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.



Nonintegrable inverse powers as distributions
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" 


