the multiplication of distributions

Distributions are wonderfully flexible: you can add them, differentiate them infinitely often, transform them, and convolve them. But there is one ordinary operation they stubbornly resist — you cannot, in general, multiply two distributions together. This is not a temporary gap waiting for a clever fix; it is a genuine, provable obstruction at the heart of the theory, and respecting it is part of using distributions correctly.

What you can always do is multiply a distribution by a smooth function. If a is infinitely smooth and T is a distribution, the product a times T is defined by handing the smooth factor over to the probe: (a T, phi) = (T, a phi), which makes sense because a phi is still a smooth test function. The trouble starts when both factors are rough. Try to square the delta: there is no consistent way to define delta times delta. A clean way to see the inconsistency is through the Fourier transform, which turns products into convolutions — and the delta transforms to the constant 1, whose convolution with itself diverges. Even the product of the Heaviside step with the delta is ambiguous, since H equals 1/2 at the jump in some conventions and the naive answer depends on that choice.

This limitation matters most for nonlinear PDEs. A nonlinear term like u^2 or u times u_x is a product, and if u is only a distribution that product may have no meaning — which is precisely why nonlinear equations resist the linear distribution machinery and demand extra structure (entropy conditions, viscosity solutions, Sobolev embeddings that keep functions regular enough to multiply). Whole theories exist to find honest substitutes; do not paper over the gap by multiplying distributions as if it were allowed.