I am working through Lieb/Loss's "Analysis", and have been stuck on one of the problems for a while;

Suppose we are on $\mathbb{R}^n$ and define $f(x) = |x|^{-n}$. This is not a locally integrable function. However if $\phi \in C_c^{\infty}(\mathbb{R}^n)$ is a function vanishing at the origin, we can still define the action of $f$ on $\phi$ as a distribution: \begin{align*} T_f (\phi) = \int_{\mathbb{R}^n} \frac{\phi(x)}{|x|^n} dx \end{align*} Which is always well defined when $\phi$ vanishes at the origin, as can be seen by converting to polar coordinates for example. However Lieb states that there are many actual distributions $T$ that agree with $T_f$ on (test) functions that vanish at the origin, and he wants the reader to find all of them.

I have tried a couple of things so far. In one-dimension, I noticed (i'm pretty sure at least) that $T_f$ agrees with the derivative of the distribution given by $-\ln(1/|x|)$. However i'm really stuck as to how to classify all such distributions, or how to generalize to $n$ dimensions.

Thanks for the help in advance.