Case 4

Consider the function $u(\xi) := \vert\xi\vert^{-n}$ for $\xi \neq 0$. (By averaging $v(\om)$ over $\bS^{n-1}$, one can see that any smooth $(-n)$-homogeneous function on $\bR^n \setminus \{0\}$ is a linear combination of $\vert\xi\vert^{-n}$ and a function in Case 3.

We can try the cutoff regularization, anyway. Let $R_f u$ be given by the recipe of ():

$$\dst{R_f u}{\phi} := \int_{\bR^n} u(\xi)\bigl(\phi(\xi) - \phi(0) f(\vert\xi\vert)\bigr) d^n\xi.$$ (22)

However, in the present case, $R_f u$ is not homogeneous!