A singular integral on the sphere

Question

Fix $\theta_{0} \in \mathbb{S}^{n}$. I need to identify $\alpha\in \mathbb{R}$ such that the integral $\;\displaystyle \int_{\mathbb{S}^{n}} \frac{d\theta}{|\theta-\theta_{0}|^{\alpha}}$ converges. My guess is the integral converges iff $\alpha<n$. How do it rigorously ?

Answer 1

I would use cylindrical coordinates on the sphere, as it is done for example in the recent book of Atkinson and Han. This allows you to reduce the integral to a one-dimensional one.

SKETCHY. Since $\mathbb S^n\subset \mathbb R^{n+1}$, we denote $X=(X_0,X_1\ldots X_n)\in\mathbb R^{n+1}$. Moreover, notice that $$|\theta-\theta_0|^2=2-2\theta\cdot \theta_0.$$ So, if we assume that $\theta_0=(1,0,\ldots, 0)$, then $$|\theta- \theta_0|^\alpha=2^{\frac\alpha 2}(1-X_0)^\frac\alpha 2.$$

Now, as shown in the linked book, the volume element of $\mathbb S^n$ can be expressed in the coordinates $X_0,\ldots X_n$ as follows: $$dS^n=(1-X_0^2)^\frac{n-2}{2}dX_0dS^{n-1}, $$
where $dS^{n-1}$ is the volume element of the $n-1$ dimensional sphere (this is why I spoke of "cylindrical" coordinates).

All of this allows us to rewrite our integral as $$\int_{\mathbb S^n} \frac{d\theta}{|\theta-\theta_0|^\alpha}=C_\alpha\int_{-1}^1\frac{(1-X_0^2)^\frac{n-2}{2}}{(1-X_0)^\frac\alpha2}\, dX_0=C_\alpha\int_{-1}^1(1-X_0)^\frac{n-\alpha-2}{2}(1+X_0)^\frac{n-2}{2}\,dX_0, $$ which converges if and only if $\alpha<n$, as you conjectured.