Integral using polar coordinates in high dimension.

Question

Let $B$ be the unit ball in $\mathbb R^d$ Let $K(x,y)=|x-y|^{-d+\alpha}$ be a function on $B\times B$, where $\alpha>0$. I want to show $\int_BK(x,y)dy$ converges for every $x\in B$. It seems that this would be easier in polar coordinates, but I'm not able to do that since I don't know how to actually do it in high dimension. Any help is appreciated.

Answer 1

Fix $x \in B$. Choose $R$ sufficiently big such that $B\subset B_R(x)$. Then we see that \begin{align} \int_B\frac{1}{|x-y|^{d-\alpha}}\ dy \leq \int_{B_R(x)} \frac{1}{|x-y|^{d-\alpha}}\ dx = \int^R_0 \int_{|x-y|=r} \frac{d\sigma(y)}{r^{d-\alpha}}\ dr = C\int^R_0 \frac{1}{r^{1-\alpha}}\ dr<\infty. \end{align}