# Distributions in Product Spaces

• Lars Hörmander
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 256)

## Summary

We were not able to define the product of arbitrary distributions in Chapter III. However, as we shall now see this can always be done when they depend on different sets of variables. Thus to arbitrary distributions ujD’,(Xj), Xj open in ℝunj (j=1,2), we define in Section 5.1 a product u1u2D’,(X1xX2) in X1xX2⊂ℝn1+n2. In case uj are functions this is the function X1 × X2∈(x1, x2) → u1(x1)u1(x2).

On the other hand, a function KC(X1 × X2) can be viewed as the kernel of an integral operator K, $$({\cal K}\ u)(x_1)=\int K(x_1,x_2)u(x_2)dx_2,$$, mapping C0(X2) to C(X1) say. It is not easy to characterize the operators having such a kernel. However, the analogue in the theory of distributions is very satisfactory. It is called the Schwartz kernel theorem and states that the distributions KD’,(X1 × X2) can be identified with the continuous linear maps K from C0∞(X2) to D’,(X1) which they define. This will be proved in Section 5.2. We shall return to this topic in Section 8.2. A rather precise classification of singularities will then allow us to discuss the regularity of K u and its definition when u is not smooth.

Convolution