If $0<s<n$ and $S$ is a measurable set with $|S| < \infty$. Then $$\int_S |x-y|^{-s} dy \leq \omega_n (n-s)^{-1} R^{n-s},$$ where $|S| = e_n R^n = | B_R(x) |$ and $e_n = \frac{\omega_n}{n}$ is the volume of the $n$-dimensional unit ball.
1) Does this inequality have a name? My lecturer keeps refering to it as the inequality of Schmidt, however a Google search returns no results.
2) How do I prove it? Are there any references (books, lecture notes, ...) that contain the proof?
Thanks for your help.
It seems kind of rearrangement inequality: $$ \int_S |x-y|^{-s} dy \leq \int_{B_R(x)} |x-y|^{-s} dy = \int_{B_R(0)} |y|^{-s} dy = \omega_n \int_0^R \rho^{-s}\rho^{n-1} d\rho = \omega_n \frac{R^{n-s}}{n-s}. $$ The first inequality can be proved, for example, using the Hardy-Littlewood inequality $$ \int_{\mathbb{R}^n} f g \leq \int_{\mathbb{R}^n} f^* g^*, $$ where $f^*$ and $g^*$ are the symmetric decreasing rearrangements of $f$ and $g$ respectively.
Namely, assume w.l.o.g. $x=0$ and let $f(y) = |y|^{-s}$, $g(y) = \chi_S(y)$. Then $f^*(y) = f(y)$ and $g^*(y) = \chi_{B_R} (y)$, hence $$ \int_{S} f = \int_{\mathbb{R}^n} f g \leq \int_{\mathbb{R}^n} f^* g^* = \int_{B_R} f. $$