Is this integral convergent?

Question

If $f\in L^2(\mathbb{R}^3)$, what can I say of the integral $$\int_{\mathbb{R}^3}dx\frac{f(x)}{\vert x\vert}$$? Is it convergent?

Answer 1

Without any additional information about $f(x)$, you cannot say anything about the integrability of $f(x)/|x|$. Suppose $f(x) = 0$ on the unit ball, and $f(x) = |x|^{-2}\log|x|$ elsewhere. Then $f \in L^2(\mathbb{R}^3)$, but $$ \int_{\mathbb{R}^3} \frac{f(x)}{|x|} \, dx = \int_1^{\infty} \frac{\log{r}}{r^3} 4 \pi r^2 \, dr = 4 \pi \int_1^{\infty} r^{-1} \log{r} \, dr = 4 \pi \int_0^{\infty} u \, du = \infty.$$