Given $f\in L^1(\mathbb{R}^3)$, define
$$\phi(x)=\int_{\mathbb{R}^3}\frac{f(y)}{|y-x|}\,dy.$$
I was able to show that $\phi$ exists for almost all $x$ (I used the Lebesgue differentiation theorem). However, I haven't been able to show that $\phi$ is integrable on compact sets... The bounds I obtained when showing $\phi$ is well defined are too weak to give me integrability.
Any hints?
Thanks
I'm guessing that the question is to show that $\phi$ is locally integrable.
This is very simple, doesn't use any fancy theorems. You have $$\phi=g*f,$$where $g(x)=1/|x|$. Now $g=g_1+g_2$, where $g_1\in L^1(\Bbb R^3)$ and $g_2\in L^\infty(\Bbb R^3)$. So $g_1*f\in L^1$, while $g_2*f$ is continuous (and bounded).