A proof of Pólya-Szegő inequality

Question

Denote by $|A|$ the $N-$dimensional Lebesgue measure of a Borel set $A \subset \mathbb{R}^N$ and define $$ A^\ast := B_{R}(0), \quad R = \left(\frac{N}{\omega_N}|A| \right)^{\frac{1}{N}}, $$ where $\omega_N$ is the surface measure of $S^{N-1}$. It is known that $|A| = |A^\ast|$. The symmetric-decreasing rearrangement of the characteristic function $\chi_A$ is defined by $$ (\chi_A)^\ast = \chi_{A^\ast}. $$ Now, for $f \in C^\infty_0(\mathbb{R}^N)$ we define, for $t > 0$ the set $A_t = \{x \in \mathbb{R}^N : |f(x)| > t\}$ and the function $$ f^\ast(x) = \int_0^\infty (\chi_{A_t})^\ast (x) dt. $$ In other words, $$ f^\ast(x) = \int_0^\infty \chi_{A_t^\ast} (x) dt. $$ It is also also known that $f^\ast$ is nonnegative, radially symmetric and nonincreasing. As a reference I am using the book Analysis from Elliott H. Lieb and Michael Loss, page 81. I am looking for a proof of the inequality $$ \|\nabla f ^\ast\|_{L^2(\mathbb{R}^N)} \leq \|\nabla f\|_{L^2(\mathbb{R}^N)} $$ whenever $f \in H^1(\mathbb{R}^N)$. Or more general, $$ \int_{\mathbb{R}^N} |\nabla f^\ast|^p \leq \int_{\mathbb{R}^N} |\nabla f|^p, $$ for any $p > 1$. Any help is welcome.

Answer 1

That is known as the Polya-Szego inequality whose proof uses the co-area formula see for instance: Wikipedia.

See also the monographs: Symmetric Decreasing Rearrangement Is Sometimes Continuous Frederick J. Almgren, Jr., Elliott H. Lieb

For more details see the book: Baernstein II, Albert: Symmetrization in analysis, 2019.