Let \(n\geq 1\) and \(p\in\RR\). Show that \[ \int_{\bB^n} {1 \over \abs{x}^p } \dx &< \infty \iff p < n \\ \\ \\ \int_{\RR^n\sm \bB^n} {1 \over \abs{x}^p } \dx &< \infty \iff p > n .\]
Solution
Let \(\omega_{n-1}\) be the surface measure of the unit sphere. In polar coordinates, \[ \int_{\abs{x}<R}\abs{x}^{-p}\,\dx = \omega_{n-1}\int_0^R r^{n-1-p}\,\dr, \qquad \int_{\abs{x}>R}\abs{x}^{-p}\,\dx = \omega_{n-1}\int_R^\infty r^{n-1-p}\,\dr. \] The first one is finite exactly when \(n-1-p>-1\), that is, \(p<n\). The second is finite exactly when \(n-1-p<-1\), that is, \(p>n\). At \(p=n\), both integrals contain \(\int r^{-1}\,\dr\) and diverge.