# Product of distributions

1. Sep 7, 2009

### parton

I have a problem concerning the product of two tempered distributions.

We have the temp. distr.
$$F(x) = \dfrac{1}{4 \pi} \mathrm{sgn}(x^{0}) \delta(x^{2}) - \dfrac{i}{4 \pi} \mathcal{P}\dfrac{1}{x^{2}}$$,
where x is a vector in Minkowski space and $$\mathcal{P}$$ is the principal value (in the sense of distributions).

The Fourier transformed expression is:

$$\hat{F}(p) = \dfrac{i}{2 \pi} \theta(p^{0}) \delta(p^{2})$$.

Now, my problem is that I don't know how to define the product $$F(x)^{2}$$. I read in a book that this expression is well-definied, because if we consider the Fourier transform of this product, which will lead to a convolution
$$(2 \pi)^{-2} \int d^{4} q \hat{F}(p-q) \hat{F}(q)$$
,we obviously see that the intersection of the two $$\hat{F}$$ is compact.

I don't "see" that this intersection is really compact. Do you know, why it is compact and why this "intersection-condition" is sufficient to prove that the product in coordinate-space $$F(x)^{2}$$ resp. the convolution is well-definied?