Understanding the performed change of variable in this integration [closed]

Question

I'm stuck on a passage I do not understand, which reads:

$$\int_{r<|y|<1} \bigg| \frac{1}{(|y|^2 - r^2)^s |y|^n} - \frac{1}{|y|^{n+2s}}\bigg|\ \text{d}y$$ $$\int_1^r \bigg| \frac{1}{(t^2 - r^2)^s} - \frac{1}{t^{2s}}\bigg|\ \frac{\text{d}t}{t}$$ $$r^{s}\int_1^{1/r} \bigg| \frac{1}{(\tau^2 - 1)^s} - \frac{1}{\tau^{2s}}\bigg|\ \frac{\text{d}\tau}{\tau}$$

I understood there is a sort of rescaling between the second and the third line, but I cannot find a way to get the passage between the first and the second line. I thought about a change of variable like $|y|^n = t$ but there is something that I miss, and it doesn't work.

This is an integral in $\mathbb{R}^n$, where $0<s<1$.

Answer 1

Note that the integrand only depends on the absolute value of $y$, and not on the direction of the vector. That means it is a good idea to substitute $t = |y|$. Let's denote $$f(t) = \bigg| \frac{1}{(t^2 - r^2)^s t^n} - \frac{1}{t^{n+2s}}\bigg|$$ for positive real $t$. Denote $t\mathbb S^n$ to be the sphere of radius $t$. Then we have $$\int_{r<|y|<1}f(|y|) dy = \int_r^1\int_{t\mathbb S^n}f(t)dudt = \int_r^1\text{Vol}(t\mathbb S^n)f(t)dt = \text{Vol}(\mathbb S^n)\int_r^1f(t)t^{n-1}dt.$$ This reduces to almost the same formula that you want. It seems to me that the volume of the unit sphere has been neglected in your text, and the integration goes from $1$ to $r$ which is strange as $r<1$.

For the third line you can substitute $\tau = \frac tr$, but it seems you already knew how to do that.