# Distributions in Product Spaces

## 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 *u*_{j}∈D’,(*X*_{j}), *X*_{j} open in ℝ*u*^{nj} (*j*=1,2), we define in Section 5.1 a product *u*_{1}⊗*u*_{2}∈D’,(*X*_{1}x*X*_{2}) in *X*_{1}x*X*_{2}⊂ℝ^{n1+n2}. In case *u*_{j} are functions this is the function *X*_{1} × *X*_{2}∈(*x*_{1}, *x*_{2}) → *u*_{1}(*x*_{1})*u*_{1}(*x*_{2}).

On the other hand, a function *K*∈*C*(*X*_{1} × *X*_{2}) 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 *C*_{0}(*X*_{2}) to *C*(*X*_{1}) 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 *K*∈D’,(*X*_{1} × *X*_{2}) can be identified with the continuous linear maps K from *C*_{0}∞(*X*_{2}) to D’,(*X*_{1}) 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.

### Keywords

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