Estimating an integral via substitution

Question

I have a question about the computation of a certain integral. Let $\chi_{[0,B]}(\vert k-l\vert)$ be the characteristic function on the interval $[0,B]$. In order to estimate the following integral in terms of the variable $F$ I computed $$ \int_{\vert k \vert \leq F} d^d k \int_{\vert l \vert \geq F} d^d l \frac{\chi_{[0,B]}(\vert k-l\vert)}{\vert l \vert-\vert k \vert + F^{-1}} \lesssim \left(k_F B\right)^{d-1} \int_{-B}^0 ds \int_0^B dr \frac{1}{r-s+F^{-1}}\lesssim F^{d-1} \ln(F)\, .$$ But I don't quite know how the handle the following, similar integral $$\int_{\vert k \vert \leq F} d^d k \left(\int_{\vert l \vert \geq F} d^d l \frac{\chi_{[0,B]}(\vert k-l\vert)}{\vert l \vert-\vert k \vert + F^{-1}}\right)^2 $$ How can I find the order of magnitude of this integral?