Nonintegrable inverse powers as distributions 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.
 A: The related topic here is the homogeneous distribution on $\mathbb{R}^n\0$ and its extension to $\mathbb{R}^n$. In your case $T_{f}$ is a homogeneous distribution on  $\mathbb{R}^n\0$ of degreee $-n$. And it's always possible to extend it to a distribution on $\mathbb{R}^n$,which may not be homogeneous any more. In fact,you can define the extension $\dot{T_{f}}$ as follows
$$
\dot{T_{f}}(\phi)=T_{f}(\psi R_{-n}\phi), \quad \phi\in C_{0}^{\infty}({\mathbb{R}^n})\\
R_{-n}\phi=<t_{+}^{-1},\phi(tx)>,\quad x\neq 0
$$
where $\psi$ is a fixed function in $C_{0}^{\infty}({\mathbb{R}^n}\0)$ satisfies that $\int_{0}^{+\infty}\frac{\psi(tx)}{t}dt=1$. 
Now one can see that when restricting test functions in $C_{0}^{\infty}({\mathbb{R}^n}\0)$, then $\dot{T_{f}}=T_{f}$, and the extension depends on the choice of $\psi$. However, all such extension has the above form. And in one dimensin, it's particularly clear. You can find them in chaper $3$ of  Hormander's book "The Analysis of Linear Parital Differential Operators" 
