Evaluate $\int_{\Bbb S^n}\int_{\Bbb S^n}|s-t|^{-\lambda}\,{\rm d}s\,{\rm d}t$

Question

I'm reading the proof for HLS inequality in Analysis written by Michael Loss and Elliott H.Lieb. The HLS inequality is $$\left|\int_{\Bbb R^n}\int_{\Bbb R^n}f(x)|x-y|^{-\lambda}g(y)\,{\rm d}x\,{\rm d}y\right|\le C(n,\lambda,p)\|f\|_p\|g\|_r,\qquad 1/p+1/r+\lambda/n=2.$$ He found the best function when $p=r=2n/(2n-\lambda)$, that is $$ h(x)=|\Bbb S^n|^{-1/p}\left(\cfrac2{1+|x|^2}\right)^{n/p}. $$ Then he said that if $f(x)=g(y)=h(x)$, you can get the integral on the left side is $$ \pi^{\lambda/2}\cfrac{\Gamma(n/2-\lambda/2)}{\Gamma(n-\lambda/2)}\left(\cfrac{\Gamma(n/2)}{\Gamma(n)}\right)^{-1+\lambda/n}. $$ It is hard for me to calculate this integral, I think it should use the fact that $$ \mathcal Sh(x)=|\Bbb S^n|^{-1/p},~~~~~\text{where $\mathcal S$ is the stereographic project.} $$ Thus we only need to calculate $$ \int_{\Bbb S^n}\int_{\Bbb S^n}|s-t|^{-\lambda}\,{\rm d}s\,{\rm d}t. $$ But I cannot solve this one, could you give me some advice? Thanks a lot.

Answer 1

I'm using the notation $\mathbb{S}^n:=\{x\in\mathbb{R}^n:|x|=1\}$ in this answer.

$J(s)=\int_{\mathbb{S}^n}|s-t|^{-\lambda}{\rm d}t$ doesn't (actually) depend on $s\in\mathbb{S}^n$, so that $$\int_{\mathbb{S}^n}\int_{\mathbb{S}^n}\frac{{\rm d}s\,{\rm d}t}{|s-t|^\lambda}=S_n J(s_0),$$ where $S_n=\dfrac{2\pi^{n/2}}{\Gamma(n/2)}$ is the measure of $\mathbb{S}^n$, and $s_0=(1,0,\dots,0)$ say.

Next, each $t\in\mathbb{S}^n$ can be written as $(x,t')$ with $x\in[-1,1]$ and $t'\in\sqrt{1-x^2}\,\mathbb{S}^{n-1}$; the corresponding decomposition of the integral over $\mathbb{S}^n$ of a function $f$ is \begin{align} \int_{\mathbb{S}^n}f(t)\,{\rm d}t &=\int_{-1}^1\int_{\sqrt{1-x^2}\,\mathbb{S}^{n-1}}f(x,t')\,{\rm d}t'\frac{{\rm d}x}{\sqrt{1-x^2}} \\&=\int_{-1}^1(1-x^2)^{(n-3)/2}\int_{\mathbb{S}^{n-1}}f(x,t\sqrt{1-x^2})\,{\rm d}t\,{\rm d}x. \end{align}

In our case, for $t=(x,t')$, the quantity $|s_0-t|^2=2(1-x)$ depends on $x$ only, thus $$\int_{\mathbb{S}^n}\int_{\mathbb{S}^n}\frac{{\rm d}s\,{\rm d}t}{|s-t|^\lambda}=S_n S_{n-1}\int_{-1}^1\frac{(1-x^2)^{(n-3)/2}}{\big(2(1-x)\big)^{\lambda/2}}\,{\rm d}x,$$ and the last integral is of beta type (after $x=1-2y$). The result is

$$\int_{\mathbb{S}^n}\int_{\mathbb{S}^n}\frac{{\rm d}s\,{\rm d}t}{|s-t|^\lambda}=\frac{2^{n-\lambda}\pi^{n-1/2}\Gamma\big((n-1-\lambda)/2\big)}{\Gamma(n/2)\Gamma(n-1-\lambda/2)}.$$