 Support and singular support rules for convolution

If at least one of $u,v\in {\mathcal{D}}^{\prime }({\mathbb{R}}^n)$ is a compactly supported distribution, thenThe sums are Minkowski sums. The first rule follows from the pairing $⟨u_x\otimes v_y,\varphi (x+y)⟩$: a test function away from the support sum gives zero. For the second rule, localize the output to a small compact set avoiding the sum of the singular supports. The relevant pairs of input points form a compact set because one input support is compact. Cover it by finitely many product neighborhoods on each of which at least one input distribution is smooth; otherwise the output would meet the sum of the singular supports. Smooth cutoffs in these neighborhoods decompose the localized pairing into finitely many convolutions with a smooth compactly supported factor. Each is smooth by smoothing convolution with a test function. This proves the second rule locally.

The same argument works when addition is a proper map on the product of the two supports. Some condition defining the convolution is necessary; arbitrary pairs of distributions cannot always be convolved.

Target: Cauchy kernel fundamental solution. Parent: Fundamental solution of a linear differential operator. Intended file: distribution-theory.bigb.