Computing a higher dimensional integral with a characteristic function

Question

I have a question on how to deal with integrating a characteristic function in higher dimensions. Concretely, I have the following inequality $$ \int_{|k|\leq K}d^d k \int_{|l|\geq K}d^d l \frac{\chi_{(0,a)}(|k-l|)}{(|l|-|k|+K^{-1})^2} \lesssim (K a)^{d-1}\int_{-a}^0 ds\int_0^a dr \frac{1}{(r-s+K^{-1})^2}\, ,$$ which holds for $d=\{2,3\}$, $K,a>0$ and $\chi_{(0,a)}(|k-l|)$ being the characteristic function of the interval $(0,a)$. How does one compute the lefthand-side to end up on the righthand-side? What does the characteristic function in this higher dimensional case? I suppose one uses polar coodinate transformation?