(\mu \left(f^{*}(x)>r\right);\mu \left(g^{*}(x)>s\right)\right)\,dr\,ds\\[8pt]&=\int _{0}^{\infty }\int _{0}^{\infty }\mu \left(\left\{\chi _{f^{*}(x)>r\cap g^{*}(x)>s}\right\}\right)\,dr\,ds\\[8pt]&=\int _{\mathbb {R} ^{n}}f^{*}(x)g^{*}(x)\,dx\end{aligned}}}$

• Desigualdade do rearranjo