Does there exist a compactly supported integrable function with infinite Coulomb energy?

Question

The title of the question pretty much says it all. I am looking for a function $f\in L^1(\Omega)$, where $\Omega \subset \mathbb{R}^3$ is a bounded domain, such that $$ E[f] = \iint\limits_{\Omega\times\Omega} \frac{f(x)f(y)dxdy}{|x-y|}\text{ is infinite}. $$ Suffice to take $f \geq 0$. Note that $f\notin L^\infty$ since the Coulomb potential is locally integrable.

Answer 1

My apologies: I meant to post this question on mathoverflow. Thank you all. Here is the answer I was able to put together: We know that $L^{6/5}$ embeds in the set of measures with finite energy, suggesting we find a function in $L^1\setminus L^{6/5}$. Before multiplying by a smooth cutoff, $f(x) = |x|^{-5/2}$ should suffice, or more generally, $|x|^{3-\epsilon}$ for epsilon small. See https://mathoverflow.net/questions/326993/does-there-exist-a-compactly-supported-integrable-function-with-infinite-coulomb and credit Willie Wong for the answer.