Original proof by a cylindrical isoperimetric inequality

The original proof by Pólya and Szegő for ${\displaystyle p=2}$ was based on an isoperimetric inequality comparing sets with cylinders and an asymptotics expansion of the area of the area of the graph of a function.^[1] The inequality is proved for a smooth function ${\displaystyle u}$ that vanishes outside a compact subset of the Euclidean space ${\displaystyle \mathbb {R} ^{n}.}$ For every ${\displaystyle \varepsilon >0}$, they define the sets

${\displaystyle {\begin{aligned}C_{\varepsilon }&=\{(x,t)\in \mathbb {R} ^{n}\times \mathbb {R} \,:\,0<t<\varepsilon u(x)\}\\C_{\varepsilon }^{*}&=\{(x,t)\in \mathbb {R} ^{n}\times \mathbb {R} \,:\,0<t<\varepsilon u^{*}(x)\}\end{aligned}}}$

These sets are the sets of points who lie between the domain of the functions ${\displaystyle \varepsilon u}$ and ${\displaystyle \varepsilon u^{*}}$ and their respective graphs. They use then the geometrical fact that since the horizontal slices of both sets have the same measure and those of the second are balls, to deduce that the area of the boundary of the cylindrical set ${\displaystyle C_{\varepsilon }^{*}}$ cannot exceed the one of ${\displaystyle C_{\varepsilon }}$. These areas can be computed by the area formula yielding the inequality

${\displaystyle \int _{u^{*}{}^{-1}((0,+\infty ))}1+{\sqrt {1+\varepsilon ^{2}|\nabla u^{*}|^{2}}}\leq \int _{u^{-1}((0,+\infty ))}1+{\sqrt {1+\varepsilon ^{2}|\nabla u|^{2}}}.}$

Since the sets ${\displaystyle u^{-1}((0,+\infty ))}$ and ${\displaystyle u{}^{*}{}^{-1}((0,+\infty ))}$ have the same measure, this is equivalent to

${\displaystyle {\frac {1}{\varepsilon }}\int _{u^{*}{}^{-1}((0,+\infty ))}{\sqrt {1+\varepsilon ^{2}|\nabla u^{*}|^{2}}}-1\leq {\frac {1}{\varepsilon }}\int _{u^{-1}((0,+\infty ))}{\sqrt {1+\varepsilon ^{2}|\nabla u|^{2}}}-1.}$

The conclusion then follows from the fact that

${\displaystyle \lim _{\varepsilon \to 0}{\frac {1}{\varepsilon }}\int _{u^{-1}((0,+\infty ))}{\sqrt {1+\varepsilon ^{2}|\nabla u|^{2}}}-1={\frac {1}{2}}\int _{\mathbb {R} ^{n}}|\nabla u|^{2}.}$

Coarea formula and isoperimetric inequality

The Pólya–Szegő inequality can be proved by combining the coarea formula, Hölder’s inequality and the classical isoperimetric inequality.^[2]

If the function ${\displaystyle u}$ is smooth enough, the coarea formula can be used to write

${\displaystyle \int _{\mathbb {R} ^{n}}|\nabla u|^{p}=\int _{0}^{+\infty }\int _{u^{-1}({t})}|\nabla u|^{p-1}\,d{\mathcal {H}}^{n-1}\,dt,}$

where ${\displaystyle {\mathcal {H}}^{n-1}}$ denotes the ${\displaystyle (n-1)}$–dimensional Hausdorff measure on the Euclidean space ${\displaystyle \mathbb {R} ^{n}}$. For almost every each ${\displaystyle t\in (0,+\infty )}$, we have by Hölder's inequality,

${\displaystyle {\mathcal {H}}^{n-1}\left(u^{-1}(\{t\})\right)\leq \left(\int _{u^{-1}(\{t\})}|\nabla u|^{p-1}\right)^{\frac {1}{p}}\left(\int _{u^{-1}(\{t\})}{\frac {1}{|\nabla u|}}\right)^{1-{\frac {1}{p}}}.}$

Therefore, we have

${\displaystyle \int _{u^{-1}(\{t\})}|\nabla u|^{p-1}\geq {\frac {{\mathcal {H}}^{n-1}\left(u^{-1}(\{t\})\right)^{p}}{\left(\int _{u^{-1}(\{t\})}{\frac {1}{|\nabla u|}}\right)^{p-1}}}.}$

Since the set ${\displaystyle u^{*}{}^{-1}((t,+\infty ))}$ is a ball that has the same measure as the set ${\displaystyle u^{-1}((t,+\infty ))}$, by the classical isoperimetric inequality, we have

${\displaystyle {\mathcal {H}}^{n-1}\left(u^{*}{}^{-1}(\{t\})\right)\leq {\mathcal {H}}^{n-1}\left(u^{-1}(\{t\})\right).}$

Moreover, recalling that the sublevel sets of the functions ${\displaystyle u}$ and ${\displaystyle u^{*}}$ have the same measure,

${\displaystyle \int _{u^{*}{}^{-1}(\{t\})}{\frac {1}{|\nabla u^{*}|}}=\int _{u^{-1}(\{t\})}{\frac {1}{|\nabla u|}},}$

and therefore,

${\displaystyle \int _{\mathbb {R} ^{n}}|\nabla u|^{p}\geq \int _{0}^{+\infty }{\frac {{\mathcal {H}}^{n-1}\left(u^{*}{}^{-1}(\{t\})\right)^{p}}{\left(\int _{u^{*}{}^{-1}(\{t\})}{\frac {1}{|\nabla u^{*}|}}\right)^{p-1}}}\,dt.}$

Since the function ${\displaystyle u^{*}}$ is radial, one has

${\displaystyle {\frac {{\mathcal {H}}^{n-1}\left(u^{*}{}^{-1}(\{t\})\right)^{p}}{\left(\int _{u^{*}{}^{-1}(\{t\})}{\frac {1}{|\nabla u^{*}|}}\right)^{p-1}}}=\int _{u^{*}{}^{-1}(\{t\})}|\nabla u^{*}|^{p-1},}$

and the conclusion follows by applying the coarea formula again.

Rearrangement inequalities for convolution

When ${\displaystyle p=2}$, the Pólya–Szegő inequality can be proved by representing the Sobolev energy by the heat kernel.^[13] One begins by observing that

${\displaystyle \int _{\mathbb {R} ^{n}}|\nabla u|^{2}=\lim _{t\to 0}{\frac {1}{t}}\left(\int _{\mathbb {R} ^{n}}|u|^{2}-\int _{\mathbb {R} ^{n}}\int _{\mathbb {R} ^{n}}K_{t}(x-y)u(x)u(y)\,dx\,dy\right),}$

where for ${\displaystyle t\in (0,+\infty )}$, the function ${\displaystyle K_{t}:\mathbb {R} ^{n}\to \mathbb {R} }$ is the heat kernel, defined for every ${\displaystyle z\in \mathbb {R} ^{n}}$ by

${\displaystyle K_{t}(z)={\frac {1}{(4\pi t)^{\frac {n}{2}}}}e^{-{\frac {|z|^{2}}{4t}}}.}$

Since for every ${\displaystyle t\in (0,+\infty )}$ the function ${\displaystyle K_{t}}$ is radial and radially decreasing, we have by the Riesz rearrangement inequality

${\displaystyle \int _{\mathbb {R} ^{n}}\int _{\mathbb {R} ^{n}}K_{t}(x-y)\,u(x)\,u(y)\,dx\,dy\leq \int _{\mathbb {R} ^{n}}\int _{\mathbb {R} ^{n}}K_{t}(x-y)\,u^{*}(x)\,u^{*}(y)\,dx\,dy}$

Hence, we deduce that

${\displaystyle {\begin{aligned}\int _{\mathbb {R} ^{n}}|\nabla u|^{2}&=\lim _{t\to 0}{\frac {1}{t}}\left(\int _{\mathbb {R} ^{n}}|u|^{2}-\int _{\mathbb {R} ^{n}}\int _{\mathbb {R} ^{n}}K_{t}(x-y)u(x)u(y)\,dx\,dy\right)\\[6pt]&\geq \lim _{t\to 0}{\frac {1}{t}}\left(\int _{\mathbb {R} ^{n}}|u|^{2}-\int _{\mathbb {R} ^{n}}\int _{\mathbb {R} ^{n}}K_{t}(x-y)u^{*}(x)u^{*}(y)\,dx\,dy\right)\\[6pt]&=\int _{\mathbb {R} ^{n}}|\nabla u^{*}|^{2}.\end{aligned}}}$