Let $B$ be a ball of radius $R$ centered at $0$. I am considering the integral $$\int_B \int_B \frac{1}{|x-y|^k}dxdy,$$ where $k$ is some real number (which can make the integral infinity). To calculate it explicitly, I want to do the obvious change of variable $r=|x-y|$. Here's my attempt:
If we only consider the inside integral, for any fixed $y$, this becomes $$\int_B\frac{1}{|x-y|^k}dx = |S^{n-1}|\int_0^{2R} \frac{1}{r^k}r^{N-1}dr$$ which is just a constant. Then the other integral just gives $|B_r|$. I think this might not be true and I am more convinced that $$\int_B\int_B\frac{1}{|x-y|^k}dx \leq |S^{n-1}||B_r|\int_0^{2R} \frac{1}{r^k}r^{N-1}dr = CR^{2n-k}.$$ since when $y$ is fixed, we are not integrating the whole sphere but only part of the sphere included in $B \times B$. I wonder if you could explain how to do this correctly?