Example of tempered distribution

Question

I am learning about tempered distributions. Today we learned that on $\mathbb{R}^n$, $\frac{1}{|x|^p}$ is a tempered distribution as long as $0 < p < n$. My question is, what goes wrong when $p \geq n$? A reference would be highly appreciated.

Answer 1

The reason that $|x|^{-p}$ defines a tempered distribution $T$ for $p<n$ is that for every test function $\phi$ from the Schwartz class, the integral $$ T[\phi]:=\int_{\mathbb{R}^n} |x|^{-p} \phi(x)\,dx \tag{1} $$ converges. (Note that the restriction $p>0$ is not needed; functions like $|x|^{3}$ are perfectly reasonable tempered distributions.)

When $p\ge n$, the integral $(1)$ diverges whenever $\phi(0)\ne 0$. Indeed, in some neighborhood of $0$ the function $\phi$ has the same sign as $\phi(0)$, and the contribution of this neighborhood to the integral is infinite. So, we can't use $(1)$ to define a tempered distribution.

This does not necessarily prevent us from defining a distribution by other means... Note that the Laplacian of $|x|^{-p}$ is a constant multiple of $|x|^{-p-2}$. And for any tempered distribution $T$, the Laplacian $\Delta T$ is also a tempered distribution defined by $$ (\Delta T)[\phi] = T[\Delta \phi] $$

So, given $p\ge n$, we can begin with $|x|^{-p+2k}$ where $k$ is a positive integer, and take the Laplacian $k$ times to arrive at a plausible definition for $|x|^{-p}$. This is sometimes called regularization. Note that the distribution obtained in this way will have an additional component at the origin that cancels out the singularity of $|x|^{-p}$. In particular, it is not a positive distribution (not always positive on positive test functions).

As a reference, I recommend Chapter 4 of Lecture notes on Distributions by Hasse Carlsson.