Suppose $f \in L^1(\mathbb{R}^2)$ and also $$F(x) = \log|x|.$$ I want to show that if $$\int |f(x)| F(x) \, dx < \infty,$$ then $f * F$ is locally integrable, i.e. either $$ \int |(f * F)(x) \phi(x)| \, dx < \infty$$ for every test function $\phi \in \mathcal{C}^\infty_c(\mathbb{R}^2)$, or $$ \int_K |(f * F)(x)| \, dx$$ for every compact set $K \subset \mathbb{R}^2$.
I see that if $|x| > 1$, then $F = |F|$. I am not sure why we don't have absolute value signs around the $|F(x)|$ in the second equation: this integral could be $-\infty$, right?