I'm studying a paper and I'm not able to prove the following inequality:
$$\int_{U(r)} \frac{1}{(2\pi)^{k+1}}\prod_{j=1}^{k+1}\bigg(2\frac{\sin(b(x_j-y_j))}{x_j-y_j}\bigg)^2\,d\sigma(y) \leq c(k)b^{2(k+1)}\bigg(\frac{1}{b}\bigg)^k=c(k)b^{k+2},$$
where $r>0$ and $b\geq \frac{\sqrt{2}}{r}$ are fixed, $U(r) = \{x\in\mathbb{R}^{k+1};\,\|x\|=r\}$, $x\in \big[-\frac{1}{b},\frac{1}{b}\big]^{k+1}$, $c(k)$ is a costant depending only on $k$ and $d\sigma(y)$ is an element of surface area.
My problem is that I don't know how to uniformly bound this expression dealing with points $x\in \big[-\frac{1}{b},\frac{1}{b}\big]^{k+1}$ and $y\in U(r)$ that may have some of the coordinates equal. I understand that in such a case the $\sin(\cdot)$ in the numerator fixes the problem with the zero in the denominator, but still I'm not able to prove it.
Any ideas?