Let $B_F:=\{k\in \mathbb{R}^d : | k| \leq F \}$ be the $d$-dimensional ball with radius $F$ and $M_k^F:= B_F^c \cap (B_F+k)$ the crescent moon in $d$ dimensions.
For $B>0$ and $\chi$ being the characteristic function I want to bound the following integral for $d\in \{2,3\}$ from above in terms of F:
$$ \int_{k\in [-B,B]^d}d^dk \int_{q\in M_{-k}^F}d^dq \int_{p\in M_k^F}d^dp \frac{\chi_{(0,B)}(|k|)}{(|p|-|p-k|+|q|-|q+k|+1)^2} $$
My problem is that I really don't know how to handle $M_k^F$ in the integral. Since $|p|-|p-k|+|q|-|q+k|$ the denominator is not necessarily positive, it is not possible to bound the integrand by 1. Is there a clever way to handle this?